diff --git a/archetypes/notes.md b/archetypes/notes.md
new file mode 100644
index 00000000..9ec01533
--- /dev/null
+++ b/archetypes/notes.md
@@ -0,0 +1,9 @@
+---
+title: "{{ .Name | humanize | title }}"
+weight: 1
+# bookFlatSection: false
+bookToc: false
+# bookHidden: false
+# bookCollapseSection: false
+# bookComments: true
+---
diff --git a/assets/_custom.scss b/assets/_custom.scss
index 3651bbfd..6653bca9 100644
--- a/assets/_custom.scss
+++ b/assets/_custom.scss
@@ -2,6 +2,21 @@
     display: none;
 }
 
-body::-webkit-scrollbar {
-    display: none;
+// body::-webkit-scrollbar {
+//     display: none;
+// }
+
+@import "plugins/scrollbars";
+
+img {
+    width: 80%;
 }
+
+@media (prefers-color-scheme: dark) {
+    :root {
+        --body-background: #212121
+    }
+    img {
+        filter: invert(87.05%);
+    }
+}
\ No newline at end of file
diff --git a/config.yaml b/config.yaml
index 14da08a8..fd36be5d 100644
--- a/config.yaml
+++ b/config.yaml
@@ -66,7 +66,7 @@ params:
   # (Optional, default docs) Specify root page to render child pages as menu.
   # Page is resoled by .GetPage function: https://gohugo.io/functions/getpage/
   # For backward compatibility you can set '*' to render all sections to menu. Acts same as '/'
-  BookSection: docs
+  BookSection: '*'
 
   # Set source repository location.
   # Used for 'Last Modified' and 'Edit this page' links.
diff --git a/content/_index.md b/content/_index.md
index 6123c282..66c6c2ad 100644
--- a/content/_index.md
+++ b/content/_index.md
@@ -1,6 +1,6 @@
 ---
 title: Introduction
-type: docs
+type: notes
 ---
 
 # Acerbo datus maxime
diff --git a/content/docs/example/_index.md b/content/docs/example/_index.md
deleted file mode 100644
index 4835b7ca..00000000
--- a/content/docs/example/_index.md
+++ /dev/null
@@ -1,71 +0,0 @@
----
-weight: 1
-bookFlatSection: true
-title: "Example Site"
----
-
-# Introduction
-
-## Ferre hinnitibus erat accipitrem dixi Troiae tollens
-
-Lorem markdownum, a quoque nutu est *quodcumque mandasset* veluti. Passim
-inportuna totidemque nympha fert; repetens pendent, poenarum guttura sed vacet
-non, mortali undas. Omnis pharetramque gramen portentificisque membris servatum
-novabis fallit de nubibus atque silvas mihi. **Dixit repetitaque Quid**; verrit
-longa; sententia [mandat](http://pastor-ad.io/questussilvas) quascumque nescio
-solebat [litore](http://lacrimas-ab.net/); noctes. *Hostem haerentem* circuit
-[plenaque tamen](http://www.sine.io/in).
-
-- Pedum ne indigenae finire invergens carpebat
-- Velit posses summoque
-- De fumos illa foret
-
-## Est simul fameque tauri qua ad
-
-Locum nullus nisi vomentes. Ab Persea sermone vela, miratur aratro; eandem
-Argolicas gener.
-
-## Me sol
-
-Nec dis certa fuit socer, Nonacria **dies** manet tacitaque sibi? Sucis est
-iactata Castrumque iudex, et iactato quoque terraeque es tandem et maternos
-vittis. Lumina litus bene poenamque animos callem ne tuas in leones illam dea
-cadunt genus, et pleno nunc in quod. Anumque crescentesque sanguinis
-[progenies](http://www.late.net/alimentavirides) nuribus rustica tinguet. Pater
-omnes liquido creditis noctem.
-
-    if (mirrored(icmp_dvd_pim, 3, smbMirroredHard) != lion(clickImportQueue,
-            viralItunesBalancing, bankruptcy_file_pptp)) {
-        file += ip_cybercrime_suffix;
-    }
-    if (runtimeSmartRom == netMarketingWord) {
-        virusBalancingWin *= scriptPromptBespoke + raster(post_drive,
-                windowsSli);
-        cd = address_hertz_trojan;
-        soap_ccd.pcbServerGigahertz(asp_hardware_isa, offlinePeopleware, nui);
-    } else {
-        megabyte.api = modem_flowchart - web + syntaxHalftoneAddress;
-    }
-    if (3 < mebibyteNetworkAnimated) {
-        pharming_regular_error *= jsp_ribbon + algorithm * recycleMediaKindle(
-                dvrSyntax, cdma);
-        adf_sla *= hoverCropDrive;
-        templateNtfs = -1 - vertical;
-    } else {
-        expressionCompressionVariable.bootMulti = white_eup_javascript(
-                table_suffix);
-        guidPpiPram.tracerouteLinux += rtfTerabyteQuicktime(1,
-                managementRosetta(webcamActivex), 740874);
-    }
-    var virusTweetSsl = nullGigo;
-
-## Trepident sitimque
-
-Sentiet et ferali errorem fessam, coercet superbus, Ascaniumque in pennis
-mediis; dolor? Vidit imi **Aeacon** perfida propositos adde, tua Somni Fluctibus
-errante lustrat non.
-
-Tamen inde, vos videt e flammis Scythica parantem rupisque pectora umbras. Haec
-ficta canistris repercusso simul ego aris Dixit! Esse Fama trepidare hunc
-crescendo vigor ululasse vertice *exspatiantur* celer tepidique petita aversata
-oculis iussa est me ferro.
diff --git a/content/docs/example/collapsed/3rd-level/4th-level.md b/content/docs/example/collapsed/3rd-level/4th-level.md
deleted file mode 100644
index aa451f19..00000000
--- a/content/docs/example/collapsed/3rd-level/4th-level.md
+++ /dev/null
@@ -1,12 +0,0 @@
-# 4th Level of Menu
-
-## Caesorum illa tu sentit micat vestes papyriferi
-
-Inde aderam facti; Theseus vis de tauri illa peream. Oculos **uberaque** non
-regisque vobis cursuque, opus venit quam vulnera. Et maiora necemque, lege modo;
-gestanda nitidi, vero? Dum ne pectoraque testantur.
-
-Venasque repulsa Samos qui, exspectatum eram animosque hinc, [aut
-manes](http://www.creveratnon.net/apricaaetheriis), Assyrii. Cupiens auctoribus
-pariter rubet, profana magni super nocens. Vos ius sibilat inpar turba visae
-iusto! Sedes ante dum superest **extrema**.
diff --git a/content/docs/example/collapsed/3rd-level/_index.md b/content/docs/example/collapsed/3rd-level/_index.md
deleted file mode 100644
index cc0100f3..00000000
--- a/content/docs/example/collapsed/3rd-level/_index.md
+++ /dev/null
@@ -1,26 +0,0 @@
-# 3rd Level of Menu
-
-Nefas discordemque domino montes numen tum humili nexilibusque exit, Iove. Quae
-miror esse, scelerisque Melaneus viribus. Miseri laurus. Hoc est proposita me
-ante aliquid, aura inponere candidioribus quidque accendit bella, sumpta.
-Intravit quam erat figentem hunc, motus de fontes parvo tempestate.
-
-    iscsi_virus = pitch(json_in_on(eupViral),
-            northbridge_services_troubleshooting, personal(
-            firmware_rw.trash_rw_crm.device(interactive_gopher_personal,
-            software, -1), megabit, ergonomicsSoftware(cmyk_usb_panel,
-            mips_whitelist_duplex, cpa)));
-    if (5) {
-        managementNetwork += dma - boolean;
-        kilohertz_token = 2;
-        honeypot_affiliate_ergonomics = fiber;
-    }
-    mouseNorthbridge = byte(nybble_xmp_modem.horse_subnet(
-            analogThroughputService * graphicPoint, drop(daw_bit, dnsIntranet),
-            gateway_ospf), repository.domain_key.mouse(serverData(fileNetwork,
-            trim_duplex_file), cellTapeDirect, token_tooltip_mashup(
-            ripcordingMashup)));
-    module_it = honeypot_driver(client_cold_dvr(593902, ripping_frequency) +
-            coreLog.joystick(componentUdpLink), windows_expansion_touchscreen);
-    bashGigabit.external.reality(2, server_hardware_codec.flops.ebookSampling(
-            ciscNavigationBacklink, table + cleanDriver), indexProtocolIsp);
diff --git a/content/docs/example/collapsed/_index.md b/content/docs/example/collapsed/_index.md
deleted file mode 100644
index e954f087..00000000
--- a/content/docs/example/collapsed/_index.md
+++ /dev/null
@@ -1,4 +0,0 @@
----
-bookCollapseSection: true
-weight: 20
----
diff --git a/content/docs/example/hidden.md b/content/docs/example/hidden.md
deleted file mode 100644
index df7cb9eb..00000000
--- a/content/docs/example/hidden.md
+++ /dev/null
@@ -1,52 +0,0 @@
----
-bookHidden: true
----
-
-# This page is hidden in menu
-
-# Quondam non pater est dignior ille Eurotas
-
-## Latent te facies
-
-Lorem markdownum arma ignoscas vocavit quoque ille texit mandata mentis ultimus,
-frementes, qui in vel. Hippotades Peleus [pennas
-conscia](http://gratia.net/tot-qua.php) cuiquam Caeneus quas.
-
-- Pater demittere evincitque reddunt
-- Maxime adhuc pressit huc Danaas quid freta
-- Soror ego
-- Luctus linguam saxa ultroque prior Tatiumque inquit
-- Saepe liquitur subita superata dederat Anius sudor
-
-## Cum honorum Latona
-
-O fallor [in sustinui
-iussorum](http://www.spectataharundine.org/aquas-relinquit.html) equidem.
-Nymphae operi oris alii fronde parens dumque, in auro ait mox ingenti proxima
-iamdudum maius?
-
-    reality(burnDocking(apache_nanometer),
-            pad.property_data_programming.sectorBrowserPpga(dataMask, 37,
-            recycleRup));
-    intellectualVaporwareUser += -5 * 4;
-    traceroute_key_upnp /= lag_optical(android.smb(thyristorTftp));
-    surge_host_golden = mca_compact_device(dual_dpi_opengl, 33,
-            commerce_add_ppc);
-    if (lun_ipv) {
-        verticalExtranet(1, thumbnail_ttl, 3);
-        bar_graphics_jpeg(chipset - sector_xmp_beta);
-    }
-
-## Fronde cetera dextrae sequens pennis voce muneris
-
-Acta cretus diem restet utque; move integer, oscula non inspirat, noctisque
-scelus! Nantemque in suas vobis quamvis, et labori!
-
-    var runtimeDiskCompiler = home - array_ad_software;
-    if (internic > disk) {
-        emoticonLockCron += 37 + bps - 4;
-        wan_ansi_honeypot.cardGigaflops = artificialStorageCgi;
-        simplex -= downloadAccess;
-    }
-    var volumeHardeningAndroid = pixel + tftp + onProcessorUnmount;
-    sector(memory(firewire + interlaced, wired));
\ No newline at end of file
diff --git a/content/docs/example/table-of-contents/_index.md b/content/docs/example/table-of-contents/_index.md
deleted file mode 100644
index c7ee0d87..00000000
--- a/content/docs/example/table-of-contents/_index.md
+++ /dev/null
@@ -1,85 +0,0 @@
----
-weight: 10
----
-
-# Ubi loqui
-
-## Mentem genus facietque salire tempus bracchia
-
-Lorem markdownum partu paterno Achillem. Habent amne generosi aderant ad pellem
-nec erat sustinet merces columque haec et, dixit minus nutrit accipiam subibis
-subdidit. Temeraria servatum agros qui sed fulva facta. Primum ultima, dedit,
-suo quisque linguae medentes fixo: tum petis.
-
-## Rapit vocant si hunc siste adspice
-
-Ora precari Patraeque Neptunia, dixit Danae [Cithaeron
-armaque](http://mersis-an.org/litoristum) maxima in **nati Coniugis** templis
-fluidove. Effugit usus nec ingreditur agmen *ac manus* conlato. Nullis vagis
-nequiquam vultibus aliquos altera *suum venis* teneas fretum. Armos [remotis
-hoc](http://tutum.io/me) sine ferrea iuncta quam!
-
-## Locus fuit caecis
-
-Nefas discordemque domino montes numen tum humili nexilibusque exit, Iove. Quae
-miror esse, scelerisque Melaneus viribus. Miseri laurus. Hoc est proposita me
-ante aliquid, aura inponere candidioribus quidque accendit bella, sumpta.
-Intravit quam erat figentem hunc, motus de fontes parvo tempestate.
-
-    iscsi_virus = pitch(json_in_on(eupViral),
-            northbridge_services_troubleshooting, personal(
-            firmware_rw.trash_rw_crm.device(interactive_gopher_personal,
-            software, -1), megabit, ergonomicsSoftware(cmyk_usb_panel,
-            mips_whitelist_duplex, cpa)));
-    if (5) {
-        managementNetwork += dma - boolean;
-        kilohertz_token = 2;
-        honeypot_affiliate_ergonomics = fiber;
-    }
-    mouseNorthbridge = byte(nybble_xmp_modem.horse_subnet(
-            analogThroughputService * graphicPoint, drop(daw_bit, dnsIntranet),
-            gateway_ospf), repository.domain_key.mouse(serverData(fileNetwork,
-            trim_duplex_file), cellTapeDirect, token_tooltip_mashup(
-            ripcordingMashup)));
-    module_it = honeypot_driver(client_cold_dvr(593902, ripping_frequency) +
-            coreLog.joystick(componentUdpLink), windows_expansion_touchscreen);
-    bashGigabit.external.reality(2, server_hardware_codec.flops.ebookSampling(
-            ciscNavigationBacklink, table + cleanDriver), indexProtocolIsp);
-
-## Placabilis coactis nega ingemuit ignoscat nimia non
-
-Frontis turba. Oculi gravis est Delphice; *inque praedaque* sanguine manu non.
-
-    if (ad_api) {
-        zif += usb.tiffAvatarRate(subnet, digital_rt) + exploitDrive;
-        gigaflops(2 - bluetooth, edi_asp_memory.gopher(queryCursor, laptop),
-                panel_point_firmware);
-        spyware_bash.statePopApplet = express_netbios_digital(
-                insertion_troubleshooting.brouter(recordFolderUs), 65);
-    }
-    recursionCoreRay = -5;
-    if (hub == non) {
-        portBoxVirus = soundWeb(recursive_card(rwTechnologyLeopard),
-                font_radcab, guidCmsScalable + reciprocalMatrixPim);
-        left.bug = screenshot;
-    } else {
-        tooltipOpacity = raw_process_permalink(webcamFontUser, -1);
-        executable_router += tape;
-    }
-    if (tft) {
-        bandwidthWeb *= social_page;
-    } else {
-        regular += 611883;
-        thumbnail /= system_lag_keyboard;
-    }
-
-## Caesorum illa tu sentit micat vestes papyriferi
-
-Inde aderam facti; Theseus vis de tauri illa peream. Oculos **uberaque** non
-regisque vobis cursuque, opus venit quam vulnera. Et maiora necemque, lege modo;
-gestanda nitidi, vero? Dum ne pectoraque testantur.
-
-Venasque repulsa Samos qui, exspectatum eram animosque hinc, [aut
-manes](http://www.creveratnon.net/apricaaetheriis), Assyrii. Cupiens auctoribus
-pariter rubet, profana magni super nocens. Vos ius sibilat inpar turba visae
-iusto! Sedes ante dum superest **extrema**.
diff --git a/content/docs/example/table-of-contents/with-toc.md b/content/docs/example/table-of-contents/with-toc.md
deleted file mode 100644
index 5345c668..00000000
--- a/content/docs/example/table-of-contents/with-toc.md
+++ /dev/null
@@ -1,64 +0,0 @@
----
-title: With ToC
-weight: 1
----
-# Caput vino delphine in tamen vias
-
-## Cognita laeva illo fracta
-
-Lorem markdownum pavent auras, surgit nunc cingentibus libet **Laomedonque que**
-est. Pastor [An](http://est.org/ire.aspx) arbor filia foedat, ne [fugit
-aliter](http://www.indiciumturbam.org/moramquid.php), per. Helicona illas et
-callida neptem est *Oresitrophos* caput, dentibus est venit. Tenet reddite
-[famuli](http://www.antro-et.net/) praesentem fortibus, quaeque vis foret si
-frondes *gelidos* gravidae circumtulit [inpulit armenta
-nativum](http://incurvasustulit.io/illi-virtute.html).
-
-1. Te at cruciabere vides rubentis manebo
-2. Maturuit in praetemptat ruborem ignara postquam habitasse
-3. Subitarum supplevit quoque fontesque venabula spretis modo
-4. Montis tot est mali quasque gravis
-5. Quinquennem domus arsit ipse
-6. Pellem turis pugnabant locavit
-
-## Natus quaerere
-
-Pectora et sine mulcere, coniuge dum tincta incurvae. Quis iam; est dextra
-Peneosque, metuis a verba, primo. Illa sed colloque suis: magno: gramen, aera
-excutiunt concipit.
-
-> Phrygiae petendo suisque extimuit, super, pars quod audet! Turba negarem.
-> Fuerat attonitus; et dextra retinet sidera ulnas undas instimulat vacuae
-> generis? *Agnus* dabat et ignotis dextera, sic tibi pacis **feriente at mora**
-> euhoeque *comites hostem* vestras Phineus. Vultuque sanguine dominoque [metuit
-> risi](http://iuvat.org/eundem.php) fama vergit summaque meus clarissimus
-> artesque tinguebat successor nominis cervice caelicolae.
-
-## Limitibus misere sit
-
-Aurea non fata repertis praerupit feruntur simul, meae hosti lentaque *citius
-levibus*, cum sede dixit, Phaethon texta. *Albentibus summos* multifidasque
-iungitur loquendi an pectore, mihi ursaque omnia adfata, aeno parvumque in animi
-perlucentes. Epytus agis ait vixque clamat ornum adversam spondet, quid sceptra
-ipsum **est**. Reseret nec; saeva suo passu debentia linguam terga et aures et
-cervix [de](http://www.amnem.io/pervenit.aspx) ubera. Coercet gelidumque manus,
-doluit volvitur induta?
-
-## Enim sua
-
-Iuvenilior filia inlustre templa quidem herbis permittat trahens huic. In
-cruribus proceres sole crescitque *fata*, quos quos; merui maris se non tamen
-in, mea.
-
-## Germana aves pignus tecta
-
-Mortalia rudibusque caelum cognosceret tantum aquis redito felicior texit, nec,
-aris parvo acre. Me parum contulerant multi tenentem, gratissime suis; vultum tu
-occupat deficeret corpora, sonum. E Actaea inplevit Phinea concepit nomenque
-potest sanguine captam nulla et, in duxisses campis non; mercede. Dicere cur
-Leucothoen obitum?
-
-Postibus mittam est *nubibus principium pluma*, exsecratur facta et. Iunge
-Mnemonidas pallamque pars; vere restitit alis flumina quae **quoque**, est
-ignara infestus Pyrrha. Di ducis terris maculatum At sede praemia manes
-nullaque!
diff --git a/content/docs/example/table-of-contents/without-toc.md b/content/docs/example/table-of-contents/without-toc.md
deleted file mode 100644
index 9b163188..00000000
--- a/content/docs/example/table-of-contents/without-toc.md
+++ /dev/null
@@ -1,59 +0,0 @@
----
-title: Without ToC
-weight: 2
-bookToc: false
----
-
-# At me ipso nepotibus nunc celebratior genus
-
-## Tanto oblite
-
-Lorem markdownum pectora novis patenti igne sua opus aurae feras materiaque
-illic demersit imago et aristas questaque posset. Vomit quoque suo inhaesuro
-clara. Esse cumque, per referri triste. Ut exponit solisque communis in tendens
-vincetis agisque iamque huic bene ante vetat omina Thebae rates. Aeacus servat
-admonitu concidit, ad resimas vultus et rugas vultu **dignamque** Siphnon.
-
-Quam iugulum regia simulacra, plus meruit humo pecorumque haesit, ab discedunt
-dixit: ritu pharetramque. Exul Laurenti orantem modo, per densum missisque labor
-manibus non colla unum, obiectat. Tu pervia collo, fessus quae Cretenque Myconon
-crate! Tegumenque quae invisi sudore per vocari quaque plus ventis fluidos. Nodo
-perque, fugisse pectora sorores.
-
-## Summe promissa supple vadit lenius
-
-Quibus largis latebris aethera versato est, ait sentiat faciemque. Aequata alis
-nec Caeneus exululat inclite corpus est, ire **tibi** ostendens et tibi. Rigent
-et vires dique possent lumina; **eadem** dixit poma funeribus paret et felix
-reddebant ventis utile lignum.
-
-1. Remansit notam Stygia feroxque
-2. Et dabit materna
-3. Vipereas Phrygiaeque umbram sollicito cruore conlucere suus
-4. Quarum Elis corniger
-5. Nec ieiunia dixit
-
-Vertitur mos ortu ramosam contudit dumque; placabat ac lumen. Coniunx Amoris
-spatium poenamque cavernis Thebae Pleiadasque ponunt, rapiare cum quae parum
-nimium rima.
-
-## Quidem resupinus inducto solebat una facinus quae
-
-Credulitas iniqua praepetibus paruit prospexit, voce poena, sub rupit sinuatur,
-quin suum ventorumque arcadiae priori. Soporiferam erat formamque, fecit,
-invergens, nymphae mutat fessas ait finge.
-
-1. Baculum mandataque ne addere capiti violentior
-2. Altera duas quam hoc ille tenues inquit
-3. Sicula sidereus latrantis domoque ratae polluit comites
-4. Possit oro clausura namque se nunc iuvenisque
-5. Faciem posuit
-6. Quodque cum ponunt novercae nata vestrae aratra
-
-Ite extrema Phrygiis, patre dentibus, tonso perculit, enim blanda, manibus fide
-quos caput armis, posse! Nocendo fas Alcyonae lacertis structa ferarum manus
-fulmen dubius, saxa caelum effuge extremis fixum tumor adfecit **bella**,
-potentes? Dum nec insidiosa tempora tegit
-[spirarunt](http://mihiferre.net/iuvenes-peto.html). Per lupi pars foliis,
-porreximus humum negant sunt subposuere Sidone steterant auro. Memoraverit sine:
-ferrum idem Orion caelum heres gerebat fixis?
diff --git a/content/docs/notes/crews2018.md b/content/docs/notes/crews2018.md
deleted file mode 100644
index e6a3d363..00000000
--- a/content/docs/notes/crews2018.md
+++ /dev/null
@@ -1,34 +0,0 @@
----
-title: "Crews (2018)"
----
-
-# Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities
-
-Shielded potential of a test electron:
-
-{{< katex display >}}
-\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D}
-{{< /katex >}}
-
-where the Debye length is {{< katex >}} \lambda_D = \sqrt{\frac{\epsilon_0 T_e}{ n_e e}} {{< /katex >}}. The mean free path between large-angle collisions is estimated as
-
-{{< katex display >}}
-\lambda_{mfp} \sim \frac{\epsilon_0 T_e ^2}{\phi_e n_e \log ( \Lambda)}
-{{< /katex >}}
-where {{< katex >}}\phi_e = e^2 / 4 \pi \epsilon_0{{< /katex >}} are the constants from the Coulomb force law.
-
-Smooth out the discreteness of particles via spatial average over small volumes:
-
-{{< katex display >}}
-\rho \rightarrow \langle \rho_c \rangle \qquad \vec E \rightarrow \langle \vec E \rangle + \delta \vec E
-{{< /katex >}}
-
-The mean field {{< katex >}}\langle \vec E \rangle{{< /katex >}} is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales {{< katex >}}\lambda_{mfp} / \lambda_D{{< /katex >}}
-
-{{< katex display >}}
-\frac{\lambda_{mfp}}{\lambda_D} \sim \frac{T_e ^{3/2}}{n_e ^{1/2}}
-{{< /katex >}}
-
-Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified.
-
-Maybe this will publish...
\ No newline at end of file
diff --git a/content/docs/shortcodes/buttons.md b/content/docs/shortcodes/buttons.md
deleted file mode 100644
index c2ef1e75..00000000
--- a/content/docs/shortcodes/buttons.md
+++ /dev/null
@@ -1,13 +0,0 @@
-# Buttons
-
-Buttons are styled links that can lead to local page or external link.
-
-## Example
-
-```tpl
-{{</* button relref="/" [class="..."] */>}}Get Home{{</* /button */>}}
-{{</* button href="https://github.com/alex-shpak/hugo-book" */>}}Contribute{{</* /button */>}}
-```
-
-{{< button relref="/" >}}Get Home{{< /button >}}
-{{< button href="https://github.com/alex-shpak/hugo-book" >}}Contribute{{< /button >}}
diff --git a/content/docs/shortcodes/columns.md b/content/docs/shortcodes/columns.md
deleted file mode 100644
index 0b8fde83..00000000
--- a/content/docs/shortcodes/columns.md
+++ /dev/null
@@ -1,45 +0,0 @@
-# Columns
-
-Columns help organize shorter pieces of content horizontally for readability.
-
-
-```html
-{{</* columns */>}} <!-- begin columns block -->
-# Left Content
-Lorem markdownum insigne...
-
-<---> <!-- magic separator, between columns -->
-
-# Mid Content
-Lorem markdownum insigne...
-
-<---> <!-- magic separator, between columns -->
-
-# Right Content
-Lorem markdownum insigne...
-{{</* /columns */>}}
-```
-
-## Example
-
-{{< columns >}}
-## Left Content
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes.
-Miseratus fonte Ditis conubia.
-
-<--->
-
-## Mid Content
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter!
-
-<--->
-
-## Right Content
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes.
-Miseratus fonte Ditis conubia.
-{{< /columns >}}
diff --git a/content/docs/shortcodes/details.md b/content/docs/shortcodes/details.md
deleted file mode 100644
index 248bafd9..00000000
--- a/content/docs/shortcodes/details.md
+++ /dev/null
@@ -1,22 +0,0 @@
-# Details
-
-Details shortcode is a helper for `details` html5 element. It is going to replace `expand` shortcode.
-
-## Example
-```tpl
-{{</* details "Title" [open] */>}}
-## Markdown content
-Lorem markdownum insigne...
-{{</* /details */>}}
-```
-```tpl
-{{</* details title="Title" open=true */>}}
-## Markdown content
-Lorem markdownum insigne...
-{{</* /details */>}}
-```
-
-{{< details "Title" open >}}
-## Markdown content
-Lorem markdownum insigne...
-{{< /details >}}
diff --git a/content/docs/shortcodes/expand.md b/content/docs/shortcodes/expand.md
deleted file mode 100644
index c62520f3..00000000
--- a/content/docs/shortcodes/expand.md
+++ /dev/null
@@ -1,35 +0,0 @@
-# Expand
-
-Expand shortcode can help to decrease clutter on screen by hiding part of text. Expand content by clicking on it.
-
-## Example
-### Default
-
-```tpl
-{{</* expand */>}}
-## Markdown content
-Lorem markdownum insigne...
-{{</* /expand */>}}
-```
-
-{{< expand >}}
-## Markdown content
-Lorem markdownum insigne...
-{{< /expand >}}
-
-### With Custom Label
-
-```tpl
-{{</* expand "Custom Label" "..." */>}}
-## Markdown content
-Lorem markdownum insigne...
-{{</* /expand */>}}
-```
-
-{{< expand "Custom Label" "..." >}}
-## Markdown content
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes.
-Miseratus fonte Ditis conubia.
-{{< /expand >}}
diff --git a/content/docs/shortcodes/hints.md b/content/docs/shortcodes/hints.md
deleted file mode 100644
index 3477113d..00000000
--- a/content/docs/shortcodes/hints.md
+++ /dev/null
@@ -1,32 +0,0 @@
-# Hints
-
-Hint shortcode can be used as hint/alerts/notification block.  
-There are 3 colors to choose: `info`, `warning` and `danger`.
-
-```tpl
-{{</* hint [info|warning|danger] */>}}
-**Markdown content**  
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-{{</* /hint */>}}
-```
-
-## Example
-
-{{< hint info >}}
-**Markdown content**  
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-{{< /hint >}}
-
-{{< hint warning >}}
-**Markdown content**  
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-{{< /hint >}}
-
-{{< hint danger >}}
-**Markdown content**  
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-{{< /hint >}}
diff --git a/content/docs/shortcodes/katex.md b/content/docs/shortcodes/katex.md
deleted file mode 100644
index c7459e51..00000000
--- a/content/docs/shortcodes/katex.md
+++ /dev/null
@@ -1,28 +0,0 @@
-# KaTeX
-
-KaTeX shortcode let you render math typesetting in markdown document. See [KaTeX](https://katex.org/)
-
-## Example
-{{< columns >}}
-
-```latex
-{{</* katex [display] [class="text-center"]  */>}}
-f(x) = \int_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,d\xi
-{{</* /katex */>}}
-```
-
-<--->
-
-{{< katex display >}}
-f(x) = \int_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,d\xi
-{{< /katex >}}
-
-{{< /columns >}}
-
-## Display Mode Example
-
-Here is some inline example: {{< katex >}}\pi(x){{< /katex >}}, rendered in the same line. And below is `display` example, having `display: block`
-{{< katex display >}}
-f(x) = \int_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,d\xi
-{{< /katex >}}
-Text continues here.
\ No newline at end of file
diff --git a/content/docs/shortcodes/mermaid.md b/content/docs/shortcodes/mermaid.md
deleted file mode 100644
index 3a617bcc..00000000
--- a/content/docs/shortcodes/mermaid.md
+++ /dev/null
@@ -1,38 +0,0 @@
-# Mermaid Chart
-
-[Mermaid](https://mermaidjs.github.io/) is library for generating svg charts and diagrams from text.
-
-## Example
-
-{{< columns >}}
-```tpl
-{{</* mermaid [class="text-center"]*/>}}
-sequenceDiagram
-    Alice->>Bob: Hello Bob, how are you?
-    alt is sick
-        Bob->>Alice: Not so good :(
-    else is well
-        Bob->>Alice: Feeling fresh like a daisy
-    end
-    opt Extra response
-        Bob->>Alice: Thanks for asking
-    end
-{{</* /mermaid */>}}
-```
-
-<--->
-
-{{< mermaid >}}
-sequenceDiagram
-    Alice->>Bob: Hello Bob, how are you?
-    alt is sick
-        Bob->>Alice: Not so good :(
-    else is well
-        Bob->>Alice: Feeling fresh like a daisy
-    end
-    opt Extra response
-        Bob->>Alice: Thanks for asking
-    end
-{{< /mermaid >}}
-
-{{< /columns >}}
diff --git a/content/docs/shortcodes/section/_index.md b/content/docs/shortcodes/section/_index.md
deleted file mode 100644
index bd5db38b..00000000
--- a/content/docs/shortcodes/section/_index.md
+++ /dev/null
@@ -1,15 +0,0 @@
----
-bookCollapseSection: true
----
-
-# Section
-
-Section renders pages in section as definition list, using title and description.
-
-## Example
-
-```tpl
-{{</* section */>}}
-```
-
-{{<section>}}
diff --git a/content/docs/shortcodes/section/page1.md b/content/docs/shortcodes/section/page1.md
deleted file mode 100644
index 96080014..00000000
--- a/content/docs/shortcodes/section/page1.md
+++ /dev/null
@@ -1 +0,0 @@
-# Page 1
diff --git a/content/docs/shortcodes/section/page2.md b/content/docs/shortcodes/section/page2.md
deleted file mode 100644
index f310be33..00000000
--- a/content/docs/shortcodes/section/page2.md
+++ /dev/null
@@ -1 +0,0 @@
-# Page 2
diff --git a/content/docs/shortcodes/tabs.md b/content/docs/shortcodes/tabs.md
deleted file mode 100644
index 096892c6..00000000
--- a/content/docs/shortcodes/tabs.md
+++ /dev/null
@@ -1,50 +0,0 @@
-# Tabs
-
-Tabs let you organize content by context, for example installation instructions for each supported platform.
-
-```tpl
-{{</* tabs "uniqueid" */>}}
-{{</* tab "MacOS" */>}} # MacOS Content {{</* /tab */>}}
-{{</* tab "Linux" */>}} # Linux Content {{</* /tab */>}}
-{{</* tab "Windows" */>}} # Windows Content {{</* /tab */>}}
-{{</* /tabs */>}}
-```
-
-## Example
-
-{{< tabs "uniqueid" >}}
-{{< tab "MacOS" >}}
-# MacOS
-
-This is tab **MacOS** content.
-
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes.
-Miseratus fonte Ditis conubia.
-{{< /tab >}}
-
-{{< tab "Linux" >}}
-
-# Linux
-
-This is tab **Linux** content.
-
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes.
-Miseratus fonte Ditis conubia.
-{{< /tab >}}
-
-{{< tab "Windows" >}}
-
-# Windows
-
-This is tab **Windows** content.
-
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat
-stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa
-protulit, sed sed aere valvis inhaesuro Pallas animam: qui _quid_, ignes.
-Miseratus fonte Ditis conubia.
-{{< /tab >}}
-{{< /tabs >}}
diff --git a/content/menu/index.md b/content/menu/index.md
index 810bcfd1..0c8fbf66 100644
--- a/content/menu/index.md
+++ b/content/menu/index.md
@@ -2,21 +2,3 @@
 headless: true
 ---
 
-- [**Example Site**]({{< relref "/docs/example" >}})
-- [Table of Contents]({{< relref "/docs/example/table-of-contents" >}})
-  - [With ToC]({{< relref "/docs/example/table-of-contents/with-toc" >}})
-  - [Without ToC]({{< relref "/docs/example/table-of-contents/without-toc" >}})
-- [Collapsed]({{< relref "/docs/example/collapsed" >}})
-  - [3rd]({{< relref "/docs/example/collapsed/3rd-level" >}})
-    - [4th]({{< relref "/docs/example/collapsed/3rd-level/4th-level" >}})
-<br />
-
-- **Shortcodes**
-- [Buttons]({{< relref "/docs/shortcodes/buttons" >}})
-- [Columns]({{< relref "/docs/shortcodes/columns" >}})
-- [Expand]({{< relref "/docs/shortcodes/expand" >}})
-- [Hints]({{< relref "/docs/shortcodes/hints" >}})
-- [Katex]({{< relref "/docs/shortcodes/katex" >}})
-- [Mermaid]({{< relref "/docs/shortcodes/mermaid" >}})
-- [Tabs]({{< relref "/docs/shortcodes/tabs" >}})
-<br />
diff --git a/content/notes/UWAA558/01-syllabus.md b/content/notes/UWAA558/01-syllabus.md
new file mode 100644
index 00000000..b618c91d
--- /dev/null
+++ b/content/notes/UWAA558/01-syllabus.md
@@ -0,0 +1,50 @@
+---
+title: Syllabus
+weight: 10
+---
+
+
+## Syllabus
+
+The course topics planned for this section are (in rough order):
+
+Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, "Frozen-In" Flux)
+
+Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces
+
+One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q
+
+Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)
+
+MHD Stability, General Concepts, Linearized MHD, Exponential (Linear) Stability, Force Operator and Properties, Variational Formulation, Energy Principle, Intuitive Form of delta W, Classification of Instabilities (internal/external, pressure-drive/current-driven, kink/interchange/ballooning)
+
+Stability of One-Dimensional Equilibria, Modal Analysis, Rayleigh-Taylor, Theta-Pinch, Z-Pinch (Kadomtsev Condition), Screw-Pinch (Kruskal-Shafranov Condition, Suydam Criterion), RFP, "Straight" Tokamak
+
+Stability of Two-Dimensional Equilibria, Tokamak, Mercier Criterion, Elmo Bumpy Torus
+
+Resistive (Tearing) Instabilities, Stability of Non-static Equilibria, Nonlinear Stability Effects
+
+## Course Motivation
+
+Plasma phenomena tend to be hard to treat because of the span of relevant scales. You have ions, electrons, and photons interacting through electromagnetic interactions. There is a tremendous variation in mass across species, which leads to a large span of both spatial and temporal scales. The species can interact through both short scale collisions and long range interactions through EM forces. In contrast, in normal gas dynamics you may consider only the short-scale interactions. As a consequence, we can describe dispersive plasma waves.
+
+For comparison, remember in gas dynamics, the speed of sound is 
+
+
+{{< katex display >}}
+\frac{\omega}{k} = v_s = \pdv{\omega}{k} \qquad \text{(gas)}
+{{< /katex >}}
+
+Here the phase velocity {{< katex >}} \frac{\omega}{k} {{< /katex >}} is equal to the phase velocity {{< katex >}} \pdv{\omega}{k} {{< /katex >}}. In a plasma, we can have non-linear dispersion relations in which the phase and group velocity are different.
+
+{{< katex display >}}
+\frac{\omega}{k} = v(\omega, k) \neq \pdv{\omega}{k} \qquad \text{(plasma)} 
+{{< /katex >}}
+
+The number of particles we typically deal with in a laboratory plasma is roughly on the order of a mole of particles
+
+{{< katex display >}}
+\text{particles} \sim O(10^{23})
+{{< /katex >}}
+
+With long-range interactions, we have a combinatorial explosion of interacting particles! It is not possible to track individual particles at such a scale, so we need much simpler plasma descriptions. These are the plasma models we will discuss in the next chapter. When simplifying our models, we need to pay careful attention to the simplifications we are making because in general, inaccurate physics lead to incorrect conclusions.
\ No newline at end of file
diff --git a/content/notes/UWAA558/02-plasma-models.md b/content/notes/UWAA558/02-plasma-models.md
new file mode 100644
index 00000000..9186507c
--- /dev/null
+++ b/content/notes/UWAA558/02-plasma-models.md
@@ -0,0 +1,124 @@
+---
+title: Plasma Models
+weight: 20
+bookToc: false
+---
+
+# Plasma Models
+
+## Working towards MHD
+
+Let's start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we'll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.
+
+The most important question to ask ourselves is "when is this model going to be useful?" The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid. This means that we can define a fluid element (some lil' box of plasma) and define the physical properties (mass, density, magnetization, etc.) of the element. We need to make some assumptions about scale in order to do this. In terms of spatial scales, we abstract properties below a discrete scale {{< katex >}} a_0 {{< /katex >}}  into the properties of a fluid element
+
+{{< katex display >}}
+\frac{a_0}{L} \rightarrow 0 \qquad a_0 = \text{discrete scale} \qquad L = \text{spatial scale of interest}
+{{< /katex >}}
+
+Length scales smaller than the discrete scale will not be properly captured by the model, so scales like the particle radius will be meaningless in our fluid model.
+
+## Plasma Definition
+
+A plasma is a quasi-neutral gas of charged and neutral particles which exhibit _collective behavior_. The particles (electrons, ions, neutrals) interact through EM fields and collisions.
+
+Mathematically you would think that plasmas could be treated as individual particles. Doing so gives an N-body problem with classical interactions through the Lorentz force (Coulomb interaction removed by assumption of quasi-neutrality) and binary collisions. Each particle {{< katex >}} i {{< /katex >}}  has a well-defined mass {{< katex >}} m_i {{< /katex >}}  and charge {{< katex >}} q_i {{< /katex >}}  which do not change in time. The governing equations are
+
+{{< katex display >}}
+\dv{\vec v_i}{t} = \frac{q_i}{m_i} (\vec E + \vec v_i \cross \vec B) + \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} (\vec r_i - \vec r_j) \right]
+{{< /katex >}}
+{{< katex display >}}
+\dv{\vec r_i}{t} = \vec v_i
+{{< /katex >}}
+
+The fields E and B are described by the Maxwell equations
+
+{{< katex display >}}
+\pdv{B}{t} = - \curl E
+{{< /katex >}}
+{{< katex display >}}
+\frac{1}{c^2} \pdv{E}{t} = \curl B - \mu_0 \sum_i q_i v_i \delta (r - r_i)
+{{< /katex >}}
+{{< katex display >}}
+\div B = 0
+{{< /katex >}}
+{{< katex display >}}
+\epsilon_0 \div E = \sum_i q_i \delta (r - r_i)
+{{< /katex >}}
+
+## Klimontovich Equation
+
+Re-writing the force relations as a statement of conservation in phase space, we get Klimontovich equation for species {{< katex >}} \alpha {{< /katex >}} 
+
+{{< katex display >}}
+\dv{N_\alpha}{t} = 0 = \pdv{N_\alpha}{t} + \pdv{}{q} \cdot (\dot{q} N_\alpha) + \pdv{}{p} \cdot (\dot{p}N_\alpha)
+{{< /katex >}}
+
+The particle phase space is defined by 
+{{< katex display >}}
+N_\alpha(p, q) = \sum_i \delta(p - p_i) \delta(q - q_i)
+{{< /katex >}}
+where {{< katex >}} p {{< /katex >}} and {{< katex >}} q {{< /katex >}} are generalized momentum and position coordinates. The resulting {{< katex >}} N(p) {{< /katex >}} looks very spiky, with nonzero values only at the exact values inhabited by particles. Unfortunately, that means that only the tools of discrete mathematics are applicable to the distribution, and we're forbidden from our favorite tool (calculus). To make the analysis possible, we can smooth over the discreteness by performing an ensemble average of the Klimontovich equation. This gives us a statistical description using smooth distribution functions:
+
+{{< katex display >}}
+f(x, v, t) \qquad \frac{(\text{no. of particles})}{(\text{unit distance})^3(\text{unit velocity})^3}
+{{< /katex >}}
+
+That is, {{< katex >}} f(x, v, t) {{< /katex >}} is the number of particles at position {{< katex >}} x {{< /katex >}} with velocity {{< katex >}} v {{< /katex >}} at time {{< katex >}} t {{< /katex >}}. We also work with normalized distributions which give the probability of finding a particle.
+
+By our definition, we can integrate to get the total number of particles at time {{< katex >}} t {{< /katex >}}.
+
+{{< katex display >}}
+\int \int \dd x \dd v f(x, v, t) = N(t)
+{{< /katex >}}
+
+The ensemble averaging process works well if the the number of particles is very large
+
+{{< katex display >}}
+N \gg 1
+{{< /katex >}}
+
+<p align="center"> <img alt="Figure 12.1" src="/r/img/12.1.png" /> </p>
+
+See a kinetic theory text (e.g. Krall and Trivelpiece) for a full description of the ensemble averaging process.
+
+Now we can write our Klimontovich equation in terms of continuous quantities
+
+{{< katex display >}}
+\pdv{f}{t} + \left[ \pdv{}{q} \cdot (\dot{q} f) + \pdv{}{p} \cdot (\dot{p}f) \right] = \text{(cross terms)}
+{{< /katex >}}
+
+The collision terms now have an infinite number of cross terms. We call this the BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy.
+
+Expressing as the Boltzmann equation (generally called the Boltzmann-Maxwell equation, since the solution requires solving for the electromagnetic fields of the Maxwell equations)
+
+{{< katex display >}}
+\pdv{f_\alpha}{t} + \vec v \cdot \pdv{f_\alpha}{\vec x} + \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll}
+{{< /katex >}}
+
+We leave the collision term in. A lot of the work of kinetic theory is coming up with an applicable form of the collision operator which is appropriate but still simple enough to solve. We often write the collision operator as the product of binary collisions
+
+{{< katex display >}}
+ \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_\beta C_{\alpha \beta}
+{{< /katex >}}
+
+Notice that the terms {{< katex >}} \vec v \cdot \pdv{f_\alpha}{\vec x} {{< /katex >}} and {{< katex >}} \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} {{< /katex >}} are advection equations, advecting in {{< katex >}} \vec x {{< /katex >}} and {{< katex >}} \vec v {{< /katex >}} respectively. If we ignore the collision term (set the RHS to zero) we have the Vlasov equation.
+
+Note that the fields E and B at any location are generated from the charges and currents of the entire plasma volume, including externally applied fields. As a result, there's an inherent integrating process taking into account the sources across the whole volume that leads to long-range smoothly varying forces. This is in contrast to the collisional effects, which by their nature lead to very short range abrupt forces. It makes sense to make a distinction between the long-range electromagnetic forces and the short range collisional forces.
+
+Because the Boltzmann-Maxwell model is inherently 6-dimensional, it is a very challenging model to implement. The B-M model provides a complete description, but it is often too detailed to solve.
+
+If we solve the Vlasov equation for two parallel opposite beams, for example, we see that the 
+
+As it turns out, the integral, centroid, and variance are all that are required to fully describe a Maxwellian distribution
+
+{{< katex display >}}
+f_{M, \alpha} (\vec v) = n_0 \left( \frac{ m_\alpha }{2 \pi T_\alpha} \right)^{3/2} \text{exp}\left[ - \frac{ \frac{1}{2} m_\alpha (v - v_\alpha)^2}{T_\alpha} \right]
+{{< /katex >}}
+
+In other words, we can write the Maxwellian distribution as {{< katex >}} f_M(n_0, \vec v_\alpha, T_\alpha ; \vec v) {{< /katex >}}. We care about Maxwellian distributions so much in plasma physics because it is the solution to the Boltzmann equation for {{< katex >}} \left. \pdv{f_\alpha}{t} \right|_{coll} = 0 {{< /katex >}} (Vlasov equation). That's not saying that there are no collisions, it is saying that there are so many collisions that the effect is isotropic and the overall force is zero.
+
+Another feature of the Maxwellian distribution is {{< katex >}} \vec v \cdot \pdv{f_\alpha}{\vec x} = 0 {{< /katex >}}. This is a famous result called the Boltzmann H-theorem, and says that any initial distribution will relax (and very quickly) to a Maxwellian distribution.
+
+By replacing our velocity distribution with the associated Maxwell distribution, we arrive at a Plasma Fluid Model 
+
diff --git a/content/notes/UWAA558/03-plasma-fluid-model.md b/content/notes/UWAA558/03-plasma-fluid-model.md
new file mode 100644
index 00000000..299a17d1
--- /dev/null
+++ b/content/notes/UWAA558/03-plasma-fluid-model.md
@@ -0,0 +1,528 @@
+---
+title: Plasma Fluid Model
+bookToc: false
+weight: 30
+---
+
+
+# Plasma Fluid Model
+
+We take velocity moments of each of the pieces of the kinetic model:
+  - distribution function, {{< katex >}} f_\alpha {{< /katex >}} {{< katex >}} \rightarrow {{< /katex >}} fluid variables
+  - Boltzmann Equation {{< katex >}} \rightarrow {{< /katex >}} governing equations describing the evolution of the fluid variables.
+
+Starting with the zeroth moment (integral) of the distribution function:
+
+{{< katex display >}}
+\int f_\alpha (\vec x, \vec v, t) \dd \vec v = n_\alpha(\vec x, t)
+{{< /katex >}}
+
+1st Moment (momentum):
+
+{{< katex display >}}
+m_\alpha \int \vec v \cdot  f_\alpha(\vec x, \vec v, t) \dd \vec v = \vec p_\alpha (\vec x, t) = m_\alpha n_\alpha \vec v_\alpha
+{{< /katex >}}
+
+which is to say that the velocity is the 1st moment divided by the zeroth moment
+
+{{< katex display >}}
+v_\alpha = \frac{\int \vec v f_\alpha}{\int f_\alpha}
+{{< /katex >}}
+
+2nd Moment:
+
+{{< katex display >}}
+\int \vec v \vec v f_\alpha (\vec x, \vec v, t) \dd \vec v = \vec E_\alpha(\vec x, t) (\text{energy tensor})
+{{< /katex >}}
+
+We can simplify the 2nd moment by taking a reduced 2nd moment. This means that we're going to insert a dot product
+
+{{< katex display >}}
+\int \vec v \cdot \vec v f_\alpha (\vec x, \vec v, t) \dd \vec v
+{{< /katex >}}
+
+Before moving forward, we want to define a "random" velocity. Note that
+
+{{< katex display >}}
+\int \vec v f_\alpha - \vec v_\alpha \int f_\alpha = 0 \rightarrow \int (\vec v - \vec v_\alpha) f_\alpha \dd \vec v = 0
+{{< /katex >}}
+
+We can define a random velocity {{< katex >}} \vec w = \vec v - \vec v_\alpha {{< /katex >}}. It is random in the sense that it is a fluctuation about the mean velocity, and when we integrate it we get zero. We can use this to define the energy tensor using the mean velocity to get a meaningful result. The pressure tensor is the second moment, using the random velocity
+
+{{< katex display >}}
+\vec P_\alpha = m_\alpha \int \vec w \vec w f_\alpha \dd \vec v = P_\alpha \overline{I} + \overline{\Pi}_\alpha
+{{< /katex >}}
+
+where we've decomposed the pressure into an isotropic value {{< katex >}} P_\alpha {{< /katex >}} and what's called the Braginskii stress tensor {{< katex >}} \overline{\Pi}_\alpha {{< /katex >}}. The average isotropic pressure is given by the reduced 2nd moment:
+
+{{< katex display >}}
+P_\alpha = n_\alpha T_\alpha = \frac{1}{3} \int m_\alpha \vec w \cdot \vec w f_\alpha
+{{< /katex >}}
+
+where the factor of {{< katex >}} 1/3 {{< /katex >}} comes from the number of degrees of freedom in our system. It is related to the thermodynamic factor {{< katex >}} \gamma {{< /katex >}} where {{< katex >}} \gamma = \frac{DOF + 2}{DOF} {{< /katex >}}. Now we can define the temperature {{< katex >}} T_\alpha {{< /katex >}} as
+
+{{< katex display >}}
+T_\alpha (\vec x, t) = \frac{1}{DOF} \frac{\int m_\alpha \vec w \cdot \vec w f_\alpha \dd \vec v}{\int f_\alpha \dd \vec v}
+{{< /katex >}}
+
+Now that we've got {{< katex >}} n_\alpha {{< /katex >}}, {{< katex >}} v_\alpha {{< /katex >}}, and {{< katex >}} T_\alpha {{< /katex >}} we have what we need to define a Maxwellian distribution. Higher moments would be required to describe non-Maxwellian distribution functions. For example, the 3rd moment is called the skewness of the distribution. The 4th moment is the kurtosis. So on and so forth. These give a measure of degree of departure from a Maxwellian distribution, in which case it is often more useful to talk about the excess kurtosis, where the excess kurtosis of a Maxwellian is defined to be zero. You can continue to calculate the moment expansion, and in general it requires an infinite number of moments to describe an arbitrary distribution function. Because of the Boltzmann H-theorem and the tendency of plasmas to quickly relax to Maxwellian, we can usually get away with using just the first three moments.
+
+Now, what are the governing equations? We get these by taking moments of the Boltzmann equation. Let's proceed carefully in sections, so we'll integrate each piece of the BE in terms.
+
+### 0th Moment of Boltzmann Equation (Conservation)
+
+{{< katex display >}}
+\int \pdv{f_\alpha}{t} \dd \vec v + \int \vec v \cdot \pdv{f_\alpha}{x} \dd v + \int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \left. \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v
+{{< /katex >}}
+
+First, we take {{< katex >}} \int \pdv{f_\alpha}{t} \dd \vec v {{< /katex >}}. Because t and v are both independent variables, with an argument of sufficient smoothness we can reverse the order of integration and differentiation
+
+{{< katex display >}}
+\int \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t}\int f_\alpha \dd v = \pdv{n_\alpha}{t}
+{{< /katex >}}
+
+For {{< katex >}}  \int \vec v \cdot \pdv{f_\alpha}{x} \dd v {{< /katex >}} we can perform an integration by parts
+
+{{< katex display >}}
+ \int \vec v \cdot \pdv{f_\alpha}{x} \dd v = \int \pdv{}{\vec x} \cdot ( \vec v f_\alpha) \dd \vec v - \int f_\alpha \pdv{}{\vec x} \cdot \vec v \dd \vec v \\
+ = \int \pdv{}{\vec x} \cdot ( \vec v f_\alpha) \dd \vec v  \\
+ = \pdv{}{\vec x} \cdot \int \vec v f_\alpha \dd \vec v \\
+ = \pdv{}{\vec x} \cdot (n_\alpha \vec v_\alpha) = \div (n_\alpha \vec v_\alpha)
+{{< /katex >}}
+
+Once again, we've switched the order of integration of {{< katex >}} x {{< /katex >}}  and {{< katex >}} v {{< /katex >}}, which we can only do because we have specified that {{< katex >}} f {{< /katex >}} is a distribution function, and as such meets the criterion of sufficient smoothness.
+
+For the last part, we can write it as a surface integral
+
+{{< katex display >}}
+\int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \oint f_\alpha \frac{q_\alpha}{m_\alpha} ( \vec E + \vec v \cross \vec B) \cdot \dd S_v - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot (\vec E + \vec v \cross \vec B) \dd \vec v
+{{< /katex >}}
+
+For {{< katex >}} f_\alpha {{< /katex >}} to be well-defined, we require {{< katex >}} \lim_{v \rightarrow \infty} v^3 f_\alpha = 0 {{< /katex >}} so the surface term vanishes and we're left with
+
+{{< katex display >}}
+\int \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot (\vec E + \vec v \cross \vec B) \dd \vec v 
+{{< /katex >}}
+
+Distributing the divergence through, 
+
+{{< katex display >}}
+\pdv{}{\vec v} \cdot \vec E = 0
+{{< /katex >}}
+because {{< katex >}} \vec E {{< /katex >}} is independent of {{< katex >}} \vec v {{< /katex >}}.
+
+{{< katex display >}}
+\pdv{}{\vec v} \cdot ( \vec v \cross \vec B) = 0
+{{< /katex >}}
+because {{< katex >}} \vec v \cross \vec B {{< /katex >}} is always orthogonal to {{< katex >}} \vec v {{< /katex >}}.
+
+We've been working through this in normal vector notation for familiarity, but there is another notation known as Einstein Tensor Notation using the lovely Levi-Civita symbol {{< katex >}} \epsilon_{ijk} {{< /katex >}}.
+
+{{< katex display >}}
+\epsilon_{ijk} = 1 \qquad \text{even permutations of i j k} \\
+ = -1 \qquad \text{odd permutations of i j k} \\
+= 0 \qquad \text{any repeated indexes}
+{{< /katex >}}
+
+
+We can write out vector products as products of indices and operators, and any repeated indices are implicitly summed:
+
+{{< katex display >}}
+\vec v \cross \vec B = \epsilon_{ijk} v_j B_k = \sum_{jk} \epsilon_{ijk} v_j B_k
+{{< /katex >}}
+
+where {{< katex >}} \epsilon_{ijk} {{< /katex >}} is defined to be 1 for even permutations of ijk, -1 for negative permutations if ijk, and 0 for any repeated indices.
+
+We can now write the derivative with respect to {{< katex >}} \vec v {{< /katex >}} as
+
+{{< katex display >}}
+\pdv{}{\vec v} \cdot \vec v \cross \vec B = \partial _{v_i} \epsilon_{ijk} v_j B_k \\
+= \epsilon_{ijk} (\partial_{v_i} v_j) B_k
+{{< /katex >}}
+
+We see that we're taking the derivative of the j-th component of velocity with respect to the i-th component of velocity, and _that's_ how we can most easily point out that the quantity is zero without relying on properties of 3-vector products.
+
+We'll also want to use the divergence in Einstein tensor notation
+
+{{< katex display >}}
+\div \vec A = \partial_i A_j \delta_{ij} = \partial_i A_i
+{{< /katex >}}
+
+Finally, we come back to the collision term in the zeroth moment of the B-M equation
+
+{{< katex display >}}
+\sum_\beta \int C_{\alpha \beta} \dd \vec v = 0
+{{< /katex >}}
+
+We can say the collision term is zero by making a physical argument, rather than a mathematical one. We assert that collisions cannot create or destroy particles. This assumption is now baked into our equations going forward, but is not always true! Ionization, recombination, fusion reactions all create/destroy species.
+
+Finally, we've got
+
+{{< katex display >}}
+\pdv{n_\alpha}{t} + \div ( n_\alpha \vec v_\alpha) = 0 \qquad \text{continuity equation}
+{{< /katex >}}
+
+So by taking the 0th moment of the Boltzmann Equation, we've arrived at the continuity equation for {{< katex >}} n_\alpha {{< /katex >}} by introducing {{< katex >}} \vec v_\alpha {{< /katex >}}.
+
+### 1st Moment of Boltzmann Equation
+
+{{< katex display >}}
+\int \vec v \pdv{f_\alpha}{t} \dd \vec v + \int \vec v \vec v  \pdv{f_\alpha}{\vec x} \dd \vec v + \int \vec v \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \left. \vec v \pdv{f_\alpha}{t} \right|_{coll}
+{{< /katex >}}
+
+Term 1:
+
+{{< katex display >}}
+\int \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v f_\alpha \dd \vec v - \int \pdv{\vec v}{t} f_\alpha \dd \vec v \\
+= \pdv{}{t} \left( n_\alpha \vec v_\alpha \right)
+{{< /katex >}}
+
+Before we get to the second term, let's have an aside about dyad math. An outer product {{< katex >}} \vec A \vec B {{< /katex >}} gives a second-rank tensor. In Cartesian coordinates, it looks like
+
+{{< katex display >}}
+\vec A \vec B = \begin{bmatrix}
+A_x B_x & A_x B_y & A_x B_z \\
+A_y B_x & A_y B_y & A_y B_z \\
+A_z B_x & A_z B_y & A_z B_z \\
+ \end{bmatrix}
+{{< /katex >}}
+
+A useful property to do with dot products:
+
+{{< katex display >}}
+\vec A \vec B \cdot \vec C = \vec A ( \vec B \cdot \vec C ) = (\vec A \vec B) \cdot \vec C
+{{< /katex >}}
+
+{{< katex display >}}
+( \vec V \cdot \grad) \vec B = \vec V \cdot ( \grad \vec B)
+{{< /katex >}}
+
+Back to term 2:
+
+{{< katex display >}}
+\int \vec v \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v = \int \vec v \cdot \pdv{}{\vec x} ( \vec v f_\alpha) \dd \vec v - \int \vec v f_\alpha \pdv{}{\vec x} \cdot \vec v \dd \vec v \\
+ = \int \pdv{}{\vec x} \cdot ( \vec v \vec v f_\alpha) \dd \vec v - \int \vec v f_\alpha \cdot \pdv{\vec v}{\vec x} \dd \vec v \\
+ = \pdv{}{\vec x} \cdot \int \vec v \vec v f_\alpha \dd \vec v
+{{< /katex >}}
+
+Re-expanding in terms of random velocity {{< katex >}} \vec v = \vec v_\alpha + \vec w {{< /katex >}}
+
+{{< katex display >}}
+= \pdv{}{\vec x} \cdot \int (\vec v_\alpha + \vec w) (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \\
+= \pdv{}{\vec x} \cdot \int (\vec v_\alpha \vec v_\alpha + \vec v_\alpha \vec w + \vec w \vec v_\alpha  + \vec w \vec w)  f_\alpha \dd \vec v \\
+= \pdv{}{\vec x} \cdot \left[ \vec v_\alpha \vec v_\alpha \int f_\alpha \dd \vec v + \vec v_\alpha \int \vec w f_\alpha \dd \vec v + \left( \int \vec w f_\alpha \dd \vec v \right) \vec v_\alpha + \int \vec w \vec w f_\alpha f_\alpha \dd \vec v \right]
+{{< /katex >}}
+
+The middle two terms are zero by the very definition of {{< katex >}} \vec v_\alpha {{< /katex >}}. The random velocity {{< katex >}} \vec w {{< /katex >}} is defined such that the integral of the random velocity over all phase space is zero.
+
+{{< katex display >}}
+= \pdv{}{\vec x} \cdot \left( \vec v_\alpha \vec v_\alpha n_\alpha \right) + \pdv{}{\vec x} \cdot \frac{ \vec P_\alpha}{m_\alpha} \\
+= \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha
+{{< /katex >}}
+
+Substituting in the pressure tensor,
+
+{{< katex display >}}
+= \div ( n_\alpha \vec v_\alpha \vec v_\alpha ) + \frac{1}{m_\alpha} \left( \grad P_\alpha  + \div \vec \Pi _\alpha \right)
+{{< /katex >}}
+
+Moving on to term 3:
+
+{{< katex display >}}
+\int \vec v \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[( \vec E + \vec v \cross \vec B) f_\alpha \right] \vec v \dd \vec v - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot \left[ \vec E + \vec v \cross \vec B \right] \vec v \dd \vec v \\
+= \int \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[ f_\alpha (\vec E + \vec v \cross \vec B) \vec v \right] \dd \vec v - \int \frac{q\alpha}{m\alpha} f ( \vec E + \vec v\alpha \cross \vec B) \cdot \pdv{\vec v}{\vec v} \dd \vec v \\
+= \frac{q\alpha}{m\alpha} \oint f\alpha ( \vec E + \vec v\alpha \cross \vec B) \vec v \cdot \dd \vec S_v - \frac{q\alpha}{m\alpha} \left[ \vec E \int f \dd \vec v\alpha - \vec B \cross \int \vec v f\alpha \dd \vec v \right] \\
+= - \frac{q\alpha}{m\alpha} n\alpha (\vec E + v\alpha \cross \vec B)
+{{< /katex >}}
+
+Finally, the 4th term gives
+
+{{< katex display >}}
+ \int \left. \vec v \pdv{f_\alpha}{t} \right|_{coll} = \sum_\beta \int \vec v C_{\alpha \beta} \dd \vec v \\
+= \int \vec v C_{\alpha \alpha} \dd \vec v + \sum_{\beta \neq \alpha} \int \vec v C_{\alpha \beta} \dd \vec v
+{{< /katex >}}
+
+Collisions of like particles {{< katex >}} \alpha {{< /katex >}} do not result in a net change of momentum of species {{< katex >}} \alpha {{< /katex >}}, so all we have left is the change in momentum due to collisions of unlike particles
+
+{{< katex display >}}
+= \sum_{\beta \neq \alpha} \left[ \vec v_{\alpha} \int C_{\alpha \beta} \dd \vec v + \int \vec w C_{\alpha \beta} \dd \vec v \right]
+{{< /katex >}}
+
+For the same reason as before, we assert that collisions between particles {{< katex >}} \alpha {{< /katex >}}  and {{< katex >}} \beta {{< /katex >}}  do not lead to the creation or destruction of any species, so the 0th moment {{< katex >}} \int C_{\alpha \beta} \dd \vec v = 0{{< /katex >}}. This leads to the conclusion that only random motion contributes to momentum transfer, not {{< katex >}} \vec v_\alpha {{< /katex >}}. Viscosity and friction are good examples of similar physical processes where bulk velocity does not transfer momentum, but random motion does.
+
+The momentum transfer from {{< katex >}} \alpha {{< /katex >}}  to {{< katex >}} \beta {{< /katex >}} must equal transfer from {{< katex >}} \beta {{< /katex >}} to {{< katex >}} \alpha {{< /katex >}}. Momentum is globally conserved.
+
+{{< katex display >}}
+\int \vec w C_{\alpha \beta} \dd \vec v = - \int \vec w C_{\beta \alpha} \dd \vec v
+{{< /katex >}}
+
+Now we can finally write out the full momentum equation
+
+{{< katex display >}}
+\pdv{}{t} (n_\alpha \vec v_\alpha ) + \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v
+{{< /katex >}}
+
+The collision term is often represented as the momentum transfer vector {{< katex >}} \vec R_{\alpha \beta} {{< /katex >}} to {{< katex >}} \alpha {{< /katex >}} from {{< katex >}} \beta {{< /katex >}}.
+
+{{< katex display >}}
+\vec R_{\alpha \beta} \equiv m_\alpha \int \vec w C_{\alpha \beta} \dd \vec v
+{{< /katex >}}
+{{< katex display >}}
+\pdv{}{t} (m_\alpha n_\alpha v_\alpha) + \div (m_\alpha n_\alpha \vec v_\alpha \vec v_\alpha) + \grad \vec P_{\alpha} + \div \vec \Pi_{\alpha} - q_\alpha n_\alpha (\vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta} R_{\alpha \beta}
+{{< /katex >}}
+
+Once again, we've written the above in a conservation law form. The terms that aren't strictly conservation terms are the source term {{< katex >}} q_\alpha n_\alpha (\vec E + \vec v _\alpha \cross \vec B) {{< /katex >}} and sink term {{< katex >}} \sum_{\beta} R_{\alpha \beta} {{< /katex >}}.
+
+Introducing the mass density
+
+{{< katex display >}}
+\rho_\alpha = m_\alpha n_\alpha \qquad \text{mass density}
+{{< /katex >}}
+
+{{< katex display >}}
+\pdv{}{t} (\rho_\alpha \vec v _\alpha) + \div (\rho_\alpha \vec v_\alpha v_\alpha + \ldots \\
+= \rho_\alpha \pdv{\vec v_\alpha}{t} + \vec v_\alpha \left( \pdv{\rho_\alpha}{t} + \div \rho_\alpha \vec v_\alpha \right) + \rho_\alpha \vec v_\alpha \cdot \grad \vec v_\alpha + \ldots
+{{< /katex >}}
+The term in parentheses is just the continuity equation, which is zero, and what's left is a more usual form of the momentum equation
+
+{{< katex display >}}
+\text{Momentum Equation:}\\
+\rho_\alpha \left(\pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) + \grad \vec P_\alpha + \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}
+{{< /katex >}}
+
+So by taking the 1st moment of the BE, we arrive at the momentum equation for {{< katex >}} \vec v_\alpha {{< /katex >}} and we have introduced {{< katex >}} \vec P_\alpha {{< /katex >}}.
+
+### Momentum transfer (collisions)
+
+Looking back at {{< katex >}} \vec R_{\alpha \beta} {{< /katex >}}, it's worth noting that the actual situation is complicated by the potential for multi-body collisions. While it is true that the probability of, e.g. three-body collisions is small, but there are many three-body collisional processes which are extremely important in plasma physics. Three-body recombination is the primary loss term in the ionization balance, for example. For now the simplest model is to only include binary collisions that result in small angel deflections. If we suppose we only have two species with distributions that look like the following:
+
+<p align="center"> <img alt="Figure 12.2" src="/r/img/12.2.png" /> </p>
+
+then we expect collisions between species to tend to drag the distributions towards each other. We should expect the collision-based momentum transfer to be proportional to the distance between the distributions, i.e. the total current density {{< katex >}} \vec j {{< /katex >}}.
+
+For an ion-electron plasma, the collision momentum transfer vector (to ions from electrons) is
+
+{{< katex display >}}
+\vec R_{ie} = - n_e e \eta \vec j
+{{< /katex >}}
+
+where
+
+{{< katex display >}}
+\eta = \frac{m_e \nu_{ei}}{n e^2} \qquad \text{resistivity}
+{{< /katex >}}
+
+{{< katex display >}}
+\vec j = n e (\vec v_i - \vec v_e) \qquad \text {current density for Z = 1}
+{{< /katex >}}
+
+### 2nd Moment of Boltzmann Equation
+
+Finally we're on to our last expansion term. In general, we would say that the second moment would be
+
+{{< katex display >}}
+\int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha
+{{< /katex >}}
+
+Usually we think of the second moment as giving us energy, a scalar, rather than a tensor equation that we have here. Since the pressure tensor is symmetric, the energy equation will contain 6 independent quantities. Together with the continuity and velocity equations, we have 10 quantities in total, so we call this expansion a 10-moment expansion.
+
+Instead of the full tensor equation, we're actually going to compute the _reduced 2nd moment_ by contracting {{< katex >}} \vec v \cdot \vec v {{< /katex >}}
+
+{{< katex display >}}
+\int \vec v \cdot \vec v \pdv{f_\alpha}{t} \dd v
+{{< /katex >}}
+
+So, let's get after it:
+
+{{< katex display >}}
+\int v^2 \pdv{f_\alpha}{t} \dd \vec v + \int v^2 \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v + \int v^2 \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \left. \int v^2 \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v
+{{< /katex >}}
+
+Simplifying... Term 1:
+
+{{< katex display >}}
+\int v^2 \pdv{f_\alpha}{t} \dd \vec v = \int \pdv{}{t} (v^2 f_\alpha) \dd \vec v - \int \pdv{v^2}{t} f_\alpha \dd \vec v \\
+= \pdv{}{t} \int (\vec v_\alpha + \vec w) \cdot (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \\
+= \pdv{}{t} \left[ \int v_\alpha ^2 f_\alpha \dd \vec v + 2 \int \vec v_\alpha \cdot \vec w f_\alpha \dd \vec v + \int w^2 f_\alpha \dd \vec v \right] \\
+= \pdv{}{t} \left[ v_\alpha ^2 \int f_\alpha \dd \vec v + 2 \vec v_\alpha \cdot \int \vec w f_\alpha \dd \vec v + 3 \frac{n_\alpha T_\alpha}{m_\alpha} \right] \\
+= \pdv{}{t} \left[ n_\alpha v_\alpha ^2 + 3 \frac{n_\alpha T_\alpha}{m_\alpha} \right]
+{{< /katex >}}
+
+Term 2:
+
+{{< katex display >}}
+\int v^2 \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v = \int \pdv{}{\vec x} \cdot (v^2 \vec v f_\alpha) \dd \vec v - \int f_\alpha \pdv{}{\vec x} \cdot (v^2 \vec v) \dd \vec v \\
+= \pdv{}{\vec x} \cdot \int ( \vec v_\alpha + \vec w) \cdot (\vec v_\alpha + \vec w) \vec w f_\alpha \dd \vec v \\
+= \pdv{}{\vec x} \cdot \int (v_\alpha ^2 + 2 \vec v_\alpha \cdot \vec w + w^2) \vec v f_\alpha \dd \vec v \\
+= \pdv{}{\vec x} \cdot \left[ \int v_\alpha ^2 (\vec v_\alpha + \vec w) f_\alpha \dd \vec v + \int 2 \vec v_\alpha \cdot \vec w (\vec v_\alpha + \vec w) f_\alpha \dd \vec v + \int w^2 (\vec v_\alpha + \vec w) f_\alpha \dd \vec v \right] \\
+= \pdv{}{\vec x} \cdot \left[ v_\alpha ^2 \vec v_\alpha \int f_\alpha \dd \vec v + v_\alpha ^2 \int \vec w f_\alpha \dd \vec v + 2 \vec v_\alpha \cdot \left(\int \vec w f_\alpha \dd \vec v \right) \vec v_\alpha \\
+\qquad + 2 \vec v_\alpha \cdot \int \vec w \vec w f_\alpha \dd \vec v + \vec v_\alpha \int w^2 f_\alpha \dd \vec v + \int \vec w w^2 f_\alpha \dd \vec v \right] \\
+= \pdv{}{\vec x} \cdot (n_\alpha v_\alpha ^2 \vec v_\alpha) + \pdv{}{\vec x} \cdot 2 \vec v_\alpha \cdot \frac{\vec P_\alpha}{m_\alpha} + \pdv{}{\vec x} \cdot \left(\frac{3 n_\alpha T_\alpha \vec v_\alpha}{m_\alpha} \right) + \pdv{}{\vec x} \cdot \left( \frac{2 \vec h_\alpha}{m_\alpha} \right)
+{{< /katex >}}
+
+In keeping with the progression so far, the very last term would be something like a full 3rd moment {{< katex >}} \int \vec w \vec w \vec w f_\alpha \dd \vec v {{< /katex >}}. Instead, we have a "contracted 3rd moment" of the distribution {{< katex >}} \vec h_\alpha {{< /katex >}} which is actually the _heat flux_, or the random energy flux of random energy.
+
+Term 3:
+
+{{< katex display >}}
+\int v^2 \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v \\
+ = \oint f_\alpha \frac{q_\alpha}{m_\alpha} v^2 ( \vec E + \vec v \cross \vec B) \cdot \dd \vec S_v - \int f_\alpha \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[ v^2 ( \vec E + \vec v \cross \vec B) \right] \dd \vec v \\
+= - \frac{q_\alpha}{m_\alpha} \int f_\alpha \left[ v^2 \left(\pdv{}{\vec v} \cdot \vec E \right) + \left(\vec E \cdot \pdv{}{\vec v} v^2 \right) \right] \dd \vec v  \\
+\qquad - \frac{q_\alpha}{m_\alpha}\int f_\alpha \left[v^2 \pdv{}{\vec v} \cdot (\vec v \cross \vec B) + \vec v \cross \vec B \cdot \pdv{}{\vec v} v^2 \right] \dd \vec v \\
+= - \frac{q_\alpha}{m_\alpha} \int f_\alpha (2 \vec E \cdot \vec v) \dd \vec v \\
+= - \frac{2 q_\alpha}{m_\alpha} n_\alpha \vec v_\alpha \cdot \vec E
+{{< /katex >}}
+
+We've ended up with the work done on a fluid by the electric field. As expected, there is no work term associated with the bulk magnetic field.
+
+Term 4:
+
+{{< katex display >}}
+\left. \int v^2 \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v = \int v^2 C_{\alpha \alpha} \dd \vec v + \sum _{\beta \neq \alpha} \int v^2 C_{\alpha \beta} \dd \vec v 
+{{< /katex >}}
+
+We can make an energy conservation argument to get rid of the first term. The first term goes to 0 since collisions of like particles results in no net energy exchange.
+
+{{< katex display >}}
+= \sum_{\beta \neq \alpha} \int ( \vec v_\alpha + \vec w) \cdot(\vec v_\alpha + \vec w) C_{\alpha \beta} \dd \vec v \\
+= \sum_{\beta \neq \alpha}\left( v_\alpha ^2 \int C_{\alpha \beta} \dd \vec v + 2 \vec v_\alpha \cdot \int \vec w C_{\alpha \beta} \dd \vec v + \int w^2 C_{\alpha \beta} \dd \vec v \right)
+{{< /katex >}}
+
+Since collisions (we assume) don't create or destroy particles, the first term {{< katex >}} v_\alpha ^2 \int C_{\alpha \beta} \dd \vec v {{< /katex >}} goes away. The second term {{< katex >}} \vec v_\alpha \cdot \int \vec w C_{\alpha \beta} \dd \vec v {{< /katex >}} we interpret as a frictional force due to the relative motion between {{< katex >}} \alpha {{< /katex >}}  and {{< katex >}} \beta {{< /katex >}} 
+
+{{< katex display >}}
+= \sum_{\beta \neq \alpha} 2 \frac{\vec v_\alpha \cdot \vec R_{\alpha \beta}}{m_\alpha} + 2 \frac{Q_{\alpha \beta}}{m_\alpha}
+{{< /katex >}}
+
+where {{< katex >}} Q_{\alpha \beta} {{< /katex >}} is the heat exchange term (heat generation term in Braginskii). It describes the heat exchange by random collisions of unlike particles, analogous to viscous heating.
+
+{{< katex display >}}
+Q_{\alpha \beta} = \frac{1}{2} \int m_\alpha w^2 C_{\alpha \beta} \dd \vec v
+{{< /katex >}}
+
+If we were to draw distribution functions for species {{< katex >}} \alpha {{< /katex >}}  and {{< katex >}} \beta {{< /katex >}} (as shown), even though they have the same centroid they will interact by viscous heating. Species {{< katex >}} \beta {{< /katex >}} will heat up and {{< katex >}} \alpha {{< /katex >}} will cool
+
+<p align="center"> <img alt="Figure 12.3" src="/r/img/12.3.png" /> </p>
+
+If we multiply by {{< katex >}} \frac{1}{2} m_\alpha {{< /katex >}} then the energy equation becomes
+
+{{< katex display >}}
+\text{Energy Equation: } \\ \qquad \pdv{}{t} \left( \frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2 \right) + \\
+ \div \left(\frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2 \right) \vec v_\alpha + \div (\vec v_\alpha \cdot \vec P_\alpha) + \div \vec h_\alpha - q_\alpha n_\alpha \vec v_\alpha \cdot \vec E \\ =  \sum_{\beta \neq \alpha} (v_\alpha \cdot \vec R_{\alpha \beta} + Q_{\alpha \beta})
+{{< /katex >}}
+
+Notice that the total energy appears
+
+{{< katex display >}}
+E_\alpha = \frac{3}{2} n_\alpha T_\alpha + \frac{1}{2} \rho_\alpha v_\alpha ^2
+{{< /katex >}}
+
+The factor of {{< katex >}} \frac{3}{2} {{< /katex >}} as usual comes from {{< katex >}} \gamma {{< /katex >}} 
+
+{{< katex display >}}
+\frac{3}{2} = \frac{1}{\gamma - 1} = \frac{1}{5/3 - 1}
+{{< /katex >}}
+
+where
+
+{{< katex display >}}
+\gamma = \frac{DOF + 2}{DOF} = \frac{5}{3}
+{{< /katex >}}
+
+We can remove the kinetic portion of the energy equation by subtracting the product of the momentum equation with {{< katex >}} \vec v_\alpha {{< /katex >}}.
+
+{{< katex display >}}
+\rho_\alpha \vec v_\alpha \cdot \pdv{v_\alpha}{t} = \frac{1}{2} \pdv{}{t} (\rho_\alpha v_\alpha ^2)
+{{< /katex >}}
+
+{{< katex display >}}
+\pdv{}{t} \left( \frac{3}{2} n_\alpha T_\alpha \right) + \div \left(\frac{3}{2} n_\alpha T_\alpha \vec v_\alpha \right) + \div \left( \vec v_\alpha \cdot \vec P_\alpha \right) - \vec v_\alpha \cdot (\div \vec P_\alpha) + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}
+{{< /katex >}}
+{{< katex display >}}
+\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \div T_\alpha \right) + \frac{3}{2} T_\alpha \left[ \pdv{n_\alpha}{t} + \div (n_\alpha \vec v_\alpha) \right] + \vec P_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}
+{{< /katex >}}
+{{< katex display >}}
+\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \div T_\alpha \right) + \vec P_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}
+{{< /katex >}}
+
+Where {{< katex >}} \vec P_\alpha \cdot \cdot \grad \vec v_\alpha {{< /katex >}} is a tensor contraction in two indices. It is a generalization of the dot product, i.e.
+{{< katex display >}}
+\vec P_\alpha \cdot \cdot \grad \vec v_\alpha = \delta_{ik} \delta_{jl} P_{ij} \partial_k v_l \\
+= P_{xx} \partial_x v_x + P_{xy} \partial_y v_x + \ldots
+{{< /katex >}}
+
+{{< katex display >}}
+\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \grad T_\alpha \right) + P_\alpha \div \vec v_\alpha + \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}
+{{< /katex >}}
+
+Now the continuity, momentum, and energy equations describe the evolution of each fluid {{< katex >}} \alpha {{< /katex >}}. The only velocity information we have retained in taking moments is the centroid {{< katex >}} \vec v_\alpha {{< /katex >}} and the width {{< katex >}} T_\alpha {{< /katex >}}. This is the only information we will include in the 5-moment model (5N-moment plasma fluid model). The higher moments (describing skewness, kurtosis) are not evolved with the 5-moment model. There are fluid models that do evolve those moments, but we won't touch on those here.
+
+### Closure Problem
+
+We still need to "close" the fluid model (i.e. solving the **Closure Problem**). As we have been finding out, each additional moment of the Boltzmann Equation introduces the next-higher moment. In calculating the 0th moment, we introduce the 1st moment, etc. We need to address the closure problem by relating higher moment variables to lower moment variables. Applying a closure scheme (relating variables that are not evolved directly) is usually equivalent to making a statement on heat flow.
+
+In our model, we are not evolving the heat flux {{< katex >}} \vec h_\alpha {{< /katex >}}, so we need to relate it to one of the other variables that we do evolve directly. We usually do that by writing down a conductivity relation
+
+{{< katex display >}}
+\vec h_\alpha = - \kappa \grad T_\alpha
+{{< /katex >}}
+
+for some conductivity {{< katex >}} \kappa {{< /katex >}}. 
+
+We're also only evolving the scalar pressure, so we need to relate the stress tensor {{< katex >}} \vec \Pi_\alpha {{< /katex >}} to the other variables. We can do that with a viscosity relation
+
+{{< katex display >}}
+\vec \Pi_\alpha = \nu \grad \vec v_\alpha
+{{< /katex >}}
+
+for some viscosity {{< katex >}} \nu {{< /katex >}}. We call the introduced quantities {{< katex >}} \kappa {{< /katex >}}  and {{< katex >}} \nu {{< /katex >}} the **transport coefficients**. They are usually derived by performing an expansion of the equations we've already discussed. They are what lead to "nondimensional" numbers that characterize the flow in our system.
+
+We can also achieve closure by constraining portions of the system. In an isothermal system
+
+{{< katex display >}}
+\text{isothermal: } \quad T = \text{const.}, p \propto n
+{{< /katex >}}
+{{< katex display >}}
+\text{adiabatic: } \quad  p \propto n^\gamma
+{{< /katex >}}
+{{< katex display >}}
+\text{force-free, cold: } \quad p = 0
+{{< /katex >}}
+
+The expressions that lead us to a closure scheme are the **equations of state**. They are not time-dependent, they simply relate fluid parameters to other properties of the system.
+
+Maxwell's equations to couple the electromagnetic terms to the fluid variables.
+{{< katex display >}}
+\pdv{\vec B}{t} = - \curl \vec E
+{{< /katex >}}
+{{< katex display >}}
+\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_{alpha} q_\alpha n_\alpha \vec v_\alpha 
+{{< /katex >}}
+{{< katex display >}}
+\epsilon_0 \div \vec E = \sum_\alpha q_\alpha n_\alpha 
+{{< /katex >}}
+{{< katex display >}}
+\div \vec B = 0
+{{< /katex >}}
+
+The fluid variables provide the source terms for Ampere's law and Gauss' law, and we've already seen how the electromagnetic fields appear in the momentum equations as source terms for the fluid forces.
+
+### Stopping after the 2nd moment (5N-Moment Plasma Fluid Model)
+
+What assumptions have we made by stopping here? It's important to know what they are and to recognize what they mean for the fluid model.
+
+ - Each species {{< katex >}} \alpha {{< /katex >}} is well-represented by a Maxwellian with a small perturbation. The pressure tensor is not strictly diagonal and the heat flux is not zero, so the small perturbations are what lead to the transport coefficients. This is due to the process by which we obtain the transport coefficients, called the Chapman-Enskog expansion.
+ - Kinetic effects (stream instabilities, counter-flow instabilities, etc.) are not captured.
+
+The variables of the 5-moment model that we _do_ evolve are 
+{{< katex display >}}
+n, v_x, v_y, v_z, T
+{{< /katex >}}
+
+We can take additional moments. The 10N-moment equation evolves, in addition to the quantities in the 5N-moment model, we evolve all of the independent terms of the pressure tensor
+{{< katex display >}}
+n, v_x, v_y, v_z, P_{xx}, P_{xy}, P_{xz} , P_{yy} , P_{yz} , P_{zz} 
+{{< /katex >}}
+
+The 13-N moment model has everything from the 10N-moment model plus the heat flux tensor {{< katex >}} h_x, h_y, h_z {{< /katex >}} 
+
diff --git a/content/notes/UWAA558/04-two-fluid-plasma-model.md b/content/notes/UWAA558/04-two-fluid-plasma-model.md
new file mode 100644
index 00000000..bfb8b368
--- /dev/null
+++ b/content/notes/UWAA558/04-two-fluid-plasma-model.md
@@ -0,0 +1,195 @@
+---
+title: Two-Fluid Plasma Model
+bookToc: false
+weight: 40
+---
+
+
+# Two-Fluid Plasma Model (ions-electrons)
+
+Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let's start with a cold plasma approximation ({{< katex >}} p = 0 {{< /katex >}}) and neglect collisions. The momentum equation reduces to
+
+{{< katex display >}}
+m_\alpha \left( \pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) - q_\alpha (\vec E + \vec v_\alpha \cross \vec B) = 0
+{{< /katex >}}
+
+From here on out we can avoid some clutter (and wrist strain) by dropping the {{< katex >}} \alpha {{< /katex >}} subscripts and acknowledging that we have sets of equations for ions and electrons. Apply a perturbation to an equilibrium {{< katex >}} g = g_0 + g_1 {{< /katex >}} 
+
+{{< katex display >}}
+m \left(\pdv{\vec v_0}{t} + \vec v_0 \cdot \grad \vec v_0 + \pdv{\vec v_0}{t} + \vec v_1 \cdot \grad \vec v_0 + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_1 \right) \\
+\qquad - q (E_0 + E_1) - q (\vec v_0 \cross \vec B_0 + \vec v_1 \cross \vec B_0 + \vec v_0 \cross \vec B_1 + \vec v_1 \cross \vec B_1) = 0
+{{< /katex >}}
+
+We can drop some terms because equilibrium has to satisfy the original equation. We can balance all of the subscript-0 terms and sum them to get zero.
+
+{{< katex display >}}
+m \left( \pdv{\vec v_1}{t} + \vec v_1 \cdot \grad \vec v_0 + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_1 \right) + q (- E_1 - \vec v_1 \cross \vec B_0 - \vec v_0 \cross \vec B_1 - \vec v_1 \cross \vec B_1)
+{{< /katex >}}
+
+Let's now make the assumption that the perturbation is small, that is {{< katex >}} g_1 \ll g_0 {{< /katex >}}. That means that nonlinear products of perturbation terms are negligible (linearization process). 
+
+{{< katex display >}}
+m \left( \pdv{\vec v_1}{t} + \vec v_0 \cdot \grad \vec v_1 + \vec v_1 \cdot \grad \vec v_0 \right) - q \vec E_1 - q (\vec v_1 \cross \vec B_0 + \vec v_0 \cross \vec B_1) = 0
+{{< /katex >}}
+
+Now, assume that the equilibrium is a **static equilibrium**, that is {{< katex >}} \vec v_0 = 0 {{< /katex >}}. If we decompose into components that are parallel and perpendicular to the equilibrium magnetic field {{< katex >}} \vec B_0 {{< /katex >}}, then 
+
+{{< katex display >}}
+\pdv{v_{1, \parallel}}{t} - \frac{q}{m} E_{1, \parallel} = 0
+{{< /katex >}}
+{{< katex display >}}
+\pdv{\vec v_{1, \perp}}{t} - \frac{q}{m} \left(\vec E_{1, \perp} + B_0 \vec v_{1, \perp} \cross \vu z \right) = 0
+{{< /katex >}}
+
+The parallel component {{< katex >}} E_{1, \parallel} {{< /katex >}} will lead us to the ordinary wave (**O-wave**). Consideration of the more general case with perpendicular components will lead to the **X-wave**. 
+
+The plasma velocity is related to the fields through the current density (Maxwell equations). Faraday's law gives
+
+{{< katex display >}}
+\pdv{\vec B}{t} = - \curl \vec E
+{{< /katex >}}
+{{< katex display >}}
+\rightarrow \curl \pdv{\vec B}{t} = - \curl \curl \vec E = - \grad (\div \vec E) + \nabla ^2 \vec E 
+{{< /katex >}}
+
+Ampere's law gives
+
+{{< katex display >}}
+\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_{\alpha} q_\alpha n_\alpha \vec v_\alpha
+{{< /katex >}}
+{{< katex display >}}
+\epsilon_0 \pdv{^2 \vec E }{t ^2} = \frac{1}{\mu_0} \curl \pdv{\vec B}{t} \\
+= \frac{1}{\mu_0} \left[ \nabla ^2 \vec E - \grad (\div \vec E) \right] - \sum_\alpha q_\alpha \pdv{}{t} (n_\alpha \vec v_\alpha) 
+{{< /katex >}}
+
+Since this is a linear system, assume that the perturbed quantities have a wave-like structure. That is, the perturbed quantities {{< katex >}} g_1 {{< /katex >}} are proportional to {{< katex >}} e^{i(\omega t + \vec k \cdot \vec r)} {{< /katex >}}. This lets us transform the spatial and temporal derivatives into factors of {{< katex >}} \omega {{< /katex >}} and {{< katex >}} \vec k {{< /katex >}} 
+
+{{< katex display >}}
+- \epsilon_0 \omega ^2 \vec E_1 = - \frac{1}{\mu_0} \left[k^2 \vec E_1 - \vec k (\vec k \cdot \vec E_1) \right] + i \omega e n_0 \vec v_1
+{{< /katex >}}
+
+Let's now consider only high frequency oscillations, assuming that only the electrons respond and the ions remain stationary. There's nothing particularly complicated about including the ion response, this just lets us drop the {{< katex >}} \alpha {{< /katex >}} subscripts and focus on a single set of equations.
+
+{{< katex display >}}
+- i \omega \frac{e n_0}{\epsilon_0} \vec v_1 = (\omega ^2 - c^2 k^2) \vec E_1 + c^2 \vec k (\vec k \cdot \vec E_1)
+{{< /katex >}}
+
+Now let's apply the perturbed form to the linearized momentum equation
+
+{{< katex display >}}
+\pdv{v_{1, \parallel}}{t} - \frac{q}{m} E_{1, \parallel} = 0 \\
+\rightarrow i \omega v_{1, \parallel} = - \frac{e}{m} E_{1, \parallel} \\
+\rightarrow i \omega \frac{e n_0}{\epsilon_0} v_{1, \parallel} = \frac{e^2 n_0}{\epsilon_0 m} E_{1, \parallel}
+{{< /katex >}}
+
+Combine the momentum equation and the Maxwell equations to eliminate {{< katex >}} \vec E_1 {{< /katex >}} and {{< katex >}} \vec v_1 {{< /katex >}}
+
+{{< katex display >}}
+\frac{e^2 n_0}{\epsilon_0 m} E_{1, \parallel} = ( \omega ^2 - c^2 k^2 ) E_{1, \parallel} + c^2 k_{\parallel} \vec (k \cdot \vec E_1)
+{{< /katex >}}
+
+Consider different possibilities for the {{< katex >}} \vec k {{< /katex >}} vector. If it is along the magnetic field {{< katex >}} \vec k = k_{\parallel} \vu{e}_\parallel {{< /katex >}} (longitudinal wave) then
+
+{{< katex display >}}
+\frac{e^2 n_0}{\epsilon_0 m} = \omega ^2 - c^2 k^2 + c^2 k^2 = \omega_{pe}^2
+{{< /katex >}}
+
+For {{< katex >}} \vec k = k_{\perp} \vu e_\perp {{< /katex >}} (transverse wave) then we get the dispersion relation for the O-wave
+
+{{< katex display >}}
+\text{dispersion relation for O-waves:} \qquad \omega^2 - c^2 k^2 = \omega_p ^2
+{{< /katex >}}
+
+The electric field is in the same direction as the magnetic field {{< katex >}} (\vec E_1 = \vec E_{1, \parallel}) {{< /katex >}}, which means the O-wave is linearly polarized. At large {{< katex >}} k {{< /katex >}} we just have regular light waves, but as we turn the frequency downwards we see a cut-off at the plasma frequency:
+
+<p align="center"> <img alt="Figure 12.4" src="/r/img/12.4.png" /> </p>
+
+It turns out that the dispersion relation for the X-wave has the same cut-off, but also has another branch with a resonance
+
+<p align="center"> <img alt="Figure 12.5" src="/r/img/12.5.png" /> </p>
+
+
+
+The two-fluid plasma model is highly reduced from the full kinetic model, but it is still too complete to be useful when studying gross plasma behavior. Further reductions of the model are possible by making asymptotic assumptions:
+
+## Low-frequency Asymptotic Assumption
+
+ - Eliminate high frequency, short wavelength phenomena by using pre-Maxwell field equations. Formally, this is {{< katex >}} \epsilon_0 \rightarrow 0 {{< /katex >}}. 
+
+The direct consequences of the low-frequency approximation are
+
+{{< katex display >}}
+c^2 = \frac{1}{\epsilon_0 \mu_0} \qquad c \rightarrow \infty
+{{< /katex >}}
+{{< katex display >}}
+\omega_p ^2 = \frac{n e^2}{\epsilon_0 m} \qquad \omega_p \rightarrow \infty
+{{< /katex >}}
+{{< katex display >}}
+\lambda_D = \frac{v_T}{\omega_p} \rightarrow 0
+{{< /katex >}}
+
+This means that all phenomena will have {{< katex >}} \omega \ll \omega_p {{< /katex >}}, limiting the frequencies we can resolve to the ion plasma frequency. The characteristic speeds will be limited by the speed of light
+
+{{< katex display >}}
+\frac{\omega}{k} \ll c
+{{< /katex >}}
+
+and all characteristic lengths will be much greater than the Debye length
+
+{{< katex display >}}
+x_0 \gg \lambda_D
+{{< /katex >}}
+
+Looking at Gauss' law,
+{{< katex display >}}
+\epsilon_0 \div \vec E = \sum_\alpha q_\alpha n_\alpha \rightarrow \sum_\alpha q_\alpha n_\alpha = 0
+{{< /katex >}}
+
+so we now have charge neutrality everywhere in the domain. For H plasma, locally we have {{< katex >}} n_e = n_i {{< /katex >}} everywhere.
+
+Looking at Ampere's law,
+
+{{< katex display >}}
+\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_\alpha q_\alpha n_\alpha \vec v_\alpha = 0 \\
+\rightarrow \vec j = \frac{1}{\mu_0} \curl \vec B
+{{< /katex >}}
+
+Things we do **not** get from this approximation are {{< katex >}} \vec E = 0 {{< /katex >}} or {{< katex >}} \pdv{\vec E}{t} = 0 {{< /katex >}}. It does mean that plasma dynamics occur on a sufficiently large spatial scale that charge separation is small, and they occur on a sufficiently long temporal scale that electrons respond quickly.
+
+## Tiny electron asymptotic assumption
+
+2nd approximation: neglect electron inertia in the momentum equation. Formally, we let the electron mass {{< katex >}} m_e \rightarrow 0 {{< /katex >}}
+
+{{< katex display >}}
+\omega_{pe}^2 = \frac{n e^2}{\epsilon_0 m_e} \rightarrow \infty
+{{< /katex >}}
+{{< katex display >}}
+\omega_{c, e} = \frac{e B}{m_e} \rightarrow \infty
+{{< /katex >}}
+{{< katex display >}}
+v_{T, e} \rightarrow \infty
+{{< /katex >}}
+The Larmor radius goes to zero
+{{< katex display >}}
+r_{l, e} = \frac{v_{T, e}}{\omega_{c, e}} \rightarrow 0
+{{< /katex >}}
+Importantly, as the gyroradius {{< katex >}} r_{l, e} {{< /katex >}} goes to 0 (because the thermal velocity goes as {{< katex >}} \sqrt{m_e} {{< /katex >}} and the cyclotron frequency goes as {{< katex >}} m_e {{< /katex >}}), this means that the electrons are tied to the magnetic field.
+The skin depth is also small.
+{{< katex display >}}
+\delta_e = \frac{c}{\omega_{p, e}} \rightarrow 0
+{{< /katex >}}
+
+So all phenomena that we capture must have {{< katex >}} \omega \ll \omega_{p, e} {{< /katex >}}, {{< katex >}} \omega \ll \omega_{c, e} {{< /katex >}}, and {{< katex >}} x_0 \gg r_{L, e} {{< /katex >}}, {{< katex >}} x_0 \gg \delta_e {{< /katex >}} .
+
+The electron momentum equation becomes
+
+{{< katex display >}}
+\grad P_e + \div \vec \Pi _e + e n_e (\vec E + \vec v \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}
+{{< /katex >}}
+
+The momentum equation is now a state equation, not an evolution equation. It now simply relates the dynamical variables to each other at any point in time.
+
+Now, note that along magnetic field lines electrons can travel long distances at very fast (finite) speeds which can produce low frequency, long wavelength phenomena. Neglecting electron inertia implies that electrons respond instantaneously, meaning we cannot capture these modes. An example of such a phenomena is drift waves.
+
+The characteristic speeds {{< katex >}} c {{< /katex >}}  and {{< katex >}} v_{T, e} {{< /katex >}} have disappeared from the model. Remaining is {{< katex >}} v_{T, i} {{< /katex >}}. This means that the ion dynamics dictate the plasma evolution.
+
diff --git a/content/notes/UWAA558/05-mhd-model.md b/content/notes/UWAA558/05-mhd-model.md
new file mode 100644
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@@ -0,0 +1,666 @@
+---
+title: Magnetohydrodynamic (MHD) Model
+bookToc: false
+weight: 50
+---
+
+
+# Magnetohydrodynamic (MHD) Model
+
+Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD).
+
+First, define the MHD variables:
+
+{{< katex display >}}
+\text{mass density:} \qquad \rho = n_i m_i + n_i m_e
+{{< /katex >}}
+
+{{< katex display >}}
+\text{fluid velocity:} \qquad \vec v = \frac{n_i m_i \vec v_i + n_e m_e \vec v_e}{n_i m_i + n_e m_e}
+{{< /katex >}}
+
+{{< katex display >}}
+\text{current density:} \qquad \vec j = q_i n_i \vec v_i - e n_e \vec v_e  \rightarrow \vec v_e = \frac{q_i n_i \vec v_i}{e n_e} - \vec j / e n_e
+{{< /katex >}}
+
+{{< katex display >}}
+\text{total pressure:} \qquad p = p_i + p_e = n_e T_e + n_i T_i
+{{< /katex >}}
+
+{{< katex display >}}
+\text{total temperature:} \qquad T = \frac{n_i T_i + n_e T_e}{(n_i + n_e)/2}
+{{< /katex >}}
+
+Now let's begin applying asymptotic approximations. For the mass density, applying the first approx (charge neutrality) we have
+
+{{< katex display >}}
+\rho \approx n (m_i + m_e)
+{{< /katex >}}
+
+Using approx 2 (vanishing electron mass) 
+
+{{< katex display >}}
+\rho \approx n m_i
+{{< /katex >}}
+where
+{{< katex display >}}
+n = n_i \qquad n_e = Z n
+{{< /katex >}}
+
+The center-of-mass velocity (charge neutrality) gives
+
+{{< katex display >}}
+\vec v \approx \frac{m_i \vec v_i + m_e Z \vec v_e}{m_i + Z m_e}
+{{< /katex >}}
+
+with small electron mass approximation:
+
+{{< katex display >}}
+\vec v \approx \vec v_i
+{{< /katex >}}
+
+The current density is (charge neutrality approx)
+
+{{< katex display >}}
+\vec j \approx Z e n (\vec v_i - \vec v_e) \qquad \vec v_e = \vec v_i - \vec j / Z e n
+{{< /katex >}}
+
+The pressure and total temperature are (with charge neutrality)
+
+{{< katex display >}}
+P \approx n (T_i + Z T_e)
+{{< /katex >}}
+
+{{< katex display >}}
+T \approx \frac{T_i + Z T_e}{(1 + Z)/2}
+{{< /katex >}}
+
+### MHD Momentum Equation
+
+Now we combine the two-fluid equations with these asymptotic approximations to obtain the governing equations for the MHD variables. Multiplying the ion continuity equation by the ion mass gives
+
+{{< katex display >}}
+\pdv{}{t} (m_i n_i) + \div (m_i n_i \vec v_i) = 0 \\
+\rightarrow \pdv{\rho}{t} + \div (\rho \vec v) = 0 \quad \text{(MHD continuity eq.)}
+{{< /katex >}}
+
+If we multiply the two-fluid continuity equations by the charge and sum them, we get
+
+{{< katex display >}}
+\pdv{}{t}(q_i n_i - e n_e) + \div (q_i n_i \vec v_i - e n_e \vec v_e) = 0 \\
+\rightarrow \div \vec j = 0 \quad \text{(no accumulation of charge)}
+{{< /katex >}}
+
+If we add the electron and ion momentum equations and apply the small electron mass approximation, and recognizing that {{< katex >}} \vec R_{ei} = - \vec R_{ie} = n e \eta \vec j {{< /katex >}} 
+
+{{< katex display >}}
+\rho_i \left(\pdv{\vec i}{t} + \vec v_i \cdot \grad \vec v_i \right) + \grad (P_i + P_e) + \div (\vec \Pi _i + \vec \Pi_e) \\
+\qquad  - (q_i n_i - e n_e) \vec E - (q_i n_i \vec v_i - e n_e \vec v_e) \cross \vec B = 0
+{{< /katex >}}
+{{< katex display >}}
+\text{MHD Momentum Equation} \\
+\rightarrow \rho \left(\pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi _i + \vec \Pi _e)
+{{< /katex >}}
+
+!!! info "MHD Momentum Equation"
+
+    {{< katex display >}}
+    \rho \left(\pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi _i + \vec \Pi _e)
+    {{< /katex >}}
+
+If we take the electron momentum equation, apply the small electron mass asymptotic approximation and introduce the current density, then we have
+
+### MHD Ohm's Law
+
+{{< katex display >}}
+e n_e (\vec E + \vec v_i \cross \vec B - \vec j \cross \vec B / Z e n) = - \grad P_e - \div \vec \Pi_e + \vec R_{ei}
+{{< /katex >}}
+
+Substituting {{< katex >}} \vec R_{ei} = e n_e \eta \vec j {{< /katex >}} 
+{{< katex display >}}
+\text{MHD Generalized Ohms Law}\\
+\vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left(\vec j \cross \vec B - \grad p_e - \div \vec \Pi_e \right)
+{{< /katex >}}
+
+!!! info "MHD Generalized Ohms Law"
+    
+    {{< katex display >}}
+    \vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left(\vec j \cross \vec B - \grad p_e - \div \vec \Pi_e \right)
+    {{< /katex >}}
+
+
+### MHD Energy Equation
+
+The energy equation is found by adding the ion and electron energies, but first we need to manipulate them into a common form. The ion energy equation is 
+
+{{< katex display >}}
+\frac{1}{\gamma - 1} \left[ \pdv{}{t} (n_i T_i) + \div (n_i T_i \vec v_i) \right] + n_i T_i \div \vec v_i = RHS_i
+{{< /katex >}}
+
+where {{< katex >}} RHS_i \equiv - \vec \Pi_i \cdot \cdot \grad \vec v_i - \div \vec h_i + Q_{ie} {{< /katex >}} 
+
+{{< katex display >}}
+\frac{1}{\gamma - 1} \left( \pdv{p_i}{t} + \vec v_i \cdot \grad p_i \right) + \frac{\gamma}{\gamma - 1} p_i (\div \vec v_i) = RHS_i
+{{< /katex >}}
+
+Multiply by {{< katex >}} \frac{\gamma - 1}{n_i ^\gamma} {{< /katex >}} and define the total derivative with respect to the ion velocity as
+
+{{< katex display >}}
+\dv{}{t} \equiv \pdv{}{t} + \vec v_i \cdot \grad
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \dv{}{t} \left( \frac{p_i}{n_i ^\gamma} \right) - p_i \dv{}{t} \left( \frac{1}{n_i ^\gamma} \right) + \frac{\gamma p_i}{n_i ^\gamma} \div \vec v_i = \frac{\gamma - 1}{n_i ^\gamma} RHS_i
+{{< /katex >}}
+
+Simplify further by recognizing that carrying out the total derivative gives 
+
+{{< katex display >}}
+\dv{}{t} \left( \frac{p_i}{n_i ^\gamma} \right) - \frac{\gamma p_i}{n_i ^{\gamma + 1}} \dv{}{t} n_i + \frac{\gamma p_i}{n_i ^\gamma} \div \vec v_i = \frac{\gamma - 1}{n_i ^\gamma} RHS_i
+{{< /katex >}}
+
+The continuity equation says
+{{< katex display >}}
+\dv{}{t} n_i = - n_i \div \vec v_i
+{{< /katex >}}
+Multiply by {{< katex >}} m_i ^{- \gamma} {{< /katex >}} and use {{< katex >}} \rho = n m_i {{< /katex >}} and the resulting ion energy equation is
+
+{{< katex display >}}
+\dv{}{t} \left( \frac{p_i}{\rho ^\gamma} \right) = \frac{\gamma - 1}{\rho ^\gamma} RHS_i
+{{< /katex >}}
+
+We want to get the electron energy equation in the same form. The steps are very similar:
+
+{{< katex display >}}
+\dv{}{t} \left( \frac{p_e}{n_e ^\gamma} \right) + \vec v_e \cdot \grad \left( \frac{p_e}{n_e ^\gamma} \right) = \frac{\gamma - 1}{n_e ^\gamma} RHS_e
+{{< /katex >}}
+
+Since we've defined our center-of-mass reference frame to be that of the ions, we can not use the same total derivative
+
+{{< katex display >}}
+\pdv{}{t} + \vec v_e \cdot \grad \neq \dv{}{t}
+{{< /katex >}}
+
+We now apply the charge neutrality approximation
+
+{{< katex display >}}
+n_e = Z n \qquad \vec v_e = \vec v_i - \frac{1}{Z e n} \vec j
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \dv{}{t} \left( \frac{p_e}{Z^\gamma n^\gamma} \right) + \vec v_i \cdot \grad \left( \frac{p_e}{Z^\gamma n^\gamma} \right) = \frac{1}{Z e n} \vec j \cdot \grad \frac{p_e}{Z^\gamma n^\gamma} + \frac{\gamma - 1}{Z^\gamma n^\gamma} RHS_e
+{{< /katex >}}
+
+Multiply by {{< katex >}} (Z / m_i)^\gamma {{< /katex >}} and the result is
+
+{{< katex display >}}
+\dv{}{t} \frac{p_e}{\rho ^\gamma} = \frac{1}{Z e n} \vec j \cdot \grad \frac{p_e}{\rho^\gamma} + \frac{\gamma - 1}{\rho^\gamma} RHS_e
+{{< /katex >}}
+
+Finally we can add the ion energy equation to the electron energy equation to get
+
+{{< katex display >}}
+\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \\
+\frac{\gamma - 1}{\rho^\gamma} \left[ Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e \right] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}
+{{< /katex >}}
+
+!!! info "MHD Energy Equation"
+
+    {{< katex display >}}
+    \dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \frac{\gamma - 1}{\rho^\gamma} \left[ Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e \right] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}
+    {{< /katex >}}
+
+Obviously, we've retained a number of terms that are specific to the behavior of the electrons. It is possible to incorporate the electron behavior by using a single-fluid MHD model with two temperatures {{< katex >}} T_i \neq T_e {{< /katex >}}. One can imagine a hierarchy of models, in which the most simplified is the single-fluid MHD model in which you evolve {{< katex >}} \rho {{< /katex >}}, {{< katex >}} \vec v {{< /katex >}}, and {{< katex >}} T {{< /katex >}}. Moving up a level, you have a MHD model with two temperatures in which you evolve {{< katex >}} \rho {{< /katex >}}, {{< katex >}} \vec v {{< /katex >}}, {{< katex >}} T_i {{< /katex >}}, and {{< katex >}} T_e {{< /katex >}}. Upwards from there you move back into the realm of multi-fluid models.
+
+Now to relate the fields back to source terms. The low-frequency Maxwell's equations are
+
+{{< katex display >}}
+\pdv{\vec B}{t} = - \curl \vec E
+{{< /katex >}}
+{{< katex display >}}
+\vec j = \frac{1}{\mu_0} \curl \vec B
+{{< /katex >}}
+{{< katex display >}}
+\div \vec B = \div \vec E = 0
+{{< /katex >}}
+
+Like in the multi-fluid plasma model, we still need to close the system by expressing some of our variables using equations of state ({{< katex >}} \vec h {{< /katex >}}, {{< katex >}} \vec \Pi {{< /katex >}}).
+
+To simplify further, we can make some assumptions about heat flow
+
+{{< katex display >}}
+\text{isothermal} \qquad \rightarrow p \propto n \qquad T = \text{const.} \qquad \gamma = 1
+{{< /katex >}} 
+{{< katex display >}}
+\text{adiabatic} \qquad \rightarrow p \propto n^\gamma
+{{< /katex >}}
+{{< katex display >}}
+\text{cols plasma / force-free} \qquad \rightarrow p = \text{const.} \qquad \gamma = 0
+{{< /katex >}}
+
+## Ideal MHD Model
+
+The extended MHD equations are simpler than the two-fluid model, but they can still be quite complicated. We can often still get useful analysis from further reductions. The ideal MHD model is such a reduction that we can get by dropping (with justification) several terms from the extended model. We justify the simplifications by comparing the magnitude of the neglected terms to the terms that are retained.
+
+Recall the characteristic speed is {{< katex >}} v_{T, i} {{< /katex >}}. If we say that the characteristic length plasma length is {{< katex >}} L {{< /katex >}}, then we can define characteristic time {{< katex >}} \tau = L / v_{T, i} {{< /katex >}}. 
+
+The derivation of the two-fluid plasma model assumed a Maxwellian velocity distribution. We need the velocity distribution to thermalize, reach local thermodynamic equilibrium, and become Maxwellian. This means that we need many collisions, in fact so many collisions occurring frequently enough that we can ignore collisional effects. There must then be many collisions during the characteristic time {{< katex >}} \tau {{< /katex >}}.
+
+For ions to be thermalized,
+
+{{< katex display >}}
+\frac{\tau_{ii}}{\tau} \ll 1
+{{< /katex >}}
+
+And similarly for electrons
+
+{{< katex display >}}
+\frac{\tau_{e}}{\tau} \ll 1
+{{< /katex >}}
+
+The continuity equation remains unchanged from the extended MHD model
+
+!!! info "Ideal MHD Continuity Equation"
+
+    {{< katex display >}}
+    \pdv{\rho}{t} + \div (\rho \vec v) = 0
+    {{< /katex >}}    
+
+The momentum equation is
+
+{{< katex display >}}
+\rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = - \div (\vec \Pi_e + \vec \Pi_e)
+{{< /katex >}}
+Drop the anisotropic pressure
+
+!!! info "Ideal MHD Momentum Equation"
+
+    {{< katex display >}}
+    \rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = 0
+    {{< /katex >}}
+
+
+The generalized Ohm's law is
+
+{{< katex display >}}
+\vec E + \vec v \cross \vec B = \eta \vec j + \frac{1}{Z e n} \left( \vec j \cross B - \grad p_e - \div \vec \Pi_e \right)
+{{< /katex >}}
+
+We're going to drop the entire right hand side
+
+!!! info "Ideal MHD Generalized Ohm's Law"
+
+    {{< katex display >}}
+    \vec E + \vec v \cross \vec B = 0
+    {{< /katex >}}
+
+For the energy equation we have
+
+{{< katex display >}}
+\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \frac{\gamma - 1}{\rho^\gamma} [Q_{ie} + Q_{ei} - \div (\vec h_i + \vec h_e) - \vec \Pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e] + \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}
+{{< /katex >}}
+
+We neglect the entire right-hand side
+
+!!! info "Ideal MHD Energy Equation"
+
+    {{< katex display >}}
+    \pdv{\rho}{t} + \div (p \vec v) = (1 - \gamma) p \div \vec v 
+    {{< /katex >}}
+
+### Collision/Pressure terms
+
+If we assume that the ions and electrons are in thermal equilibrium{{< katex >}} T_i = T_e {{< /katex >}}, we can relate the collision times
+
+{{< katex display >}}
+\tau_{ee} : \tau_{ii} : \tau_{ei} = 1 : \left( \frac{m_i}{m_e} \right) ^{1/2} : \frac{m_i}{m_e}
+{{< /katex >}}
+
+The collision times are specifically collisional relaxation times of the Boltzmann equation
+
+{{< katex display >}}
+\left. \pdv{f}{t} \right|_{coll} = \frac{f - f_{\text{Maxwellian}}}{\tau_{\alpha \beta}}
+{{< /katex >}}
+
+For electrons, the thermalization condition is much stricter for the ions
+
+{{< katex display >}}
+\frac{\tau_{ee}}{\tau} = \left( \frac{m_e}{m_i} \right)^{1/2} \frac{\tau_{ii}}{\tau} \ll 1
+{{< /katex >}}
+
+Neglect the anisotropic pressure tensor in the momentum and generalized Ohm's law, {{< katex >}} \div \vec \Pi {{< /katex >}}. {{< katex >}} \vec \Pi {{< /katex >}} is primarily the shear stress tensor. The ion thermal speed gives us a characteristic velocity for the plasma, so we use it to characterize the shear stress
+
+{{< katex display >}}
+\vec \Pi_{i, max} \sim 2 \mu \left( \pdv{u}{x} - \frac{1}{3} \div \vec v \right) \sim \mu \frac{v_{T, i}}{L}
+{{< /katex >}}
+
+Standard treatments of the viscosity (Braginskii, etc.) show that viscosity scales with the number density, temperature, and collision time
+
+{{< katex display >}}
+\mu \sim n T_i \tau_{ii} \sim p_i \tau_{ii}
+{{< /katex >}}
+
+{{< katex display >}}
+\Pi_{i, max} \sim p_i \frac{\tau_{ii} v_{T, i}}{L}
+{{< /katex >}}
+
+The specific term we want to get rid of is {{< katex >}} \div \vec \Pi {{< /katex >}}, so let's compare it to a term we want to keep {{< katex >}} \grad P {{< /katex >}} 
+
+{{< katex display >}}
+\frac{\div \vec \Pi}{\grad p} \sim \frac{p_i \tau_{ii} v_{T, i} / L^2}{p_i / L} \sim \frac{\tau_{ii}}{\tau}
+{{< /katex >}}
+
+So, to neglect the anisotropic pressure term in the momentum equation, once again we require
+
+{{< katex display >}}
+\frac{\tau_{ii}}{\tau} \ll 1
+{{< /katex >}}
+
+In other words, as long as the plasma is collision-dominated, we can drop the ion anisotropic pressure term. What about associated the electron term? If you can assume {{< katex >}} T_i \approx T_e {{< /katex >}}. Then {{< katex >}} p_i \approx p_e {{< /katex >}} for a neutral plasma, and
+
+{{< katex display >}}
+\frac{\div \vec \Pi_e}{\grad p} \sim \frac{p_i \tau_{ee} v_{T, i} / L^2}{p_i / L} \sim \frac{\tau_{ee}}{\tau} \ll 1
+{{< /katex >}}
+
+### Magnetic terms
+
+In the generalized Ohm's law, the diamagnetic drift term is
+
+{{< katex display >}}
+\frac{\grad p_e}{Z e n} \sim \frac{n T_e / L}{Z en} \sim \frac{T_i / L}{Ze} \sim \frac{m_i v_{T, i}^2}{L Z e}
+{{< /katex >}}
+
+Compare {{< katex >}} \grad p_e / Z e n {{< /katex >}} to a term that we're going to keep, which is the dynamo term {{< katex >}} \vec v \cross \vec B {{< /katex >}}
+
+{{< katex display >}}
+\frac{ \grad p_e / Z en}{|\vec v \cross \vec B|} \sim \frac{m_i v_{T, i}^2}{LZe}{v_{T, i} B} \sim \frac{m_i}{ZeB}\frac{v_{T, i}}{L} = \frac{v_{T, i}}{\omega_{c, i}} \frac{1}{L} \sim \frac{r_{L, i}}{L} \ll 1
+{{< /katex >}}
+
+So to neglect the diamagnetic drift term, we need the plasma to be well-magnetized. This means the Larmor radius must be much less than the plasma characteristic length {{< katex >}} r_{L, i} \ll L {{< /katex >}}
+
+Now what can we do with the Hall term {{< katex >}} \frac{\vec j \cross \vec B}{Zen} {{< /katex >}}. For a static plasma (or one with subsonic flows):
+
+{{< katex display >}}
+\rho (\pdv{\vec v}{t} + \vec v \cdot \grad \vec v) - \grad p - \vec j \cross \vec B = 0
+{{< /katex >}}
+
+so by "subsonic" we mean that the static pressure is much larger than the dynamic pressure and we can discard the {{< katex >}} \vec v {{< /katex >}} terms.
+
+{{< katex display >}}
+\vec j \cross \vec B \approx \grad p
+{{< /katex >}}
+
+Comparing the Hall term to the dynamo term {{< katex >}} \vec v \cross \vec B {{< /katex >}} also gives the same requirement
+
+{{< katex display >}}
+\frac{r_{L, i}}{L} \ll 1
+{{< /katex >}}
+
+How well do some real plasmas hold up to the requirements of ideal MHD? Consider field nulls in a Z-pinch or an FRC, and weakly magnetized plasmas (such as those in a Hall thruster). Clearly, the magnetization requirement does not hold up at all points in space, and ideal MHD does not necessarily apply across the whole domain.
+
+### Resistivity
+
+We neglect resistivity and the resistive electric field {{< katex >}} \eta \vec j {{< /katex >}} in the generalized Ohm's law
+
+{{< katex display >}}
+\eta \vec j \sim \eta \frac{\grad p}{B} \sim \frac{ \eta n T}{LB}
+{{< /katex >}}
+
+{{< katex display >}}
+\eta = \frac{m_e \nu_{ei}}{n e^2} = \frac{m_e}{n e^2 \tau_{ei}} = \frac{m_e ^2/ m_i}{n e^2 \tau_{ee}}
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \eta \vec j \sim \frac{m_e ^2}{e^2 L B} \frac{v_{T, i} ^2}{\tau_{ee}}
+{{< /katex >}}
+
+How does it compare to {{< katex >}} \vec v \cross \vec B {{< /katex >}}
+
+{{< katex display >}}
+\frac{\eta \vec j}{|\vec v \cross \vec B|} \sim \left(\frac{m_e}{m_i} \right)^2 \frac{m_i}{e ^2 B^2} \frac{v_{T, i} ^2}{L^2} \frac{L}{v_{T, i} \tau_{ee}} \sim \left( \frac{m_e}{m_i}\right)^2 \frac{v_{T, i} ^2}{\omega_{c, i} ^2} \frac{1}{L^2} \sim \left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^{2} \frac{\tau}{\tau_{ii}} \ll 1
+{{< /katex >}}
+
+This now places a lower limit on {{< katex >}} \tau_{ii} {{< /katex >}}; to neglect resistivity {{< katex >}} \tau_{ii} {{< /katex >}} cannot be too low
+
+{{< katex display >}}
+\left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^{2} \ll  \frac{\tau_{ii}}{\tau} \ll 1
+{{< /katex >}}
+
+### Heating sources
+
+Neglect the collisional heating sources {{< katex >}} Q_{ei} {{< /katex >}}, {{< katex >}} Q_{ie} {{< /katex >}} in the energy equation. We do that by assuming that they are equal and opposite, which only happens when the temperatures are equal. In other words, we are again assuming local thermodynamic equilibrium _between_ electrons and ions.
+
+{{< katex display >}}
+1 \gg \frac{\tau_{ei}}{\tau} = \left( \frac{m_i}{m_e} \right)^{1/2} \frac{\tau_{ii}}{\tau} \rightarrow \frac{\tau_{ii}}{\tau} \ll \left( \frac{m_e}{m_i} \right) ^{1/2}
+{{< /katex >}}
+
+This is a much more restrictive condition than ion collisionality. Alternatively, we could track two temperature independently.
+
+We also neglect the heat flux terms {{< katex >}} \div (\vec h_i + \vec h_e) {{< /katex >}} in the energy equation. Consider parallel (to the magnetic field) heat conduction which dominates since {{< katex >}} \kappa_\perp \ll \kappa_\parallel {{< /katex >}}, so
+
+{{< katex display >}}
+\div \vec h \approx \grad_\parallel \cdot (\kappa_\parallel \grad_\parallel T)
+{{< /katex >}}
+
+{{< katex display >}}
+\grad_\parallel \cdot \left[ ( \kappa_{\parallel, i} + \kappa_{\parallel, e}) \grad_\parallel T \right]
+{{< /katex >}}
+
+{{< katex display >}}
+\kappa_{\parallel, e} \sim \frac{n T_e}{m_e} \tau_{ee} \qquad \text{ and } \qquad \kappa_{\parallel, i} \sim \frac{n T_i}{m_i} \tau_{ii}
+{{< /katex >}}
+
+{{< katex display >}}
+\frac{\kappa_{\parallel, e}}{\kappa_{\parallel, i}} \sim \frac{\tau_{ee}}{\tau_{ii}} \frac{m_i}{m_e} \sim \left( \frac{m_i}{m_e} \right) ^{1/2}
+{{< /katex >}}
+
+Compare the thermal conductivity to the rate of pressure change {{< katex >}} \pdv{p}{t} {{< /katex >}} 
+
+{{< katex display >}}
+\frac{\grad_\parallel \cdot (\kappa_{\parallel, e} \grad_\parallel T)}{\pdv{p}{t}} \sim \frac{n T^2 \tau_{ee}/m_e L^2}{nT/\tau} \sim \left( \frac{m_i}{m_e} \right)^{1/2} \frac{\tau_{ii}}{\tau} \ll 1
+{{< /katex >}}
+
+We're back to the same requirement that we have ion-electron local thermodynamic equilibrium.
+
+### Anisotropic pressure terms in energy equation
+
+Neglect the anisotropic pressure terms in the energy equation. That is, {{< katex >}} \vec \Pi_i \cdot \cdot \grad \vec v_i {{< /katex >}} and {{< katex >}} \vec \Pi_e \cdot \cdot \grad \vec v_e {{< /katex >}}.  Skipping ahead, the result is
+
+{{< katex display >}}
+\frac{\tau_{ii}}{\tau} \ll 1 
+{{< /katex >}}
+{{< katex display >}}
+\left( \frac{m_e}{m_i} \right)^{1/2} \frac{\tau_{ii}}{\tau} \frac{r_{L, i}}{L} \ll 1
+{{< /katex >}}
+
+### Electron convection term
+
+Lastly, we neglect the electron convection term in the energy equation {{< katex >}} \frac{\vec j}{Zen} \cdot \grad \frac{P_e}{\rho^\gamma} {{< /katex >}}. The result is
+
+{{< katex display >}}
+\frac{r_{L, i}}{L} \ll 1
+{{< /katex >}}
+
+
+## Conservation Law Form of MHD
+
+{{< katex display >}}
+\pdv{}{t} q + \div \vec f = 0
+{{< /katex >}}
+
+We can express momentum in conservation law form:
+
+{{< katex display >}}
+\pdv{}{t} (\rho \vec v) + \div \left[ \rho \vec v \vec v - \frac{ \vec B \vec B}{\mu_0} + (p + \frac{B^2}{2 \mu_0})\vec 1 \right] = 0
+{{< /katex >}}
+
+The conservation law form for the magnetic field looks like
+
+{{< katex display >}}
+\pdv{ \vec B}{t} + \div \left[ \vec v \vec B - \vec B \vec v \right ] = 0
+{{< /katex >}}
+
+And of course the energy equation for
+
+{{< katex display >}}
+E = \frac{1}{\gamma - 1} p + \frac{1}{2} \rho v^2 + \frac{ B^2}{2 \mu_0}
+{{< /katex >}}
+
+{{< katex display >}}
+\pdv{E}{t} + \div \left[ \left( E + p + \frac{B^2}{2 \mu_0} \right) - \left( \frac{ \vec B \cdot \vec v}{\mu_0} \right) \vec B \right] = 0
+{{< /katex >}}
+
+Conservation law forms are particularly useful when considering equilibrium steady-state force balance. This means that in steady-state equilibrium we have
+
+{{< katex display >}}
+\div \left[ \rho \vec v \vec v - \frac{ \vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0} \right) \vec 1 \right] = 0
+{{< /katex >}}
+
+We can use this relationship and integrate over various volumes to determine the relationship between the various force balance terms
+
+### Examples of equilibrium plasma confinement
+
+For a plasma in static equilibrium {{< katex >}} \vec v = 0 \rightarrow \vec j \cross \vec B = \grad p {{< /katex >}} 
+
+{{< katex display >}}
+ \div \vec T = 0 = \div \left( \right) 
+ {{< /katex >}}
+
+{{< katex display >}}
+\vec j \cross \vec B = \frac{1}{\mu_0} (\curl \vec B) \cross \vec B = \frac{1}{\mu_0} (\vec B \cdot \grad \vec B - \frac{1}{2} \grad B^2) \\
+= \frac{1}{\mu_0} ( B^2 \vu e_B \cdot \grad \vu e_B + \frac{1}{2} \vu e_B \vu e_B \cdot \grad B^2 - \frac{\grad B^2}{2} )
+{{< /katex >}}
+
+where {{< katex >}} \vu e_B = \vec B / B {{< /katex >}} 
+
+{{< katex >}} \vu e_B \cdot \grad \vu e_B {{< /katex >}} is the curvature of {{< katex >}} \vec B {{< /katex >}}. Write it like a curvature
+
+{{< katex display >}}
+\vec K = - \vu r / R_c
+{{< /katex >}}
+
+{{< katex display >}}
+\vu e_B \vu e_B \cdot \grad B^2
+{{< /katex >}}
+is gradient of {{< katex >}} B^2 {{< /katex >}} that is parallel to B. Multiply by {{< katex >}} e_B {{< /katex >}} gives the component of gradient along {{< katex >}} e_B {{< /katex >}}. The difference between that and {{< katex >}} \grad B^2 / 2 {{< /katex >}} gives you the perpendicular gradient
+
+{{< katex display >}}
+\vec j \cross \vec B = \frac{1}{\mu_0} (B^2 \vec \kappa - \frac{1}{2} \grad _\perp B^2) = \grad p = \grad_\perp p
+{{< /katex >}}
+
+identify {{< katex >}} B^2 \vec \kappa {{< /katex >}} is magnetic tension resulting from having a bent magnetic field line. {{< katex >}} \frac{1}{2} \grad_\perp B^2  {{< /katex >}} is magnetic pressure. They have to balance the plasma pressure at equilibrium.
+
+For example, consider a cylindrical plasma that's in equilibrium with a helical magnetic field
+{{< katex display >}}
+\vec B = B_\theta (r) \vu \theta + B_z (r) \vu z
+{{< /katex >}}
+
+How is plasma pressure profile determined by the different components of the magnetic field? If we want to maximize the amount of pressure we confine, what should be maximized/minimized?
+
+{{< katex display >}}
+\frac{B^2}{\mu_0} \vu \kappa = \grad _\perp ( p + \frac{B^2}{2 \mu_0} ) 
+{{< /katex >}}
+
+{{< katex >}} B_z {{< /katex >}} is straight and has no curvature, so the only magnetic tension comes from {{< katex >}} B_\theta {{< /katex >}}, so the magnetic tension from {{< katex >}} B_\theta {{< /katex >}} must balance the total pressure.
+
+<p align="center"> <img alt="Figure 12.6" src="/r/img/12.6.png" /> </p>
+
+The role of {{< katex >}} B_z {{< /katex >}} is displacing plasma pressure. The utility in defining {{< katex >}} \beta {{< /katex >}} as
+
+{{< katex display >}}
+\beta = \frac{\text{plasma pressure}}{\text{magnetic pressure}}
+{{< /katex >}}
+
+### Conditions of Ideal MHD Validity
+
+The conditions for ideal MHD to be valid are
+
+1. High Ion Collisionality: {{< katex >}} \frac{\tau_{ii}}{\tau} \ll 1 {{< /katex >}} 
+2. Small ion Larmor radius: {{< katex >}} \frac{r_{L, i}}{L} \ll 1 {{< /katex >}} 
+3. Low resistivity: {{< katex >}} \left(\frac{m_e}{m_i} \right)^{3/2} \left( \frac{r_{L, i}}{L} \right)^2 \ll 1 {{< /katex >}} 
+
+For a given plasma in force balance, we can relate the plasma pressure to the magnetic pressure
+
+{{< katex display >}}
+\beta = \frac{n T}{B^2 / 2 \mu_0} = 4 \times 10^{-16} \frac{n_{cm^{-3}} T_{keV}}{B^2 _{T}}
+{{< /katex >}}
+
+Ion collision time is (Spitzer collisionality)
+
+{{< katex display >}}
+\tau_{ii} = 2.09 \times 10^{7} \frac{T_{eV} ^{3/2} \mu ^{1/2}}{\ln \Lambda n_{cm ^{-3}}} \left[\text{s}\right]
+{{< /katex >}}
+where {{< katex >}} \mu \equiv m_i / m_p {{< /katex >}}.
+
+Putting the conditions for ideal MHD in terms of {{< katex >}} \beta {{< /katex >}}  and {{< katex >}} \tau_{ii} {{< /katex >}}, 
+
+1. High collisionality:
+{{< katex display >}}
+\frac{\tau_{ii}}{\tau} = 2.14 \times 10^{12} \frac{T^2}{n L} \ll 1 
+{{< /katex >}}
+{{< katex display >}}
+\rightarrow T_{eV} \ll 6.8 \times 10^{-7} L_{cm} ^{1/2} n_{cm^{-3}} ^{1/2}
+{{< /katex >}}
+2. Small gyroradius:
+{{< katex display >}}
+\frac{r_{L, i}}{L} = 2.3 \times 10^7 \frac{1}{Z L} \sqrt{ \frac{\mu B}{n}} \ll 1
+{{< /katex >}}
+{{< katex display >}}
+\rightarrow n_{cm^{-3}} \gg 5.3 \times 10^{14} \frac{\mu B_{T}}{Z^2 L_{cm} ^2}
+{{< /katex >}}
+3. Low resistivity:
+{{< katex display >}}
+\left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^2 \frac{\tau}{\tau_{ii}} = 5.65 \frac{\mu B}{Z^2 L T^2}
+{{< /katex >}}
+{{< katex display >}}
+\rightarrow T_{eV} \gg 2.4 \frac{\mu ^{3/2} \beta^{1/2}}{Z L _{cm} ^{1.2}}
+{{< /katex >}}
+
+<p align="center"> <img alt="Figure 12.7" src="/r/img/12.7.png" /> </p>
+
+
+## Perpendicular MHD
+
+For most configurations for magnetic fusion confinement, we are able to satisfy 2. and 3. but often have densities much too low to meet the high collisionality constraint. However, in practice ideal MHD does accurately model macroscopic behavior of many plasmas. At the same time, magnetized / fusion plasmas are often largely collisionless. We can understand why by re-writing the collisionality requirement as
+
+{{< katex display >}}
+1 \gg \frac{\tau_{ii}}{\tau} = \frac{\tau_{ii} v_{T, i}}{\tau v_{T, i}} \sim \frac{\lambda}{L}
+{{< /katex >}}
+
+where {{< katex >}} \lambda {{< /katex >}} is the mean free path. The ratio {{< katex >}} \lambda / L {{< /katex >}} is also known as the **Knudsen number**. In magnetized plasmas, the mean free path is often very long, but the path between collisions can cover a great distance only by following magnetic field lines. Motion {{< katex >}} \perp {{< /katex >}} to the magnetic field is constrained by {{< katex >}} r_{L, i} {{< /katex >}}. This suggests an approach wherein we divide the plasma model into a 1-D kinetic model to be solved along the magnetic field and a 2-D MHD model {{< katex >}} \perp {{< /katex >}} to the magnetic field. 
+
+We consider diffusivity (terms like viscosity, conductivity) parallel and perpendicular to {{< katex >}} \vec B {{< /katex >}} 
+{{< katex display >}}
+k_\parallel \sim \frac{\lambda ^2}{\tau_{ii}}
+{{< /katex >}}
+{{< katex display >}}
+k_\perp \sim \frac{r_{L, i} ^2}{\tau_{ii}}
+{{< /katex >}}
+{{< katex display >}}
+\frac{k_\parallel}{k_\perp} \sim \left( \frac{\lambda}{r_{L, i}} \right)^2 \sim (\omega_{c, i} \tau_{ii} )^2
+{{< /katex >}}
+
+This changes one of our conditions for validity. Specifically, we can write
+
+{{< katex display >}}
+\frac{r_{L, i}}{L} \frac{1}{ \omega_{c, i} \tau_{i i}} = 254 \frac{\mu \beta}{Z^2 L_{cm} T_{eV} ^2}
+{{< /katex >}}
+
+or
+
+{{< katex display >}}
+T_{eV} \gg 16 \frac{\mu ^{1/2} \beta^{1/2}}{Z L^{1/2}}
+{{< /katex >}}
+
+This is now only slightly more restrictive than the low-resistivity condition. In fact, most of the plasmas that we looked at before (large tokamaks & toruses, propulsion systems) which had temperatures above the validity range now fall comfortably within the region of validity. The perpendicular MHD model is equivalent to a collisionless model, giving a much wider applicability than the collisional MHD model.
+
+What about the parallel component? Our collisionality condition still isn't valid parallel to the field. Basically, we ignore the parallel component, which is the same as assuming that {{< katex >}} \rho {{< /katex >}}, {{< katex >}} T {{< /katex >}}, {{< katex >}} p {{< /katex >}} are constant along magnetic field lines, with {{< katex >}} \div \vec v = 0 {{< /katex >}} . Let's write out the expressions for collision-less MHD and for ideal MHD and compare:
+
+|  | Collisionless MHD | Ideal MHD |
+| --- | --- | --- |
+| Continuity | {{< katex >}} \pdv{\rho}{t} = 0 {{< /katex >}} | {{< katex >}} \pdv{\rho}{t} = - \rho \div \vec v {{< /katex >}} |
+| Momentum | {{< katex >}} \rho \dv{\vec v_\perp}{t} = \vec j \cross \vec B - \grad_\perp p {{< /katex >}} | {{< katex >}} \rho \dv{\vec v}{t} = \vec j \cross \vec B - \grad p {{< /katex >}} |
+| Parallel constraint  | {{< katex >}} \vec B \cdot \grad \frac{v_\parallel}{B} = - \div \vec v _\perp {{< /katex >}} | |
+| Energy | {{< katex >}} \dv{p}{t} = 0 {{< /katex >}} | {{< katex >}} \dv{p}{t} = - \gamma p \div \vec v {{< /katex >}} |
+
+Collisionless MHD reproduces many of the effects of ideal MHD but has a wider region of validity. Corollary: ideal MHD is accurate beyond its region of validity, unless results lead to parallel gradients. For example, we know that MHD is not valid when representing confinement of a plasma confined in a magnetic mirror, which is an inherently kinetic phenomenon. But ideal MHD _can_ generate parallel gradients within its region of validity, and we need to be careful. Ideal MHD does not require different models {{< katex >}} \parallel {{< /katex >}} and {{< katex >}} \perp {{< /katex >}} to the magnetic field, and is therefore preferred. We will continue to use ideal MHD outside of its region of validity.
diff --git a/content/notes/UWAA558/06-boundary-conditions.md b/content/notes/UWAA558/06-boundary-conditions.md
new file mode 100644
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+++ b/content/notes/UWAA558/06-boundary-conditions.md
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+---
+title: Boundary Conditions
+bookToc: false
+weight: 60
+---
+
+
+# Boundary Conditions
+
+Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region.
+
+### Perfectly Conducting Wall
+
+For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:
+
+{{< katex display >}}
+\left. \vec E_t \right|_{wall} = 0 \quad \rightarrow \quad \left. \vu n \cross \vec E \right| _{wall} = 0
+{{< /katex >}}
+
+Applying Faraday's law at the wall,
+
+{{< katex display >}}
+\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = \left. - \vu n \cdot \curl \vec E \right|_{wall} = \left. \div (\vu n \cross \vec E) \right| _{wall} = 0
+{{< /katex >}}
+{{< katex display >}}
+\pdv{}{t} \vu n \cdot \vec B |_{wall} = 0
+{{< /katex >}}
+If initially there is no normal magnetic field, then
+{{< katex display >}}
+\vu n \cdot \vec B|_{wall} = 0 \quad \text{if initially true}
+{{< /katex >}}
+
+And of course, for an impermeable wall,
+
+{{< katex display >}}
+\vu n \cdot \vec v |_{wall} = 0
+{{< /katex >}}
+
+Is this a sufficient set of boundary conditions? Think back to the governing equations in conservation form
+{{< katex display >}}
+\pdv{}{t} \vec Q + \div \vec F = 0
+{{< /katex >}}
+The boundary conditions come into play when defining {{< katex >}} \vec F {{< /katex >}} at the boundary. In particular, we need to know what {{< katex >}} \dd \vec S \cdot \vec F |_{wall} {{< /katex >}} is. In our governing equations, this will involve {{< katex >}} \vec E {{< /katex >}}, {{< katex >}} \vec B {{< /katex >}}, and {{< katex >}} \vec v {{< /katex >}}.
+
+### Insulating Boundary
+
+As a slight modification, an insulating boundary can have a tangential electric field. Consider a simple geometry of parallel electrodes with an insulating wall between them.
+
+<p align="center"> <img alt="Figure 12.8" src="/r/img/12.8.png" /> </p>
+
+From Ohm's law
+{{< katex display >}}
+\vec E + \vec v \cross \vec B = 0
+{{< /katex >}}
+so the only way an electric field tangential to the wall can exist is if {{< katex >}} \vu n \cdot \vec v \neq 0 {{< /katex >}}.
+
+For either a perfectly conducting or an insulating boundary, the other variables are arbitrary: {{< katex >}} \rho {{< /katex >}} , {{< katex >}} p {{< /katex >}}, {{< katex >}} \vec v_t {{< /katex >}}, {{< katex >}} \vec B_t {{< /katex >}}.
+
+### Vacuum Region
+
+The plasma (radius {{< katex >}} R_p {{< /katex >}})  is supported by a region of vacuum out to a perfectly conducting wall at some radius {{< katex >}} R_w {{< /katex >}}. We assume that there is no plasma in the vacuum region. The governing equations in vacuum are just Maxwell's equations
+
+{{< katex display >}}
+\curl \vec B_{vac} = 0 \qquad \text{and} \qquad \div \vec B_{vac} = 0
+{{< /katex >}}
+
+At the wall, 
+{{< katex display >}}
+\vu n \cross \vec E |_{wall} = 0
+{{< /katex >}}
+{{< katex display >}}
+\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = 0
+{{< /katex >}}
+
+What happens at the plasma-vacuum interface? We need to specify jump conditions and continuity conditions. Let's use square brackets to signify a jump:
+
+{{< katex display >}}
+\left[ X \right] = \left. X \right|_{R_p + dr} - \left. X \right|_{R_p - dr}
+{{< /katex >}}
+
+The normal magnetic field has to be continuous.
+
+{{< katex display >}}
+[\vu n \cdot \vec B]_{R_p} = 0
+{{< /katex >}}
+
+The tangential magnetic field jump is given by the surface current density at the jump
+
+{{< katex display >}}
+\left[ \vu n \cross \vec B \right] _{R_p} = \mu_0 \vec K 
+{{< /katex >}}
+
+Integrating {{< katex >}} \grad_\perp (p + \frac{B^2}{2 \mu_0}) = \frac{B^2}{\mu_0} \vec \kappa {{< /katex >}} over a differential volume across the surface gives
+
+{{< katex display >}}
+\left[ p + \frac{B^2}{2 \mu_0} \right] _{R_p} = 0
+{{< /katex >}}
+
+The plasma shape is determined self-consistently by the wall shape and surface current. This is a free-boundary problem. Another option is to specify the plasma shape, and then determine the required wall shape. This is a fixed-boundary problem.
+
+The most realistic case includes externally applied magnetic fields coming from source coils, perhaps computed by Biot-Savart law. The vacuum magnetic field is then {{< katex >}} \vec B_{vac} = \vec B_{ext} + \vec B_{plasma} {{< /katex >}}. The crazy coil shapes in the stellarator design come from the 3D geometry computations solving this problem.
+
+### Conservation of Magnetic Flux ("Frozen-In" Flux)
+
+Locally, {{< katex >}} \vec E + \vec v \cross \vec B = 0 {{< /katex >}} with Faraday's law
+{{< katex display >}}
+\pdv{B}{t} = - \curl \vec E = - B \div \vec v + \vec B \cdot \grad \vec v - \vec v \cdot \grad B
+{{< /katex >}}
+From the continuity equation,
+{{< katex display >}}
+\pdv{\rho}{t} + \vec v \cdot \grad \rho = - \rho \div \vec v
+{{< /katex >}}
+Combining we find that
+{{< katex display >}}
+\dv{\vec B}{t} = \frac{\vec B}{\rho} \dv{\rho}{t} + \vec B \cdot \grad \vec v
+{{< /katex >}}
+{{< katex display >}}
+\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v
+{{< /katex >}}
+
+This says that the field and plasma density move together. Locally, if the magnetic field increases then mass density increases, such that the ratio {{< katex >}} \vec B / \rho {{< /katex >}} remains constant. In the direction parallel to the magnetic field we have a term that involves field line twisting, which is a bit more complicated, but in the perpendicular direction
+{{< katex display >}}
+\dv{}{t} \left( \frac{\vec B}{\rho} \right) _\perp = 0
+{{< /katex >}}
+
+If we consider globally the magnetic flux through a moving surface S at velocity {{< katex >}} \vec u {{< /katex >}}. The magnetic flux penetrating the surface is
+{{< katex display >}}
+\Psi = \int \vec B \cdot \dd \vec S
+{{< /katex >}}
+or
+{{< katex display >}}
+\dv{\Psi}{t} = \int \dv{\vec B}{t} \cdot \vu n \dd S
+{{< /katex >}}
+{{< katex display >}}
+= \int \pdv{\vec B}{t}\cdot \vu n \dd S + \oint \vec B \cross \vec u \dd \vec l
+{{< /katex >}}
+Using Faraday's law
+
+{{< katex display >}}
+\dv{\Psi}{t} = \int - \curl \vec E \cdot \vu n \dd S + \oint \vec B \cross \vec u \cdot \dd \vec l
+{{< /katex >}}
+{{< katex display >}}
+= \oint (- \vec E + \vec B \cross \vec u) \cdot \dd \vec l
+{{< /katex >}}
+Using the electric field from Ohm's law
+{{< katex display >}}
+\dv{\Psi}{t} = \oint(\vec v - \vec u) \cross \vec B \cdot \dd \vec l
+{{< /katex >}}
+This tells us that if the surface moves with the plasma {{< katex >}} \vec u = \vec v {{< /katex >}} then 
+{{< katex display >}}
+\dv{\Psi}{t} = 0
+{{< /katex >}}
+the flux through the surface is constant, and the flux is a constant of the topology. This is a direct consequence of ideal MHD. If we add even a small amount of resistivity, we dramatically alter the results in a process called "tearing" where the magnetic field "tears" and reconnects with itself.
\ No newline at end of file
diff --git a/content/notes/UWAA558/07-equilibrium-for-fusion.md b/content/notes/UWAA558/07-equilibrium-for-fusion.md
new file mode 100644
index 00000000..50e5900d
--- /dev/null
+++ b/content/notes/UWAA558/07-equilibrium-for-fusion.md
@@ -0,0 +1,242 @@
+---
+title: Equilibrium for Fusion
+bookToc: false
+weight: 70
+---
+
+
+# Equilibrium for Fusion ({{< katex >}} \beta {{< /katex >}})
+
+For a fusion device we would like to determine a magnetic configuration that confines plasma while it fuses. At fusion temperatures, the power required to maintain the equilibrium will be substantial. For a device to be useful, the power required to sustain the equilibrium must be less than the power released from fusion. Important loss terms for a confined plasma are transport (thermal conduction primarily) and radiation terms. The scaling factors are {{< katex >}} P_{Brem} \sim n^2 T^{1/2} {{< /katex >}} and {{< katex >}} P_{cycl} \sim n^2 T^2 {{< /katex >}} for radiation, and {{< katex >}} P_L \sim \frac{3nT}{\tau_E} {{< /katex >}} for thermal losses.
+
+We know that the fusion source term will primarily come from the DT fusion reaction
+
+{{< katex display >}}
+\text{D} + \text{T} \rightarrow \text{He}^4 (3.5\, MeV) + \text{n} (14.1\, MeV)
+{{< /katex >}}
+
+The primary fusion reaction releases an {{< katex >}} \alpha {{< /katex >}}-particle and a high-energy neutron. The concept of ignition is that the neutron leaves the plasma, and the {{< katex >}} \alpha {{< /katex >}} (with energy {{< katex >}} E_\alpha = 3.5 MeV) {{< /katex >}} remains to heat the plasma.
+
+{{< katex display >}}
+P_\alpha = \frac{1}{4} n^2 \langle \sigma v \rangle E_\alpha \qquad \text{(assuming} \quad n_D = n_T = n/2 \text{)}
+{{< /katex >}}
+
+
+{{< katex display >}}
+P_\alpha > P_L \quad \rightarrow \quad n \tau_E > \frac{12 T}{E_\alpha \langle \sigma v \rangle}
+{{< /katex >}}
+
+To sustain fusion, we set the fusion heating term above the thermal loss term. The reaction cross-section {{< katex >}} \sigma {{< /katex >}} can be maximized to give the Lawson criterion
+
+{{< katex display >}}
+n \tau_E > 10^{14} s / cm^3
+{{< /katex >}}
+
+The Lawson criterion only applies at fusion temperatures, but it is a useful parameter even outside of ignition since it gives a ratio of fusion power to lost power
+
+| {{< katex >}} T_i (keV) {{< /katex >}} | {{< katex >}} \langle \sigma v \rangle (cm^3 / s) {{< /katex >}} | Required {{< katex >}} n \tau_E (s / cm^3) {{< /katex >}} |
+| --- | --- | ---|
+| 1 | {{< katex >}} 7 \cdot 10^{-21} {{< /katex >}} | {{< katex >}} 5 \cdot 10^{17} {{< /katex >}} |
+| 5 | {{< katex >}} 1.4 \cdot 10^{-17} {{< /katex >}} | {{< katex >}} 1.2 \cdot 10^{15} {{< /katex >}} |
+| 20 | {{< katex >}} 4.3 \cdot 10^{-16} {{< /katex >}} | {{< katex >}} 1.6 \cdot 10^{14} {{< /katex >}} |
+| 60 | {{< katex >}} 8.7 \cdot 10^{-16} {{< /katex >}} | {{< katex >}} 2.4 \cdot 10^{14} {{< /katex >}} |
+
+We can see that the required {{< katex >}} n \tau_E {{< /katex >}} actually has a minimum around {{< katex >}} 20 keV {{< /katex >}} (at least, as far as the data in the table goes). Even though the maximum cross-section is at a much higher temperature, what we're really concerned with is the ratio of the fusion source term to the thermal loss term, which is linear in temperature.
+
+MHD equilibrium does not place a limit on the density {{< katex >}} n {{< /katex >}}. Instead, it places a limit on {{< katex >}} \beta {{< /katex >}} in order to achieve equilibrium force-balance {{< katex >}} (\beta = 1) {{< /katex >}}
+
+{{< katex display >}}
+\beta = \frac{n (T_e + T_i)}{B^2 / 2 \mu_0} \rightarrow n = \frac{ \beta B^2}{4 \mu_0 T}
+{{< /katex >}}
+
+In this form, we can more clearly see what our options are to achieve MHD equilibrium. Some devices (large-scale tokamaks) are able to achieve the requisite confinement time at a low {{< katex >}} \beta {{< /katex >}} by making use of very strong magnetic fields. Other devices are able to make use of more modest magnetic fields by working at a higher {{< katex >}} \beta {{< /katex >}}.
+
+Therefore,
+
+{{< katex display >}}
+\tau_E > \frac{1}{\beta B^2} \frac{48 \mu_0}{E_\alpha} \frac{T^2}{\langle \sigma v \rangle}
+{{< /katex >}}
+
+The term {{< katex >}} \frac{T^2}{\langle \sigma v \rangle} {{< /katex >}} has a minimum at {{< katex >}} 10-20 keV {{< /katex >}}. At 15 keV and a magnetic field of {{< katex >}} 5T {{< /katex >}} (many actual components cannot reasonably exceed such magnetic fields) then
+
+{{< katex display >}}
+\tau_E > \frac{0.1}{\beta} \text{s}
+{{< /katex >}}
+
+For a large-scale toroidal device with {{< katex >}} \beta = 1\% {{< /katex >}}, the confinement time {{< katex >}} \tau_E > 10s {{< /katex >}}. If we consider a common diffusivity (how fast energy will leave due to thermal conductivity) {{< katex >}} D_E \approx 1 m^2 / s{{< /katex >}}, so for a characteristic radius {{< katex >}} a {{< /katex >}} 
+
+{{< katex display >}}
+\tau_E \approx \frac{a^2}{4 D_E} \rightarrow a > 6.3 \text{m}
+{{< /katex >}}
+
+This gives you a sense of why low-{{< katex >}} \beta {{< /katex >}} devices need to be so large. Instead, if we consider {{< katex >}} \beta \sim 50\% {{< /katex >}}, {{< katex >}} \tau_E > 0.2 \text{s} {{< /katex >}} and
+
+{{< katex display >}}
+\beta \sim 50\% \rightarrow a > 0.9 \text{m}
+{{< /katex >}}
+
+When you consider that the cost of a device (to first order) scales with the volume of the device, achieving a high {{< katex >}} \beta {{< /katex >}} is very important for fusion equilibrium. However, when we consider MHD stability we are generally forced into lower {{< katex >}} \beta {{< /katex >}} to avoid destructive instabilities. Configuration optimization is the process of balancing this trade-off.
+
+## Virial Theorem
+
+Application of the virial theorem to energy balance for the stress tensor {{< katex >}} \vec T {{< /katex >}} tells us that MHD equilibria must be supported by externally supplied currents.  Many times you'll hear of theoretical designs for compact toroid devices which can maintain stability under their own currents, but they are the MHD stability equivalent of a perpetual motion machine. A compact toroid cannot exist unsupported.
+
+Writing static equilibrium:
+
+{{< katex display >}}
+\div \left[ - \frac{ \vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0}\right) \vec I \right] = \div \vec T = 0
+{{< /katex >}}
+
+If we define the direction of the magnetic field to be {{< katex >}} \vu e _B = \vu z {{< /katex >}} then
+
+{{< katex display >}}
+\vec T = p_\perp ( \vec I - \vu e_B \vu e_B ) + p_\parallel \vu e_B \vu e_B \\
+= \begin{bmatrix} p_\perp & 0 & 0 \\ 0 & p_\perp & 0 \\ 0 & 0 & p_\parallel \end{bmatrix}
+{{< /katex >}}
+where
+
+{{< katex display >}}
+p_\perp = p + \frac{B^2}{2 \mu_0}
+{{< /katex >}}
+
+and
+{{< katex display >}}
+p_\parallel = p - \frac{B^2}{2 \mu_0}
+{{< /katex >}}
+
+A gradient vector identity gives
+
+{{< katex display >}}
+\div (\vec r \cdot \vec T) = \vec r \cdot ( \div \vec T) + \vec T \cdot \cdot \grad \vec r
+{{< /katex >}}
+
+Integrating this expression over a volume and assuming that the volume contains a confined MHD equilibrium that is self-contained and self-supported:
+
+<p align="center"> <img alt="Figure 12.10" src="/r/img/12.10.png" /> </p>
+
+{{< katex display >}}
+\int_V \div ( \vec r \cdot \vec T) \dd V = \int_V (\vec r \cdot \overbrace{\cancel{(\div \vec T)}}^{\text{MHD equil.}} + \vec T \cdot \cdot \grad \vec r) \dd V
+{{< /katex >}}
+
+{{< katex display >}}
+\grad \vec r = \vec I
+{{< /katex >}}
+
+so 
+{{< katex display >}}
+\vec T \cdot \cdot \grad \vec r = p_\perp + p_\perp + p_\parallel \\
+= 3p + \frac{B^2}{2 \mu_0}
+{{< /katex >}}
+
+{{< katex display >}}
+\int_V (3p + \frac{B^2}{2 \mu_0} ) \dd V = \int _V \div ( \vec r \cdot \vec T) \dd V = \oint _S (\vec r \cdot \vec T) \cdot \vu n \dd S \\
+= \oint _S  \left[ \vec r \cdot \vec I p_\perp + \vec r \cdot \vu e_B \vu e_B (p_\parallel - p_\perp) \right]\cdot \vu n \dd S \\
+=\oint \left[ \left( \cancel{p} + \frac{B^2}{2 \mu_0} \right) \vu r \cdot \vu n - \frac{B^2}{\mu_0} (\vec r \cdot \vu e_B)(\vu e_B \cdot \vu n) \right] \dd S
+{{< /katex >}}
+
+Beyond where the plasma is contained, the pressure does not contribute {{< katex >}} p = 0 {{< /katex >}}. If all current sources are contained in the configuration, the magnetic field {{< katex >}} \sim 1/r^3 {{< /katex >}} for a dipole, {{< katex >}} \sim 1/r^4 {{< /katex >}} for a quadrupole, etc. Therefore the right-hand side will fall off like
+
+{{< katex display >}}
+RHS \propto \oint_S B^2 r \dd S \propto \left( \frac{1}{r^3} \right) ^2 r r^2 \propto \frac{1}{r^3} \text{(dipole)}
+{{< /katex >}}
+so {{< katex >}} RHS \rightarrow 0 {{< /katex >}} as {{< katex >}} r \rightarrow \infty {{< /katex >}}. But what about the left-hand side? Both of the terms in the volume integral are positive definite, so the LHS must be positive finite and the equality can't possibly hold. The assumption that the plasma is self-contained must be invalid. This tells us that we must have external currents. 
+
+## Magnetic Flux Surfaces
+
+The vast difference in thermal conductivity parallel and perpendicular to the magnetic field in a plasma confinement configuration leads to an avoidance of any open field lines. Magnetic equilibria are generally toroidal to eliminate end losses from open configurations. In general fusion confinement devices, magnetic field lines lie on a set of closed nested toroidal surfaces. This means that we can no longer describe any equilibria in a solely 1D geometry. The minor radius is no longer the only important scale length.
+
+From {{< katex >}} \vec j \cross \vec B = \grad p {{< /katex >}}, we know that the pressure gradient is perpendicular to {{< katex >}} \vec j {{< /katex >}} and {{< katex >}} \vec B {{< /katex >}}, and therefore both {{< katex >}} \vec B {{< /katex >}} and {{< katex >}} \vec j {{< /katex >}} lie on surfaces of uniform pressure. We call these toroidal surfaces either magnetic surfaces or **flux surfaces**. We can use these surfaces to build a 1-dimensional description.
+
+<p align="center"> <img alt="Figure 12.11" src="/r/img/12.11.png" /> </p>
+
+As a brief aside, some geometrical vocabulary will be useful when describing toroidal geometry. A toroid is any surface of revolution with a hole in the middle. A torus is the particular case of a toroid in which the revolved figure is a circle.
+
+We will define our global toroidal coordinate system to consist of the major axis {{< katex >}} (z) {{< /katex >}}, the distance from the major axis {{< katex >}} (R) {{< /katex >}}, and the azimuthal angle around the major axis {{< katex >}} (\phi) {{< /katex >}}.
+
+<p align="center"> <img alt="Figure 12.13" src="/r/img/12.13.png" /> </p>
+
+We will also make use of a poloidal coordinate system measured by minor radius (distance from the minor axis) {{< katex >}} r {{< /katex >}} and the poloidal angle from the minor axis {{< katex >}} \theta {{< /katex >}}. We will generally refer to a point on the torus relative to the major axis {{< katex >}} (R, z, \phi) {{< /katex >}}, or relative to the minor axis {{< katex >}} (r, \theta, \phi) {{< /katex >}}, or in spherical coordinates {{< katex >}} (R, \theta, \phi) {{< /katex >}}. The major radius {{< katex >}} R_0 {{< /katex >}} is the distance from the major axis to the minor axis. The minor radius {{< katex >}} a {{< /katex >}} is the characteristic distance from the minor axis to the exterior of the revolved figure. Usually we will find symmetry under {{< katex >}} \phi {{< /katex >}}.
+
+The aspect ratio of a torus is the ratio of the major radius to the minor radius.
+
+{{< katex display >}}
+A = \frac{R_0}{a}
+{{< /katex >}}
+
+When we talk about a "toroidal surface," we mean a cross-section of the toroidal rotation. When we talk about a "poloidal surface" we mean a surface which is coplanar with the minor axis:
+
+<p align="center"> <img alt="Figure 12.14" src="/r/img/12.14.png" /> </p>
+
+The poloidal flux is determined by the size of the poloidal surface and the poloidal magnetic field:
+
+{{< katex display >}}
+\Psi _p = \int_{S_p} \vec B \cdot \dd \vec S
+{{< /katex >}}
+
+and the toroidal flux is determined by the size of a toroidal surface and the toroidal magnetic field:
+
+{{< katex display >}}
+\Psi_t = \int_{S_t} \vec B \cdot \dd \vec S
+{{< /katex >}}
+
+Considering the poloidal flux, we can see that if we expand the size of the surface towards the minor radius, the flux will increase until eventually we come to a point where the flux begins to decrease. The position of this maximum is called the **magnetic axis**, which does not necessarily correspond to the minor axis. In fact, it is generally displaced from the minor axis.
+
+<p align="center"> <img alt="Figure 12.15" src="/r/img/12.15.png" /> </p>
+
+To refer back to something more familiar, we'll define the same terms for a cylindrical geometry {{< katex >}}(r, \theta, z)  {{< /katex >}}. An axial surface corresponds with a toroidal surface, and an azimuthal surface corresponds with a poloidal surface:
+
+<p align="center"> <img alt="Figure 12.16" src="/r/img/12.16.png" /> </p>
+
+If we consider the trajectory of a single field line, what sorts of surfaces will it trace out? What surface will contain the field line? As it turns out, there are three options:
+
+1. Rational surface - the field line closes on itself, and it does so after a finite number of revolutions. One way to quickly visualize such a surface is to draw a Poincaré puncture plot. Choose a toroidal plane and plot a point wherever the field line punctures the surface. A Poincaré puncture plot of a rational surface contains a finite number of points and no continuous curves.
+2. Ergodic surface - the field line completely covers an entire surface, which is to say the field line punctures any toroidal surface an infinite number of times. In other words, it never closes on itself and defines an irrational curve.
+3. Stochastic region - In this case, there is no definite surface and the field line fills a volume.
+
+<p align="center"> <img alt="Figure 12.17" src="/r/img/12.17.png" /> </p>
+
+Generally rational surfaces and ergodic surfaces are largely equivalent, but by introducing a small amount of resistivity a rational surface can lead to magnetic islands. One can imagine the addition of resistivity equivalent to allowing a small degree of motion of the magnetic field lines. In an ergodic surface, a flux surface is defined by a single (irrational) field line. If it moves toward itself in one location it will necessarily move away from itself in another location. But in a rational surface, different field lines can lie on the same constant pressure surface and will tend to move towards each other. By concentrating into magnetic islands, the flux surfaces are now more closely spaced, and the pressure gradient increases (a bad thing!)
+
+<p align="center"> <img alt="Figure 12.18" src="/r/img/12.18.png" /> </p>
+
+**Surface quantities**: Since pressure, current (not current density!), and _flux_ (not field!) are constant along a flux surface, it is convenient to use flux {{< katex >}} \Psi_p {{< /katex >}} as a coordinate. A particular poloidal flux itself uniquely determines a poloidal surface with constant pressure and current. The flux surface quantities are {{< katex >}} p, \, \Psi_p, \, \Psi_t, \, I_p = \int_{S_p} \vec j \cdot \dd \vec S, \, I_t = \int_{S_t} \vec j \cdot \dd \vec S{{< /katex >}}.
+
+Surface quantities are not independent. The poloidal current {{< katex >}} I_p {{< /katex >}} affects the toroidal field {{< katex >}} B_t {{< /katex >}} and toroidal flux {{< katex >}} \Psi_t {{< /katex >}}. The toroidal current affects the poloidal field {{< katex >}} B_p {{< /katex >}} and poloidal flux {{< katex >}} \Psi_p {{< /katex >}}.
+
+{{< katex display >}}
+B_t = \vec B \cdot \vu \phi
+{{< /katex >}}
+
+{{< katex display >}}
+\vec B_p = B_\theta \vu \theta + B_z \vu z \\
+= B_r \vu R + B_z \vu z
+{{< /katex >}}
+
+## Toroidal Force Balance
+
+
+Toroidal equilibria solves the end losses of linear equilibrium, but generates a new force which must be balanced. This is a result of the virial theorem.
+
+### Poloidal Fields (Tire Tube Pressure Force)
+
+If you think of a flexible bike tire being inflated, as the pressure within the inner tube increases, the major radius will increase! Why is this? The pressure within the tire is isotropic. As we pump up the tire, the pressure increase causes a force imbalance with the atmospheric pressure that causes the tube to expand. To simplify, we can consider a square tube with inner "radius" {{< katex >}} a {{< /katex >}}. The radial force scales with the pressure over the outer and inner surfaces
+
+{{< katex display >}}
+F_r = p S_{outer} - p S_{inner} \\
+\sim p (R_0 + a) a - p (R_0 - a) a \sim 2 a^2 p
+{{< /katex >}}
+
+!!! error "Missing content here on how to balance radial forces for purely poloidal fields (axisymmetric) with a conducting wall"
+
+
+Alternatively, we can use vertical external field coils to increase the field strength near the outer wall. They will also tend to center the plasma between the coils, so they have the advantage of preventing the plasma from drifting upwards or downwards
+
+<p align="center"> <img alt="Figure 12.19" src="/r/img/12.19.png" /> </p>
+
+### Toroidal Fields
+
+Now let's consider driving a toroidal field in the plasma using a central coil on the major axis. Driving a toroidal field {{< katex >}} B_\phi {{< /katex >}} will also create a poloidal current {{< katex >}} j_\theta {{< /katex >}}. We know that the driven field will decay as {{< katex >}} 1/R {{< /katex >}} through the plasma. In actuality, the relationship is not perfectly {{< katex >}} 1/R {{< /katex >}} because of the generated poloidal current, which will modify the field within the plasma. Depending on the orientation of {{< katex >}} j_\theta {{< /katex >}}, it could either increase or decrease the poloidal field within the plasma. These are two different operations by which the plasma interacts with the magnetic field. When the induced plasma current tends to increase the field, we have a paramagnetic effect. In the opposite case (decrease) there is a diamagnetic effect. Generally, plasmas are diamagnetic, but there are certain situations where they become paramagnetic.
+
+<p align="center"> <img alt="Figure 12.20" src="/r/img/12.20.png" /> </p>
+
+Here again, we have a larger field on the in-bore side than on the out-bore side, so there will be a force imbalance tending to push the plasma towards larger radii. We might consider surrounding the plasma with a conducting wall, as we did previously, but we run into a difficulty determining a current distribution in the wall which would balance the radial effect. Any current distribution must be circular; because of the geometry of a torus such a current distribution would have the same force contribution on both the inside and outside edges. There is no way to stabilize the distribution with purely toroidal fields.
diff --git a/content/notes/UWAA558/08-1d-equilibria.md b/content/notes/UWAA558/08-1d-equilibria.md
new file mode 100644
index 00000000..cbd68a2b
--- /dev/null
+++ b/content/notes/UWAA558/08-1d-equilibria.md
@@ -0,0 +1,397 @@
+---
+title: 1-D Equilibria
+bookToc: false
+weight: 80
+---
+
+
+# 1-Dimensional Equilibria
+
+## The {{< katex >}} \theta {{< /katex >}}-pinch
+
+In a {{< katex >}} \theta {{< /katex >}} pinch, we have an applied axial field generated by a driven azimuthal current distribution. The way these usually work is that you begin with a plasma generated by some pre-ionization process and zero field. Then you crank up the current to drive an azimuthal current in the plasma (in the opposite direction as the external current).
+
+{{< katex display >}}
+\vec j \cross \vec B = \grad p \quad \rightarrow \quad j_\theta B_z = \dv{p}{r}
+{{< /katex >}}
+
+{{< katex display >}}
+j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r}
+{{< /katex >}}
+
+{{< katex display >}}
+\dv{p}{r} = - \frac{1}{\mu_0} B_z \dv{B_z}{r} = - \dv{}{r} \left( \frac{B_z ^2}{2 \mu_0} \right)
+{{< /katex >}}
+{{< katex display >}}
+p + \frac{B_z ^2}{2 \mu_0} = \text{constant} = \frac{B_0 ^2}{2 \mu_0}
+{{< /katex >}}
+
+<p align="center"> <img alt="Figure 12.21" src="/r/img/12.21.png" /> </p>
+
+At equilibrium, the magnetic pressure balances the plasma pressure. If we say that the pressure is
+
+{{< katex display >}}
+p = p_0 e^{- r^2 / a^2}
+{{< /katex >}}
+
+with {{< katex >}} p_0 {{< /katex >}} the pressure on-axis, then we can solve for the axial field
+
+{{< katex display >}}
+B_z = B_0 (1 - B_0 e^{-r^2/a^2})^{1/2}
+{{< /katex >}}
+
+We can define the peak {{< katex >}} \beta {{< /katex >}} to be the ratio of the on-axis pressure to the maximum magnetic field
+
+{{< katex display >}}
+\beta_0 = \frac{p_0}{B_0 ^2 / 2 \mu_0}
+{{< /katex >}}
+
+By definition, the peak {{< katex >}} \beta {{< /katex >}}  will always be {{< katex >}} \leq  1 {{< /katex >}}. We can define {{< katex >}} \langle \beta \rangle {{< /katex >}} 
+
+{{< katex display >}}
+\langle \beta \rangle = \frac{ \langle p \rangle }{B_a ^2 / 2 \mu_0}
+{{< /katex >}}
+
+where {{< katex >}} B_a {{< /katex >}} is a characteristic field value, typically taken to be at the plasma edge.
+
+{{< katex display >}}
+\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \, \dd r
+{{< /katex >}}
+{{< katex display >}}
+= \frac{2}{a^2} \int_0 ^a \frac{ rp}{B_0 ^2/2 \mu_0} \dd r = \frac{2}{a^2} \int_0 ^a \left( 1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r
+{{< /katex >}}
+
+In this form, we can see that because {{< katex >}} B_z {{< /katex >}} will everywhere be less than {{< katex >}} B_0 {{< /katex >}}, we can increase {{< katex >}} \langle \beta \rangle {{< /katex >}} by driving {{< katex >}} B_z {{< /katex >}} as low as possible. In this particular example, {{< katex >}} \langle \beta \rangle / \beta_0 = 63\% {{< /katex >}}. 
+
+## Z-pinch
+
+In the case of a Z-pinch, we only have an applied axial current.
+
+{{< katex display >}}
+\vec j = j_z (r) \vu z
+{{< /katex >}}
+
+For force balance
+
+{{< katex display >}}
+\vec j \cross \vec B = \grad p \quad \rightarrow \quad - j_z B_\theta = \dv{p}{r}
+{{< /katex >}}
+{{< katex display >}}
+j_z = \frac{1}{\mu_0} \frac{1}{r} \dv{}{r} ( r B_\theta) 
+{{< /katex >}}
+
+{{< katex display >}}
+- \dv{p}{r} = \frac{B_\theta}{\mu_0 r} \dv{}{r} ( r B_\theta)
+{{< /katex >}}
+
+If we find it convenient we can separate this into a magnetic pressure term
+
+{{< katex display >}}
+- \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r}
+{{< /katex >}}
+
+### Bennett Profile
+
+An example of an achievable distribution is the Bennett profile, which has a diffuse form
+
+{{< katex display >}}
+B_\theta = \frac{\mu_0 I}{2 \pi} \frac{r}{r^2 + a^2}
+{{< /katex >}}
+{{< katex display >}}
+j_z = \frac{I}{\pi} \frac{a^2}{(r^2 + a^2) ^2}
+{{< /katex >}}
+
+{{< katex display >}}
+p = \frac{\mu_0 I^2}{8 \pi ^2} \frac{a^2}{(r^2 + a^2)^2}
+{{< /katex >}}
+
+Interestingly, {{< katex >}} j \propto p {{< /katex >}}. For a uniform temperature, {{< katex >}} j \propto n {{< /katex >}}. Since current density is the product of {{< katex >}} \vec v {{< /katex >}} and {{< katex >}} n {{< /katex >}}, this says that we have a uniform drift velocity and all particles are drifting with the same velocity at all points along the profile. If we consider what the equilibrium profile looks like for a Bennett profile:
+
+<p align="center"> <img alt="Figure 12.22" src="/r/img/12.22.png" /> </p>
+
+So for {{< katex >}} r < a {{< /katex >}} we have magnetic tension and pressure which balance the plasma pressure. For {{< katex >}} r \geq a {{< /katex >}} we have magnetic tension which balances both plasma pressure and magnetic pressure.
+
+The Z-pinch {{< katex >}} \langle \beta \rangle {{< /katex >}} 
+
+{{< katex display >}}
+\langle \beta \rangle \equiv \frac{ \langle p \rangle}{B_a ^2 / 2 \mu_0} \\
+= \frac{2 \mu_0}{B_a ^2 \pi a^2} \int_0 ^a 2 \pi r p \dd r
+{{< /katex >}}
+
+If we multiply the force balance by {{< katex >}} r^2 {{< /katex >}} and integrate
+
+{{< katex display >}}
+\int_0 ^a r^2 \dv{p}{r} \dd r + \frac{1}{\mu_0} \int_0 ^a r B_\theta \dv{}{r} (r B_\theta) \dd r = 0 \\
+0 = \left[ r^2 p \right] _0 ^a - \int_0 ^a p \dd (r^2) + \left[ \frac{(r B_\theta)^2}{2 \mu_0} \right] _0 ^a
+{{< /katex >}}
+
+If we have a discrete pinch such that {{< katex >}} p(a) = 0 {{< /katex >}} then the first term vanishes.
+
+{{< katex display >}}
+\int_0 ^a 2 r p \dd r = \frac{(a B_a)^2}{2 \mu_0}
+{{< /katex >}}
+
+If we substitute our definition of {{< katex >}} \langle \beta \rangle {{< /katex >}}, we find {{< katex >}} \langle \beta \rangle = 1 {{< /katex >}}. For a diffuse pinch in which {{< katex >}} p(a) \neq 0 {{< /katex >}} we end up with {{< katex >}} \langle \beta \rangle \leq 1 {{< /katex >}} and we have a wall-supported plasma. Ideal confinement ({{< katex >}} \langle \beta \rangle = 1 {{< /katex >}}) is a very nice property and is what makes the Z-pinch configuration so interesting.
+
+
+
+## Stability Considerations
+
+Instability results if there exists a plasma displacement that leads to a lower energy state. There are several ways to provide stability in the context of MHD. The two most common are **magnetic shear** and **magnetic well**.
+
+### Magnetic Shear
+
+In ideal MHD, magnetic field lines can not break or tear. Let's consider some flux surface containing field lines {{< katex >}} \vec B_3 {{< /katex >}}. Behind it, we have another flux surface containing field lines {{< katex >}} B_2 {{< /katex >}} which are not parallel to {{< katex >}} \vec B_3 {{< /katex >}}, and the same for {{< katex >}} \vec B_1 {{< /katex >}}.
+
+<p align="center"> <img alt="Figure 12.23" src="/r/img/12.23.png" /> </p>
+
+Because the field lines are a different angles to each other, these flux surfaces can _not_ interpenetrate. In other words, if the flux surface pressures are {{< katex >}} P_1 > P_2 > P_3 {{< /katex >}}, we can maintain the pressure gradient and prevent the flux surfaces from moving each other. What prevents the surfaces from achieving a lower energy state is the magnetic shear between flux surfaces.
+
+Without shear, the surfaces can interpenetrate and exchange positions. In the case of a toroidal geometry, magnetic shear is defined by the rotational transform {{< katex >}} \iota {{< /katex >}}, or by the safety factor
+
+{{< katex display >}}
+q = \frac{2 \pi}{\iota} 
+{{< /katex >}}
+
+Generally speaking, {{< katex >}} q {{< /katex >}} is generally referenced for tokamaks and {{< katex >}} \iota {{< /katex >}} is referenced for stellarators. Another way of picturing the safety factor in a toroidal geometry is
+
+{{< katex display >}}
+q \equiv \frac{\text{no. of windings long way}}{\text{no. of windings short way}} \\
+= \frac{ \dv{\psi_t}{V}}{\dv{\phi_p}{V}} = \dv{\phi_t}{\phi_p} \\
+= \frac{n}{m} = \frac{\text{toroidal transits}}{\text{poloidal transits}}
+{{< /katex >}}
+
+In a cylindrical (1D) geometry it is just
+
+{{< katex display >}}
+q = \frac{\text{longitudinal transits}}{\text{azimuthal transits}}
+{{< /katex >}}
+
+Let's calculate the safety factor for a toroidal geometry:
+
+<p align="center"> <img alt="Figure 12.24" src="/r/img/12.24.png" /> </p>
+
+{{< katex display >}}
+\dv{\phi_p}{V} = \frac{ B_\theta 2 \pi R \dd r}{2 \pi R_0 2 \pi r \dd r} \\
+= \frac{ B_\theta}{2 \pi r} \frac{R}{R_0}
+{{< /katex >}}
+
+{{< katex display >}}
+\dv{\phi_t}{V} = \frac{B_\phi 2 \pi r \dd r}{2 \pi R_0  2 \pi r \dd r} \\
+= \frac{B_\phi}{2 \pi R_0}
+{{< /katex >}}
+
+{{< katex display >}}
+q = \frac{r B_\phi}{R B_\theta}
+{{< /katex >}}
+
+In a cylindrical geometry the analysis is even simpler
+
+{{< katex display >}}
+q = 2 \pi \frac{r B_z}{L B_\theta}
+{{< /katex >}}
+
+As a note, it would appear that {{< katex >}} q \rightarrow 0 {{< /katex >}} at the magnetic axis as {{< katex >}} r \rightarrow 0 {{< /katex >}}, but in general {{< katex >}} B_\theta \rightarrow 0 {{< /katex >}} as well, and the safety factor is generally bounded at {{< katex >}} r \rightarrow 0 {{< /katex >}}
+
+{{< katex >}} q {{< /katex >}} is a flux surface quantity.
+
+We care about magnetic shear. How does that relate to the safety factor? Magnetic shear is defined as
+
+{{< katex display >}}
+s \equiv 2 \frac{\dd q / q}{\dd V /V} = 2 \dv{\ln (q)}{\ln(V)}
+{{< /katex >}}
+
+Even a uniform {{< katex >}} B_z {{< /katex >}} or {{< katex >}} B_\theta {{< /katex >}} produces a finite magnetic shear because of the way that {{< katex >}} r {{< /katex >}} and {{< katex >}} B_\theta {{< /katex >}} change. The safety factor is often considered synonymous with magnetic shear, and often we don't even compute {{< katex >}} s {{< /katex >}}.
+
+Shear is generally a stabilizing effect. Interchange between flux surfaces can be prevented/inhibited by shear, or by making it energetically unfavorable. Shear tends to stabilize current-driven instabilities.
+
+### Magnetic Well
+
+As before, we can consider two adjacent flux surfaces {{< katex >}} B_1, P_1 {{< /katex >}} and {{< katex >}} B_2, P_2 {{< /katex >}}. If {{< katex >}} B_2 > B_1 {{< /katex >}} and {{< katex >}} P_2 > P_1 {{< /katex >}}, the interchange is energetically favorable. But if {{< katex >}} B_2 > B_1 {{< /katex >}} and {{< katex >}} P_2 < P_1 {{< /katex >}} then the interchange may be unfavorable without any magnetic shear.
+
+Consider a plasma confined by an externally applied magnetic field generated by a coil {{< katex >}} I {{< /katex >}} 
+
+<p align="center"> <img alt="Figure 12.25" src="/r/img/12.25.png" /> </p>
+
+On the left side, the magnetic field gradient is in the same direction as the plasma pressure gradient, which is a destabilizing configuration. Flux surfaces are able to interchange easily, and the magnetic field is described as having bad curvature. On the right side, the gradients are in the same direction and the magnetic field has a good curvature.
+
+We can define the "wellness" {{< katex >}} W {{< /katex >}} as
+
+{{< katex display >}}
+W \equiv \frac{ \text{total pressure change relative to mag. pressure}}{\text{relative volume change}} \\
+= \frac{\dd \langle p + B^2/2 \mu_0 \rangle / \langle B^2/2 \mu_0 \rangle}{\dd V / V}
+{{< /katex >}}
+where the angle brackets indicate a quantity integrated along a field line
+{{< katex display >}}
+\langle Q \rangle \equiv \frac{\int_0 ^L \frac{ Q \dd l}{|B|}}{\int_0 ^l \frac{\dd l}{|B|}}
+{{< /katex >}}
+
+For a stabilizing effect, the wellness must be greater than 0. This means that the magnetic pressure must increase faster than the pressure decreases to prevent pressure-driven instabilities.
+
+Since {{< katex >}} W {{< /katex >}} is evaluated along a field line, it is also a surface quantity.
+
+### Application to 1D Equilibria
+
+{{< katex >}} \theta {{< /katex >}}-pinch: Since {{< katex >}} B_\theta = 0 {{< /katex >}}, {{< katex >}} q \rightarrow \infty {{< /katex >}}, which really just means {{< katex >}} q {{< /katex >}} is not well defined for a {{< katex >}} \theta {{< /katex >}}-pinch. If we consider some small {{< katex >}} \delta B_\theta {{< /katex >}}, we get a very large {{< katex >}} q {{< /katex >}}. From a magnetic shear perspective, a {{< katex >}} \theta {{< /katex >}}-pinch has very large values of shear and very good stability properties.
+
+The wellness is
+
+{{< katex display >}}
+W = \frac{V}{\langle B^2 \rangle} \dv{}{V} \langle 2 \mu_0 p + B^2 \rangle \\
+ = \frac{\pi r^2 L}{B_z ^2} \frac{1}{2 \pi r L} \dv{}{r} (2 \mu_0 p + B_z ^2) \\
+ = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p + \frac{B_z ^2}{2 \mu_0} \right) = 0 
+{{< /katex >}}
+so a {{< katex >}} \theta {{< /katex >}}-pinch has neutral magnetic well.
+
+Vacuum case:
+
+{{< katex display >}}
+W = \frac{\mu_0 r}{ B_z ^2} \dv{}{r} \left( \frac{ B_z ^2}{2 \mu_0} \right) = 0
+{{< /katex >}}
+So vacuum magnetic fields also have neutral wellness. This leads to a general result sometimes referred to as "a plasma cannot dig its own well." In other words, by introducing plasma to a magnetic configuration, it cannot make the configuration more stable than it was. Plasmas make stability more challenging, not less.
+
+Z-pinch:
+
+Since {{< katex >}} B_z = 0 {{< /katex >}}, {{< katex >}} q = 0 {{< /katex >}} and there is no magnetic shear. Even for a small value of {{< katex >}} \delta B_z {{< /katex >}} you still get a small {{< katex >}} q {{< /katex >}}. The magnetic well properties of a Z-pinch are
+
+{{< katex display >}}
+W = \frac{V}{\langle B^2 \rangle} \dv{}{V} \langle 2 \mu_0 p + B^2 \rangle \\
+= \frac{\mu_0 r}{B_\theta ^2} \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) \\
+= \frac{\mu_0 r}{B_\theta ^2} \left( - \frac{ B_\theta ^2}{ \mu_0 r} \right) = -1
+{{< /katex >}}
+
+Recall that {{< katex >}} W > 0 {{< /katex >}} for stability, so the Z-pinch has negative magnetic well and provides no pressure stability.
+
+In summary, 
+
+ - Both {{< katex >}} \theta {{< /katex >}}- and Z-pinch have high {{< katex >}} \beta {{< /katex >}}
+ - {{< katex >}} \theta {{< /katex >}}-pinch is stable
+ - Z-pinch is unstable
+ - End losses in a {{< katex >}} \theta {{< /katex >}} pinch enormous since {{< katex >}} k_\parallel \gg k_\perp {{< /katex >}} 
+
+## Screw Pinch
+
+A natural extension is to combine a moderate toroidal field and a moderate poloidal field to produce a screw pinch configuration.
+
+<p align="center"> <img alt="Figure 12.26" src="/r/img/12.26.png" /> </p>
+
+{{< katex display >}}
+\curl \vec B = \mu_0 \vec j
+{{< /katex >}}
+{{< katex display >}}
+\rightarrow j_\theta = - \frac{1}{\mu_0 } \dv{B_z}{r} 
+{{< /katex >}}
+{{< katex display >}}
+j_z = \frac{1}{\mu_0 r} \dv{}{r} (r B_\theta)
+{{< /katex >}}
+
+For static MHD equilibrium
+
+{{< katex display >}}
+\vec j \cross \vec B = \grad p
+{{< /katex >}}
+{{< katex display >}}
+\dv{}{r} \left( p + \frac{ B_\theta ^2 + B_z ^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r}
+{{< /katex >}}
+
+We can define a toroidal {{< katex >}} \beta {{< /katex >}} where {{< katex >}} B_0 = B_z (a) {{< /katex >}} 
+{{< katex display >}}
+\beta_t = \frac{\langle p \rangle }{B_0 ^2 / 2 \mu_0} \\
+= \frac{2 \mu_0}{ B_0 ^2} \frac{1}{\pi a^2} \int _0 ^a 2 \pi r p \dd r
+{{< /katex >}}
+
+and in the poloidal direction with {{< katex >}} B_{\theta, a} = B_\theta (a) =  \frac{\mu_0 I}{2 \pi a}  {{< /katex >}} 
+
+{{< katex display >}}
+\beta_p = \frac{\langle p \rangle}{B_{\theta, a} ^2 / 2 \mu_0} \\
+= \frac{8 \pi ^2 a^2}{ \mu_0 I_0 ^2} \left( \frac{1}{\pi a^2} \int _0 ^a 2 \pi r p \dd r \right) \\
+= \frac{16 \pi ^2}{\mu_0 I_0 ^2} \int_0 ^a r p \dd r
+{{< /katex >}}
+
+To proceed we can multiply the force balance by {{< katex >}} r^2 {{< /katex >}} and integrate
+
+{{< katex display >}}
+\underbrace{\int_0 ^a r^2 \pdv{p}{r} \dd r}_{(1)} + \underbrace{\int _0 ^a \pdv{}{r} \left( \frac{ B_\theta ^2 + B_z ^2 }{2 \mu_0} \right) \dd r}_{(2)} + \underbrace{\int_0 ^a r^2 \frac{B_\theta ^2}{\mu_0 r} \dd r}_{(3)} = 0
+{{< /katex >}}
+
+{{< katex display >}}
+(1) = \int_0 ^a r^2 \dd p = \left. r^2 p \right|_0 ^a - \int_0 ^a p \dd (r^2) = - \int_0 ^a 2 r p \dd r
+{{< /katex >}}
+
+{{< katex display >}}
+(2) = \int_0 ^a r^2 \dd \left( \frac{B_\theta ^2 + B_z ^2}{2 \mu_0 } \right) \\
+= \left. r^2 \left( \frac{ B_\theta ^2 + B_z ^2}{2 \mu_0 } \right) \right|_0 ^a - \int_0 ^a r \left( \frac{ B_\theta ^2 + B_z ^2}{\mu_0} \right) \dd r
+{{< /katex >}}
+
+{{< katex display >}}
+(3) = \int_0 ^a r \frac{B_\theta ^2}{\mu_0} \dd r
+{{< /katex >}}
+
+Combining we have
+
+{{< katex display >}}
+- \int_0 ^a 2 r p \dd r + \frac{a^2 B_{\theta, a}^2}{2 \mu_0} + \overbrace{\frac{a^2 B_0 ^2}{2 \mu_0}}^{B_0 = B_z(r = a)} - \int_0 ^a r \frac{B_z ^2}{\mu_0} \dd r = 0 \\
+- \int_0 ^a 2 r p \dd r + \frac{ \mu_0 I_0 ^2}{8 \pi ^2} + \int_0 ^a r \left( \frac{ B_\theta ^2 - B_z ^2}{\mu_0} \right) \dd r = 0
+{{< /katex >}}
+
+Dividing {{< katex >}} \int_0 ^a 2 r p \dd r {{< /katex >}} gives
+
+{{< katex display >}}
+\left[ \frac{16 \pi^2}{\mu_0 I^2} \int_0 ^a r p \dd r \right] ^{-1} + \left[\frac{4 \mu_0}{B_0 ^2 a^2} \int_0 ^a r p \dd r \right] ^{-1} \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r = 1
+{{< /katex >}}
+or
+
+{{< katex display >}}
+\frac{1}{\beta_p} + \frac{\alpha_t}{\beta_t} = 1
+{{< /katex >}}
+where
+{{< katex display >}}
+\alpha_t = \frac{2}{a^2} \int_0 ^a \left( 1 - \frac{B_z^2}{B_0 ^2} \right) r \dd r
+{{< /katex >}}
+is the diamagnetism.
+
+{{< katex display >}}
+\beta _p = \left( 1 - \frac{\alpha_t}{\beta_t} \right) ^{-1}
+{{< /katex >}}
+
+If we have a diamagnetic current, then {{< katex >}} \alpha > 0 {{< /katex >}}. This maximizes confinement, since we have confinement in the azimuthal field, as well as the axial field. The limit where you have a skin current such that {{< katex >}} B_z = 0 {{< /katex >}} inside the plasma results in the best confinement and {{< katex >}} \alpha _t = 1 {{< /katex >}}.
+
+Looking at the safety factor,
+
+{{< katex display >}}
+q = \frac{2 \pi r B_z}{L B_\theta}
+{{< /katex >}}
+
+If we look at the edge {{< katex >}} r = a {{< /katex >}}, 
+{{< katex display >}}
+q_a = \frac{2 \pi a B_0}{L B_{\theta, a}}
+{{< /katex >}}
+
+As it turns out, this value of the edge safety factor is critically important, and for stability we require that {{< katex >}} q_a > 1 {{< /katex >}}.
+
+The magnetic shear of a screw pinch is
+
+{{< katex display >}}
+s = 2 \frac{V}{q} \dv{q}{V} \\
+V = \pi r^2 L \\
+\dd V = 2 \pi L r \dd r = \frac{2 V}{r} \dd r \\
+s = \frac{r}{q} \dv{q}{r}
+{{< /katex >}}
+
+The shear can be adjusted by changing the applied axial field.
+
+The magnetic well is
+
+{{< katex display >}}
+W = \frac{V}{B^2} \dv{(2 \mu_0 p + B^2)}{V} \\
+= \frac{\mu_0 r}{B^2} \dv{}{r} \left( p + \frac{B^2}{2 \mu_0} \right) \\
+= \frac{\mu_0 r}{B^2} \left( - \frac{B_\theta ^2}{\mu_0 r} \right) \\
+= - \frac{B_\theta ^2}{B^2} \\
+= - \frac{B_\theta ^2}{B_\theta ^2 + B_z ^2} \\
+= - \left( 1 + \frac{B_z ^2}{B_\theta ^2} \right) ^{-1}
+{{< /katex >}}
+
+So the well is always less than zero, but adding {{< katex >}} B_z {{< /katex >}} improves the well.
+
+By combining the properties of {{< katex >}} \theta {{< /katex >}}-pinch and Z-pinch, we are able to sacrifice some {{< katex >}} \beta {{< /katex >}}  to achieve better stability properties. Of course, we have not addressed the end losses in any way; to do that, we need to connect the ends.
+
diff --git a/content/notes/UWAA558/09-2d-equilibria.md b/content/notes/UWAA558/09-2d-equilibria.md
new file mode 100644
index 00000000..8bdb184c
--- /dev/null
+++ b/content/notes/UWAA558/09-2d-equilibria.md
@@ -0,0 +1,185 @@
+---
+title: 2D Equilibria
+bookToc: false
+weight: 90
+---
+
+
+# 2D Equilibria
+
+Let's connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:
+
+<p align="center"> <img alt="Figure 12.27" src="/r/img/12.27.png" /> </p>
+
+{{< katex display >}}
+\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\
+= \grad p = \dv{p}{r}
+{{< /katex >}}
+
+we move to an axisymmetric 2-dimensional torus, replacing our cylindrical coordinate system with a toroidal one
+
+<p align="center"> <img alt="Figure 12.28" src="/r/img/12.28.png" /> </p>
+
+{{< katex display >}}
+\vec j \cross \vec B = \vec j_\theta \cross \vec B_\phi + \vec j_\phi \cross \vec B_\theta = \grad p
+{{< /katex >}}
+
+Eventually, the toroidal force balance will lead to the Grad-Shafranov Equation, which tells us how we can solve for a general equilibrium that solves {{< katex >}} \vec j \cross \vec B = \grad p {{< /katex >}}.
+
+Let's consider how we might achieve such a configuration. A toroidal magnetic field can be achieved by driving current through a poloidal coil. A more complicated problem is how to drive toroidal current. In general this is done by means of a transformer, where the plasma itself is the secondary circuit. Driving a time-varying current through the primary induces a toroidal current through the plasma. This is called a transformer drive for current.
+
+## Grad-Shafranov equation
+
+Computing {{< katex >}} j_\theta {{< /katex >}} and {{< katex >}} B_\phi {{< /katex >}} can be computationally difficult in a toroidal geometry, so let's do some work towards simplifying our force balance expression. The **toroidal magnetic vector potential** is defined as
+
+{{< katex display >}}
+\vec B_\theta = \curl \vec A_\phi
+{{< /katex >}}
+
+If we integrate {{< katex >}} B_\theta {{< /katex >}} over a poloidal surface, Stokes' theorem gives
+
+{{< katex display >}}
+\int _{S_p} \curl \vec A_\phi \cdot \dd \vec S = \oint \vec A_\phi \cdot \dd \vec l \\
+= \int _{S_p} B_\theta \cdot \dd \vec S = \Psi _p 
+{{< /katex >}}
+
+If the equilibrium is axisymmetric, {{< katex >}} A_\phi {{< /katex >}} must be uniform along {{< katex >}} \dd l {{< /katex >}}, so
+
+{{< katex display >}}
+A_\phi \vu \phi \cdot \oint \dd \vec l = A_\phi 2 \pi R = \Psi _p \\
+\rightarrow A_\phi = \frac{ \Psi_p}{R} \vu \phi
+{{< /katex >}}
+
+where we absorb the factor of {{< katex >}} 2 \pi {{< /katex >}} into the poloidal flux {{< katex >}} \Psi _p {{< /katex >}}. After some manipulation, we can relate {{< katex >}} B_\theta {{< /katex >}} to the poloidal flux
+
+{{< katex display >}}
+\vec B_\theta = \curl \vec A_\phi = - \frac{ \vu R}{R} \pdv{\Psi}{z} + \frac{\vu z}{R} \pdv{\Psi}{R}
+{{< /katex >}}
+
+{{< katex display >}}
+\mu_0 j_\phi \cross B_\theta = (\curl \vec B_\theta) \cross \vec B_\theta \\
+= \curl \vec B_\theta \cross \left( \grad \Psi \cross \frac{\vu \phi}{R} \right) \\
+= - \left[ \pdv{}{R} \left( \frac{1}{R} \pdv{\Psi}{R} \right) + \frac{1}{R} \pdv{\Psi ^2}{z^2} \right] \cdot \left[\frac{ \grad \Psi}{R} ( \vu \phi \cdot \vu \phi) - \frac{ \phi}{R} \cancel{(\grad \Psi \cdot \vu \phi)} \right]
+{{< /katex >}}
+
+That gives the first component of {{< katex >}} \grad p {{< /katex >}}, now let's do the other one
+
+{{< katex display >}}
+\mu_0 \vec j_\theta \cross \vec B_\phi = ( \curl \vec B_\phi) \cross \vec B_\phi \\
+ = \left[ - \vu R \pdv{B_\phi}{z} + \vu z \frac{1}{R} \pdv{}{R} ( R B_\phi) \right] \cross \vec B_\phi \\
+ = - \frac{B_\phi}{R} \left[ \vu R \pdv{}{R} (R B_\phi) + \vu z \pdv{}{z} (R B_\phi) \right] \\
+ = - \frac{B_\phi}{R} \grad (R B_\phi)
+{{< /katex >}}
+
+Finally, since pressure is a flux surface quantity we can write
+
+{{< katex display >}}
+\grad p = \dv{p}{\Psi} \grad \Psi = p' \grad \Psi
+{{< /katex >}}
+
+The toroidal force balance now looks like
+
+{{< katex display >}}
+\mu_0 p' \grad \Psi = - \frac{1}{R} \left( \pdv{}{R} \frac{1}{R} \pdv{\Psi}{R} + \frac{1}{R} \pdv{^2 \Psi}{z^2} \right) \grad \Psi - \frac{B_\phi}{R} \grad(R B_\phi)
+{{< /katex >}}
+
+We notice that the only vector quantities here are {{< katex >}} \grad \Psi {{< /katex >}} and {{< katex >}} \grad (R B_\phi) {{< /katex >}}, so {{< katex >}} \grad (R B_\phi) {{< /katex >}} must be parallel to {{< katex >}} \grad \Psi {{< /katex >}} and is a flux surface quantity. We can define our new flux surface quantity as
+
+{{< katex display >}}
+F(\Phi) \equiv R B_\phi = \frac{\mu_0 I_\theta}{2 \pi} = \frac{\mu_0}{2 \pi} \int_{S_p} \vec j_\theta \cdot \dd \vec S
+{{< /katex >}}
+
+{{< katex display >}}
+\grad F = \dv{F}{\Psi} \grad \Psi = F' \grad \Psi
+{{< /katex >}}
+
+Now each term in the toroidal force balance has a factor of {{< katex >}} \grad \Psi {{< /katex >}} attached. Let's multiply through by {{< katex >}} R^2 {{< /katex >}} and factor out the gradient to arrive at the **Grad-Shafranov equation**:
+
+{{< katex display >}}
+R^2 \mu_0 p' = - \Delta ^\star \Psi - F F'
+{{< /katex >}}
+
+where
+{{< katex display >}}
+\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} + \pdv{^2}{z^2}
+{{< /katex >}}
+
+To solve the Grad-Shafranov equation, you solve for {{< katex >}} \Psi(R, z) {{< /katex >}}, which determines {{< katex >}} p(\Psi) {{< /katex >}} and {{< katex >}} F(\Psi) {{< /katex >}}, which directly gives you {{< katex >}} p(R, z) {{< /katex >}} and {{< katex >}} F(R, z) {{< /katex >}} and completely defines the equilibrium.
+
+You can solve for the other terms as well. Since {{< katex >}} \vec B_\theta = \frac{\grad \Psi}{R} \cross \vu \phi {{< /katex >}} 
+
+{{< katex display >}}
+\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \Psi \vu \phi
+{{< /katex >}}
+
+and since {{< katex >}} \vec B_\phi = \frac{F}{R} \vu \phi {{< /katex >}} 
+
+{{< katex display >}}
+\vec j_\theta = - \frac{1}{\mu_0 R} \grad (R B_\phi) \cross \vu \phi
+{{< /katex >}}
+
+For the G-S equation to be solvable, you need to specify the equilibrium by specifying {{< katex >}} p(\Phi) {{< /katex >}} and {{< katex >}} F(\Phi) {{< /katex >}}. In practice, this is usually done by making experimental measurements to determine {{< katex >}} p {{< /katex >}} and {{< katex >}} F {{< /katex >}}. A common code that does this is called EFIT, which takes the boundary conditions of the magnetic field and measurements of temperature, density to perform a least-squares fit to solve the G-S equation.
+
+In general, the Grad-Shafranov equation leads to a matrix equation 
+
+{{< katex display >}}
+\overline \vec A \vec \Psi + \vec f(\Psi) = \vec g
+{{< /katex >}}
+
+Depending on the conditions we place on {{< katex >}} \Psi {{< /katex >}}, {{< katex >}} \vec f(\Psi) {{< /katex >}} can be a nonlinear function.
+
+## Solutions to the Grad-Shafranov equation
+
+In the limit that {{< katex >}} \vec j \parallel \vec B {{< /katex >}}, then {{< katex >}} \vec j \cross \vec B = 0 = \grad p \rightarrow p' = 0 {{< /katex >}}. These are called force-free states. In the G-S equation, the pressure term vanishes and we're left with 
+
+{{< katex display >}}
+\Delta ^\star \Psi + F F' = 0
+{{< /katex >}}
+
+Spheromaks and RFPs are examples of nearly force-free states in which the current is nearly parallel to the magnetic field. Notice that in completely force-free states, {{< katex >}} \langle \beta \rangle = 0 {{< /katex >}}.
+
+Another interesting limit is the case where {{< katex >}} F F' \gg \Delta ^\star \Psi {{< /katex >}}. Now we have
+{{< katex display >}}
+\grad p \approx \vec j_\theta \cross \vec B_\phi
+{{< /katex >}}
+which looks like a {{< katex >}} \theta {{< /katex >}}-pinch which has been connected at the ends. Remember from the previous section that we can not maintain radial force balance with purely toroidal fields, so the toroidal current is not zero (hence the {{< katex >}} \approx {{< /katex >}}) but is just high enough to maintain radial force balance. This sort of configuration is called a high-{{< katex >}} \beta {{< /katex >}} tokamak.
+
+The other limit is {{< katex >}} F F' \ll \Delta ^\star \Psi {{< /katex >}} 
+{{< katex display >}}
+\grad p \approx \vec j_\phi \cross \vec B_\theta
+{{< /katex >}}
+which looks like an end-connected z-pinch. This configuration is usually called an Ohmically heated Tokamak, and the majority of currently operating tokamaks operate this way. As we know, a purely poloidal field has very bad stability properties, so {{< katex >}} \vec B_\phi {{< /katex >}} needs to be added to provide stability. The toroidal {{< katex >}} \beta {{< /katex >}} is very small
+{{< katex display >}}
+\beta _t \ll 1 \qquad \beta _p \approx 1
+{{< /katex >}}
+
+## Stability Considerations
+
+The same stability factors exist in 2D equilibria that we found for 1D equilibria:
+
+Magnetic shear - the safety factor {{< katex >}} q(\Psi) = \frac{\Delta \phi}{\Delta \theta} {{< /katex >}} for {{< katex >}} \Delta \theta = 2 \pi {{< /katex >}}. We can calculate {{< katex >}} q {{< /katex >}} more easily by integrating along a flux surface in the poloidal plane:
+
+{{< katex display >}}
+q(\Psi) = \frac{1}{2 \pi} \int_0 ^{\Delta \phi} \dd \phi \\
+ = \frac{1}{2 \pi} \int _0 ^{2 \pi} \dv{\phi}{\theta} \dd \theta \\
+ = \frac{1}{2 \pi} \int_0 ^{2 \pi} \dd \theta \left. \frac{r B_\phi}{R B_\theta} \right|_{\text{along flux surf.}}  \\
+ = \frac{F(\Psi)}{2 \pi} \oint \frac{ \dd l_p}{R^2 B_\theta} \qquad \dd l_p = r \dd \theta
+{{< /katex >}}
+
+Magnetic well: similarly we can get the magnetic well factor by integrating around a flux surface in the poloidal plane
+
+{{< katex display >}}
+\langle Q \rangle = \frac{\oint \frac{Q \dd l_p}{B_\theta}}{\oint \frac{ \dd l_p}{B_\theta}}
+{{< /katex >}}
+
+## Shafranov Shift
+
+Remember that when we had an equilibrium which had a toroidal current and a corresponding poloidal magnetic field, and a poloidal magnetic field, then radial force balance will tend to shift the configuration outwards away from the major axis and a conducting wall or external coil will be required to maintain the equilibrium. The radial force balance is really achieved by {{< katex >}} \vec j_\phi \cross B_p {{< /katex >}} 
+
+As we move towards the magnetic axis, {{< katex >}} B_p \rightarrow 0 {{< /katex >}} by definition. With less poloidal field to balance the radial force imbalance, there is more radial expansion. This means that inner portion of the plasma (inner flux surfaces) must shift radially further to achieve radial force balance.
+
+<p align="center"> <img alt="Figure 12.29" src="/r/img/12.29.png" /> </p>
+
+The shift increases with plasma pressure. This effect is further enhanced if we have low poloidal fields, for example in the high-{{< katex >}} \beta {{< /katex >}} tokamak configurations.
+
+Low aspect ratios also enhance the effect. Recall that the radial force imbalance increases with smaller aspect ratio, leading to a larger shift.
\ No newline at end of file
diff --git a/content/notes/UWAA558/10-equilibrium-of-3d-configurations.md b/content/notes/UWAA558/10-equilibrium-of-3d-configurations.md
new file mode 100644
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--- /dev/null
+++ b/content/notes/UWAA558/10-equilibrium-of-3d-configurations.md
@@ -0,0 +1,45 @@
+---
+title: Equilibrium of 3D Configurations
+bookToc: false
+weight: 100
+---
+
+
+# Equilibrium of 3D Configurations
+
+In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated.
+
+Some general features of 3D equilibria are:
+
+ 1. No net toroidal current. This means that they tend to be steady-state configurations.
+ 2. Radial confinement is accomplished by toroidal fields, as in the end-connected {{< katex >}} \theta {{< /katex >}}-pinch. As we saw, toroidal fields cannot provide radial confinement in a purely axisymmetric configuration, but radial variation with {{< katex >}} \phi {{< /katex >}} _can_ provide confinement.
+ 3. Toroidal effect (radial force balance) is generated by helical magnetic fields. You can do an expansion of the magnetic field into a toroidal component, and a helical component that traces out a twisted shape as you move around the torus. These twisted shapes are what lead to radial confinement.
+
+
+## ELMO Bumpy Torus (EBT)
+
+In contrast to most other 3D configurations, even though the EBT is a 3D equilibrium, it has no helical windings.
+
+<p align="center"> <img alt="Figure 12.30" src="/r/img/12.30.png" /> </p>
+
+Since there are no helical windings, we have to provide radial stability in another way. In the EBT configuration, you also drive hot poloidal electron rings (driven by electron cyclotron resonance) to provide both stability and heating.
+
+## Stellarator
+
+The stellarator configuration is composed of a number of helical current lines (generated by helical coils with _alternating_ currents), and a net toroidal field driven by poloidal coils. The direction of the currents alternate, for a total of {{< katex >}} 2l {{< /katex >}} current lines.
+
+<p align="center"> <img alt="Figure 12.31" src="/r/img/12.31.png" /> </p>
+
+The result is a net magnetic field with a ratio such that {{< katex >}} B_{\text{helical}} \gg B_\phi {{< /katex >}} 
+
+Stellarators raise some very complicated engineering challenges both in the design and construction of the complicated geometry. It is also very difficult to maintain no net current within the plasma, especially during start-up. As you add plasma, you raise from zero {{< katex >}} \beta {{< /katex >}} to a finite {{< katex >}} \beta {{< /katex >}}, introducing things like bootstrap currents that need to be balanced.
+
+## Torsatron
+
+Similar to a stellarator, the torsatron does not have alternating currents. All of the helical current lines are in the same direction. There are also no toroidal field coils.
+
+The engineering is slightly simpler, but it is slightly less efficient at generating the helical magnetic field.
+
+The flux surfaces in stellarators and torsatrons have geometrical cross-sections depending on the number {{< katex >}} l {{< /katex >}} of helical current lines. About the current lines, the flux surfaces are nearly circular. The flux surfaces within the plasma volume are determined by the separatrix of the helical coil fields.
+
+<p align="center"> <img alt="Figure 12.32" src="/r/img/12.32.png" /> </p>
diff --git a/content/notes/UWAA558/11-mhd-stability.md b/content/notes/UWAA558/11-mhd-stability.md
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,185 @@
+---
+title: MHD Stability
+bookToc: false
+weight: 110
+---
+
+
+# MHD Stability
+
+Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear. We can define the initial deviation from equilibrium to be a linear phenomenon. As usual, we perform this linearization by letting {{< katex >}} Q(r, t) = Q_0 + Q_1(r, t) {{< /katex >}} with {{< katex >}} Q_1 {{< /katex >}} being a small first-order perturbation. Since the equilibrium is both time and space independent, the general form of the perturbation is
+
+{{< katex display >}}
+Q_1(r, t) = \vu Q_1 e^{-i (\omega t - \vec k \cdot \vec r)}
+{{< /katex >}}
+
+{{< katex display >}}
+\grad p_0 = \vec j_0 \cross \vec B_0
+{{< /katex >}}
+{{< katex display >}}
+p = p_0 + p_1 \qquad \rho = \rho_0 + \rho_1
+{{< /katex >}}
+{{< katex display >}}
+\vec j  = \vec j_0 + \vec j_1 \qquad \vec B = \vec B_0 + \vec B_1
+{{< /katex >}}
+
+and for a static equilibrium
+{{< katex display >}}
+\vec v = \vec v_1
+{{< /katex >}}
+
+In our momentum equations of the perturbed quantities, we assume that the static equilibrium holds, so most of the equilibrium terms drop out. We can define a velocity displacement {{< katex >}} \vec \xi = \int_0 ^t \vec v_1 \dd t {{< /katex >}}. As we integrate the field and pressure in time,
+
+{{< katex display >}}
+\pdv{B_1}{t} = \curl (\vec v_1 \cross \vec B_0)
+{{< /katex >}}
+{{< katex display >}}
+\int \pdv{B_1}{t} = \vec B_1 = \curl \int \vec v_1 \cross \vec B_0 \dd t \\
+ = \curl (\vec \xi \cross \vec B_0)
+{{< /katex >}}
+
+If we do the same for the pressure equation, we get
+
+{{< katex display >}}
+p_1 = - \vec \xi \cdot \grad p_0 - \Gamma p_0 \div \vec \xi
+{{< /katex >}}
+where {{< katex >}} \Gamma {{< /katex >}} is the ratio of specific heats, to avoid confusion with typical perturbation growth rate {{< katex >}} \gamma {{< /katex >}}.
+
+If we combine all of these together, substituting into the momentum equation, we can express the perturbation entirely in terms of {{< katex >}} \vec \xi {{< /katex >}} and the equilibrium properties:
+
+{{< katex display >}}
+\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \grad (\vec \xi \cdot \grad p_0 + \Gamma p_0 \div \vec \xi)\\
+ + \frac{1}{\mu_0} \left[(\curl \vec B_0) \cross \curl (\vec \xi \cross \vec B_0) \right] \\
+ + \frac{1}{\mu_0} \left[ \curl \curl ( \vec \xi \cross \vec B_0) \cross \vec B_0 \right]
+{{< /katex >}}
+
+We define the right-hand-side as the linearized forcing function of our equilibrium
+{{< katex display >}}
+\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \vec F(\vec \xi _i , p_0, \vec B_0)
+{{< /katex >}}
+
+For a linear force function, we can also write it in terms of a spring constant tensor
+
+{{< katex display >}}
+\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \vec F(\vec \xi) = - \overline \vec K \cdot \vec \xi
+{{< /katex >}}
+
+We can determine the stability behavior of a configuration by specifying an initial condition 
+
+{{< katex display >}}
+\vec \xi (t = 0) = 0 \qquad \text{and} \qquad \left. \pdv{\xi}{t} \right| _{t = 0} = f(\vec r)
+{{< /katex >}}
+and boundary conditions. A boundary condition may be a rigid wall
+{{< katex display >}}
+\vec \xi \cdot \vu n |_{wall} = 0
+{{< /katex >}}
+
+One way we can tell whether a given solution is unstable is to assume a variation of the form
+{{< katex display >}}
+\vec \xi \propto e^{-i \omega t}
+{{< /katex >}}
+If {{< katex >}} \omega^2 > 0 {{< /katex >}}, the displacement will oscillate in time without growth, and if {{< katex >}} \omega^2 < 0 {{< /katex >}} then the displacement will grow. In other words, if {{< katex >}} \omega {{< /katex >}} is real, then the mode is stable, and if {{< katex >}} \omega {{< /katex >}} is imaginary then the mode is unstable. The eigenvalue equation to be solved is
+
+{{< katex display >}}
+- \omega ^2 \rho_0 \vec \xi = \vec F(\vec \xi)
+{{< /katex >}}
+which we can write as a matrix equation
+{{< katex display >}}
+\overline \vec A\, \overline X = \lambda \overline X
+{{< /katex >}}
+{{< katex display >}}
+\frac{1}{\rho_0} \vec F (\vec \xi) = - \omega ^2 \vec \xi
+{{< /katex >}}
+
+For any arbitrary linear forcing function, we might get an infinite number of eigenvalues. How do we know which ones to look at? It turns out that the linearized force function {{< katex >}} \vec F(\vec \xi) {{< /katex >}} has the property of being self-adjoint, so
+{{< katex display >}}
+\int \vec \eta \cdot \vec F(\vec \xi) \dd V = \int \xi \cdot \vec F( \vec \eta) \dd V
+{{< /katex >}}
+where {{< katex >}} \vec \eta {{< /katex >}} and {{< katex >}} \vec \xi {{< /katex >}} are arbitrary displacements that satisfy the same boundary conditions. If {{< katex >}} \vec F {{< /katex >}} is self-adjoint, then the system is Hermitian, which guarantees that we get real eigenvalues ({{< katex >}} \omega^2 {{< /katex >}}) , orthogonal eigenfunctions, and most importantly we are guaranteed to have an ordered spectrum of eigenvalues. That is to say {{< katex >}} \omega_0 ^2 < \omega _1 ^2 < \omega _2 ^2 < \ldots {{< /katex >}}. This means that the eigenvalue of the lowest mode is guaranteed to be the most negative, and therefore dictates the stability of the system. If the lowest eigenvalue is negative, then the system is necessarily unstable, and if the lowest eigenvalue is positive, then we are guaranteed that all modes are stable.
+
+Because {{< katex >}} \vec F {{< /katex >}} is self-adjoint, we can make use of the energy principle to write the variation in the sum of the kinetic and potential energy as:
+
+{{< katex display >}}
+0 = \dv{}{t} \left[ \frac{1}{2} \int \rho_0 \left( \pdv{\vec \xi}{t} \right) ^2 \dd V - \frac{1}{2} \int \vec \xi \cdot \vec F ( \vec \xi) \dd V \right]
+{{< /katex >}}
+The kinetic energy term will always be positive, so we can formulate the stability based on the potential energy, often called a {{< katex >}} \delta W {{< /katex >}} approach
+{{< katex display >}}
+\delta W = - \frac{1}{2} \int \vec \xi \cdot \vec F ( \vec \xi) \dd V
+{{< /katex >}}
+is the change in potential energy due to a displacement {{< katex >}} \xi {{< /katex >}}. If the potential energy decreases due to a displacement {{< katex >}} \xi {{< /katex >}}, then the kinetic energy must necessarily increase, so {{< katex >}} \delta W < 0 {{< /katex >}} indicates instability.
+
+We can write the change in kinetic energy for our normal mode decomposition as
+{{< katex display >}}
+\delta T = \frac{1}{2} \int \rho _0 \left( \pdv{\xi}{t} \right) ^2 \dd V = - \frac{1}{2} \omega ^2 \int \rho _0 \vec \xi ^ \star \cdot \vec \xi \dd V \\
+= - \delta W = \frac{1}{2} \int \vec \xi ^\star \cdot \vec F(\vec \xi) \dd V
+{{< /katex >}}
+{{< katex display >}}
+\omega^2 = \frac{- \int \xi ^\star \cdot \vec F \dd V}{\int \rho_0 \xi ^\star \cdot \xi \dd V} = \frac{\delta W}{\frac{1}{2} \int \rho_0 \xi ^\star \cdot \xi \dd V}
+{{< /katex >}}
+
+The denominator is strictly positive, so the sign of {{< katex >}} \omega^2 {{< /katex >}} is determined by the sign of {{< katex >}} \delta W {{< /katex >}} 
+
+{{< katex display >}}
+\delta W < 0 \rightarrow \omega^2 < 0 \rightarrow  \text{unstable} \\
+\delta W > 0 \rightarrow \omega^2 > 0 \rightarrow \text{stable}
+{{< /katex >}}
+
+Analyzing the form of {{< katex >}} \delta W {{< /katex >}} (within the plasma volume)
+
+{{< katex display >}}
+\delta W = \frac{1}{2} \int_{plasma} \dd V \Gamma p_0(\div \vec \xi) ^2 + \vec \xi \cdot \grad p_0 (\div \vec \xi) \qquad \qquad  \\
+\qquad \qquad + \frac{1}{\mu_0} \left[ \curl ( \vec \xi \cross \vec B_0) \right]^2 \\
+\qquad \qquad - \frac{1}{\mu_0} \left[\vec \xi \cross ( \curl \vec B_0) \right] \cdot \left[ \curl ( \vec \xi \cross \vec B_0) \right]
+{{< /katex >}}
+
+Generally speaking, the plasma volume does not extend to infinity, and we care very much about the boundary. The total {{< katex >}} \delta W {{< /katex >}} is the sum of that in the plasma volume {{< katex >}} \delta W_F {{< /katex >}}, the surface {{< katex >}} \delta W_S {{< /katex >}}, and the vacuum region {{< katex >}} \delta W_V {{< /katex >}}. The vacuum term looks like
+
+{{< katex display >}}
+\delta W_V = \frac{1}{2} \int _{vac} \dd V \frac{ (\curl ( \vec \xi \cross \vec B_0))^2}{\mu_0} = \int_{vac} \dd V \frac{\vec B_1 ^2}{\mu_0} > 0
+{{< /katex >}}
+
+so the vacuum term is always positive, and has a stabilizing influence. The surface contribution offsets this
+
+{{< katex display >}}
+\delta W_S = \frac{1}{2} \oint \dd S ( \vu n \cdot \vec \xi) ^2 \left[ \left[ \grad \left( p_0 + \frac{B_0^2}{2 \mu_0} \right) \right] \right] \cdot \vu n
+{{< /katex >}}
+
+Instabilities can be characterized as:
+
+- Internal/fixed boundary {{< katex >}} \delta W = \delta W_F {{< /katex >}} 
+- External/free boundary {{< katex >}} \delta W = \delta W_F + \delta W_S + \delta W_V {{< /katex >}} 
+
+The plasma portion can be re-written slightly as
+
+{{< katex display >}}
+\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\
+ + \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0}  - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\
+   + \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\
+   + \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1  \quad \leftarrow \text{Current-driven (kink)}  \\
+- 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)}
+{{< /katex >}}
+
+where {{< katex >}} \vec \kappa {{< /katex >}} is the curvature vector {{< katex >}} \vu e_B \cdot \grad \vu e_B {{< /katex >}}. If we look at each of these terms, the first three terms are all going to be stabilizing effects, which means that all instability is going to come from the last two terms, the current-driven instability term and the pressure-driven instability term.
+
+Going back to the screw pinch, 
+{{< katex display >}}
+\dv{p}{r} = j_\theta B_z - j_z B_\theta
+{{< /katex >}}
+we have current in the same direction as magnetic field ({{< katex >}} j_\theta {{< /katex >}} with {{< katex >}} B_\theta {{< /katex >}} and {{< katex >}} j_z {{< /katex >}} with {{< katex >}} B_z {{< /katex >}}), so kink instabilities are possible. We also have a pressure gradient, so interchange instabilities are also possible.
+
+As a concrete example, look at the pressure driven instability term in a Z-pinch. 
+{{< katex display >}}
+\kappa = - \frac{ \vu r}{r}
+{{< /katex >}}
+{{< katex display >}}
+\vec \xi = \xi _r \vu r
+{{< /katex >}}
+{{< katex display >}}
+\delta W_{F, pressure} = \int \dd V \xi _r \dv{p_0}{r} \frac{\xi_r}{r} \\
+ = \int \dd V \frac{2 \xi _r ^2}{r} \dv{p_0}{r}
+{{< /katex >}}
+
+In a Z-pinch, it is always the case that {{< katex >}} \dv{p_0}{r} < 0 {{< /katex >}}. As shown by Kadomtsev (1965) it turns out that these modes can be stabilized by adding {{< katex >}} B_z {{< /katex >}}, but this also introduces kink modes.
+
+Going back to our stabilizing quantities of wellness and shear, current-driven instabilities are generally managed through shear, and pressure-driven instabilities are stabilized by well.
+
diff --git a/content/notes/UWAA558/_index.md b/content/notes/UWAA558/_index.md
new file mode 100644
index 00000000..31dae7d1
--- /dev/null
+++ b/content/notes/UWAA558/_index.md
@@ -0,0 +1,5 @@
+---
+bookFlatSection: false
+bookCollapseSection: true
+title: MHD Theory
+---
diff --git a/content/notes/UWAA558/formulary.md b/content/notes/UWAA558/formulary.md
new file mode 100644
index 00000000..b296c76e
--- /dev/null
+++ b/content/notes/UWAA558/formulary.md
@@ -0,0 +1,464 @@
+---
+title: Formulary
+weight: 999
+bookToc: false
+---
+
+{{< katex >}} {{< /katex >}}
+
+# Formulary
+
+{{< hint info >}}
+**"Kinetic Description"**
+
+
+{{< katex display >}}
+\dv{\vec v}{t} = \frac{q_i}{m_i} (\vec E + \vec v_i \cross \vec B) + \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} \delta(\vec r_i - \vec r_j) \right]
+{{< /katex >}}
+
+{{< katex display >}}
+\pdv{\vec B}{t} = - \curl \vec E
+{{< /katex >}}
+
+{{< katex display >}}
+\frac{1}{c^2} \pdv{\vec E}{t} = \curl \vec B - \mu_0 \sum_i q_i \vec v_i \delta (\vec r - \vec r_i)
+{{< /katex >}}
+
+{{< katex display >}}
+\div \vec{B} = 0
+{{< /katex >}}
+
+{{< katex display >}}
+\div \vec E = \frac{1}{\epsilon_0} \sum_i q_i \delta (\vec r - \vec r_i)
+{{< /katex >}}
+
+Klimontovich equation:
+
+{{< katex display >}}
+\dv{N}{t} = 0 = \pdv{N}{t} + \pdv{}{q_i} \cdot (\dot{q_i} N) \\
+N \equiv \sum_i \delta (p - p_i) \delta(q - q_i)
+{{< /katex >}}
+
+{{< /hint >}}
+
+{{< hint info >}}
+**"Plasma Fluid Description"**
+    
+Boltzmann Equation
+
+{{< katex display >}}
+\pdv{f_\alpha}{t} + \vec v \cdot \pdv{f_\alpha}{t} + \frac{q_\alpha}{m_\alpha} (\vec E + \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_{\beta \neq \alpha} C_{\alpha \beta}
+{{< /katex >}}
+
+Maxwellian distribution:
+
+{{< katex display >}}
+f_\alpha (\vec v) = n_\alpha \left( \frac{m_\alpha}{2 \pi T} \right)^{3/2} e^{- \frac{m_\alpha(\vec v - \vec v_\alpha)^2}{2T}}
+{{< /katex >}}
+Moments of fluid model (moments of distribution \\( \rightarrow \\) moments of Boltzmann equation:
+
+{{< katex display >}}
+\text{Continuity:} \qquad n_\alpha = \int f_\alpha \dd \vec v \\
+\quad \rightarrow \pdv{n_\alpha}{t} + \div (n_\alpha \vec v_\alpha) = 0
+{{< /katex >}}
+
+{{< katex display >}}
+\text{Momentum:} \qquad n_\alpha \vec v_\alpha = \int \vec v f_\alpha \dd v \\
+\quad \rightarrow \quad \pdv{}{t} (n_\alpha \vec v_\alpha ) + \div (n_\alpha \vec v_\alpha \vec v_\alpha) + \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E + \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \rho_\alpha \left(\pdv{\vec v_\alpha}{t} + \vec v_\alpha \cdot \grad \vec v_\alpha \right) + \grad \vec P_\alpha + \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E + \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}
+{{< /katex >}}
+
+{{< katex display >}}
+\text{Energy:} \qquad \int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \quad \frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} + \vec v_\alpha \cdot \grad T_\alpha \right) + P_\alpha \div \vec v_\alpha + \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha + \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}
+{{< /katex >}}
+Closure relations
+
+{{< katex display >}}
+\vec h_\alpha = - \kappa \grad T_\alpha
+{{< /katex >}}
+
+{{< katex display >}}
+\overline \Pi_ \alpha = \nu \grad \vec v_\alpha
+{{< /katex >}}
+
+{{< /hint >}}
+
+## Ideal MHD
+
+{{< hint info >}}
+**"Ideal MHD"**
+
+Continuity:
+
+{{< katex display >}}
+\pdv{\rho}{t} + \div (\rho \vec v) = 0
+{{< /katex >}}
+Momentum:
+
+{{< katex display >}}
+\rho \left( \pdv{\vec v}{t} + \vec v \cdot \grad \vec v \right) + \grad p - \vec j \cross \vec B = 0
+{{< /katex >}}
+Generalized Ohm's Law
+
+{{< katex display >}}
+\vec E + \vec v \cross \vec B = \frac{1}{Zen}\cancel{(\vec j \cross \vec B - \grad p_e)} = 0
+{{< /katex >}}
+Energy
+
+{{< katex display >}}
+\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = 0
+{{< /katex >}}
+
+{{< /hint >}}
+
+{{< hint info >}}
+"Lawson Criterion"
+
+
+{{< katex display >}}
+n \tau_E > 10^4 s/m^3
+{{< /katex >}}
+
+{{< /hint >}}
+
+{{< hint info >}}
+**"Conservation Law Form of Ideal MHD"**
+
+Continuity:
+
+{{< katex display >}}
+\pdv{\rho}{t} + \div (\rho \vec v) = 0
+{{< /katex >}}
+Momentum:
+
+{{< katex display >}}
+\pdv{(\rho \vec v)}{t} + \div \left[ \rho \vec v \vec v - \frac{\vec B \vec B}{\mu_0} + \left( p + \frac{B^2}{2 \mu_0} \right) \overline{I} \right] = 0
+{{< /katex >}}
+Energy:
+
+{{< katex display >}}
+\pdv{\epsilon}{t} + \div \left[ \left( \epsilon + p + \frac{B^2}{2 \mu_0} \right) \vec v - (\vec B \cdot \vec v) \frac{\vec B}{\mu_0} \right] = 0
+{{< /katex >}}
+
+{{< katex display >}}
+\pdv{\vec B}{t} + \div ( \vec v \vec B - \vec B \vec v) = 0
+{{< /katex >}}
+where
+
+{{< katex display >}}
+\epsilon = \frac{1}{\gamma - 1} p + \frac{1}{2} \rho v^2 + \frac{B^2}{2\mu_0}
+{{< /katex >}}
+
+Static Equilibrium:
+
+{{< katex display >}}
+\vec j \cross \vec B = \grad p
+{{< /katex >}}
+
+{{< katex display >}}
+\frac{B^2}{\mu_0} \vec K = \grad_\perp (p + \frac{B^2}{2 \mu_0})
+{{< /katex >}}
+
+{{< katex display >}}
+\vec K \equiv \frac{\vec B}{|B|} \cdot \grad \frac{ \vec B}{|B|}
+{{< /katex >}}
+
+Conservation of flux:
+
+{{< katex display >}}
+\vec E + \vec v \cross \vec B = 0
+{{< /katex >}}
+
+{{< katex display >}}
+\pdv{\vec B}{t} = - \curl \vec E
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v
+{{< /katex >}}
+
+{{< /hint >}}
+
+## 1D Equilibria
+
+{{< hint info >}}
+**"\\( \theta \\)-pinch"**
+
+
+{{< katex display >}}
+B_\theta = 0
+{{< /katex >}}
+
+{{< katex display >}}
+j_\theta B_z = \dv{p}{r}
+{{< /katex >}}
+
+{{< katex display >}}
+j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r}
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow p + \frac{B_z ^2}{2 \mu_0} = \frac{B_0 ^2}{2 \mu_0}
+{{< /katex >}}
+
+{{< katex display >}}
+\langle \beta \rangle = \frac{2}{a^2} \int_0 ^a \frac{r p}{B_0 ^2 / 2 \mu_0} \dd r
+{{< /katex >}}
+
+{{< katex display >}}
+q = \infty
+{{< /katex >}}
+
+{{< katex display >}}
+W = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p + \frac{B_z ^2}{2 \mu_0} \right) = 0
+{{< /katex >}}
+
+{{< /hint >}}
+
+{{< hint info >}}
+**"Z-pinch"**
+
+
+{{< katex display >}}
+B_z =0
+{{< /katex >}}
+
+{{< katex display >}}
+\grad p = \dv{p}{r} = - j_z B_\theta
+{{< /katex >}}
+
+{{< katex display >}}
+- \dv{}{r} \left( p + \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r}
+{{< /katex >}}
+
+{{< katex display >}}
+\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \dd r = 1 \quad \text{ if } \quad p(a) = 0
+{{< /katex >}}
+
+{{< katex display >}}
+q = S = 0
+{{< /katex >}}
+
+{{< katex display >}}
+W = 1
+{{< /katex >}}
+
+{{< /hint >}}
+
+{{< hint info >}}
+**"Screw pinch"**
+
+
+{{< katex display >}}
+\dv{}{r} \left( p + \frac{B^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r}
+{{< /katex >}}
+
+{{< katex display >}}
+\beta_t = \frac{2 \mu_0}{B_z (a) ^2} \left( \frac{1}{\pi a^2} \int_0 ^a 2 \pi r p \dd r \right)
+{{< /katex >}}
+
+{{< katex display >}}
+\beta_p = \left( 1 - \frac{\alpha_t}{\beta _t} \right)^{-1} \qquad \alpha_t \equiv \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r
+{{< /katex >}}
+
+{{< katex display >}}
+q = \frac{2 \pi r B_z}{L B_\theta}
+{{< /katex >}}
+
+{{< katex display >}}
+q_a = \frac{4 \pi ^2 a^2 B_0}{\mu_0 I_a}
+{{< /katex >}}
+
+{{< katex display >}}
+S = \frac{r}{q} \dv{q}{r}
+{{< /katex >}}
+
+{{< katex display >}}
+W = - \frac{B_\theta ^2}{B_\theta ^2 + B_z ^2}
+{{< /katex >}}
+
+{{< /hint >}}
+
+{{< hint info >}}
+**"Stability"**
+
+Shear:
+
+{{< katex display >}}
+S = 2 \frac{ dq / q}{dV / V} = 2 \frac{d \ln q}{d \ln V}
+{{< /katex >}}
+
+{{< katex display >}}
+q = \frac{\text{\# long windings}}{\text{\# short windings}} = \dv{\psi_t}{\psi_p}
+{{< /katex >}}
+Shear for toroid
+
+{{< katex display >}}
+q = \frac{r B_\phi}{R B_\theta}
+{{< /katex >}}
+Shear for cylinder
+
+{{< katex display >}}
+q = 2 \pi \frac{r B_z}{L B_\theta}
+{{< /katex >}}
+Well
+
+{{< katex display >}}
+W = \frac{ d \langle p + B^2 / 2 \mu_0 \rangle / \langle B^2 / 2 \mu_0 \rangle}{dV / V}
+{{< /katex >}}
+For stabilization, \\( B^2 / 2 \mu_0 \\) should increase faster than \\( p \\) decreases
+{{< /hint >}}
+
+## 2D Equilibria
+
+{{< hint info >}}
+**"Grad-Shafranov Equation: Static toroidal equilibrium"**
+
+
+{{< katex display >}}
+\grad p = \vec j_\theta \cross \vec B_\phi + \vec j_\phi \cross \vec B_\theta
+{{< /katex >}}
+
+{{< katex display >}}
+A_\phi = \frac{\phi}{R} \vu \phi
+{{< /katex >}}
+
+{{< katex display >}}
+\phi = \frac{\phi_p}{2 \pi}
+{{< /katex >}}
+
+{{< katex display >}}
+\vec B_\theta = - \frac{\vu R}{R} \pdv{\psi}{z} + \frac{ \vu z}{R} \pdv{\psi}{R} = \frac{ \grad \psi}{R} \cross \vu \phi
+{{< /katex >}}
+
+{{< katex display >}}
+F \equiv R B_\phi
+{{< /katex >}}
+
+{{< katex display >}}
+\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} + \pdv{^2}{z^2}
+{{< /katex >}}
+
+{{< katex display >}}
+\Delta ^\star \psi = \pdv{^2 \psi}{z^2} + \pdv{^2 \psi}{R^2} - \frac{1}{R} \pdv{\psi}{R}
+{{< /katex >}}
+
+{{< katex display >}}
+\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \psi \vu \phi
+{{< /katex >}}
+
+{{< katex display >}}
+\vec j_\theta = \frac{1}{\mu_0 R} \grad (F) \cross \vu \phi
+{{< /katex >}}
+
+{{< katex display >}}
+R^2 \mu_0 \pdv{p}{\psi} = - \Delta ^\star \psi - F \pdv{F}{\psi}
+{{< /katex >}}
+
+{{< katex display >}}
+q(\psi) = \frac{F(\psi)}{2 \pi} \oint_{p} \frac{r \dd \theta}{R^2 B_\theta}
+{{< /katex >}}
+Limits:
+
+{{< katex display >}}
+\text{Force-free:} \qquad \vec j \parallel \vec B
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \Delta ^\star \psi + F F' = 0
+{{< /katex >}}
+
+{{< katex display >}}
+\text{Connected $\theta$ pinch:} \qquad FF' \gg \Delta ^\star \psi
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \grad p \approx \vec j_\theta \cross \vec B_\phi
+{{< /katex >}}
+
+{{< katex display >}}
+\text{Connected Z-pinch:} \qquad FF' \ll \Delta ^\star \phi
+{{< /katex >}}
+
+{{< katex display >}}
+\rightarrow \grad p \approx j_\phi \cross B_\theta
+{{< /katex >}}
+
+{{< /hint >}}
+
+## MHD Stability
+
+{{< hint info >}}
+**"Linear stability"**
+    
+
+{{< katex display >}}
+\pdv{\rho_1}{t} = - \vec v_1 \grad \rho_0 - \rho_0 \div \vec v_1
+{{< /katex >}}
+
+{{< katex display >}}
+\pdv{\vec B_1}{t} = \curl ( \vec v_1 \cross \vec B_0)
+{{< /katex >}}
+
+{{< katex display >}}
+\rho_0 \pdv{\vec v_1}{t} = - \grad p_1 + \vec j_0 \cross \vec B_1 - \vec j_1 \cross \vec B_0
+{{< /katex >}}
+
+{{< katex display >}}
+\pdv{p_1}{t} = - \vec v_1 \cdot \grad p_0 - \gamma p_0 \div \vec v_1
+{{< /katex >}}
+For linear perturbation \\( \vec \xi = \int_0 ^t \vec v_1 \dd t \\) the momentum equation becomes
+
+{{< katex display >}}
+\rho_0 \pdv{^2 \xi}{t^2} = \vec F(\xi)
+{{< /katex >}}
+where
+
+{{< katex display >}}
+F(\xi) = \grad (\xi \cdot \grad p_0 + \gamma p_0 \div \xi) + \frac{1}{\mu_0} \left[ ( \curl \vec B_0) \cross \curl (\xi \cross \vec B_0) + \curl \curl (\xi \cross \vec B_0) \cross \vec B_0 \right]
+{{< /katex >}}
+
+Eigenvalues of \\( \frac{1}{\rho_0} \vec F (\xi) = \omega^2 \xi \\) are real and ordered. Only need to check \\( n=0 \\) to determine stability/instability of configuration.
+
+{{< /hint >}}
+
+{{< hint info >}}
+**"\\( \delta W \\) Approach"**
+
+\\( \delta W = \\) change in potential energy due to a displacement \\( \xi \\)
+
+{{< katex display >}}
+\delta W < 0  \rightarrow \text{instability}
+{{< /katex >}}
+
+{{< katex display >}}
+\delta W = - \frac{1}{2} \int \xi \cdot F(\xi) \dd V = \delta W_F + \delta W_S + \delta W_V
+{{< /katex >}}
+Surface term:
+
+{{< katex display >}}
+\delta W_s = \frac{1}{2} \oint \dd S (\vu n \cdot \xi) ^2 \left( \vu n \cdot \grad p_0 + \left[ \vu n \cdot \grad \frac{B_0 ^2}{2 \mu_0} \right]_{jump} \right)
+{{< /katex >}}
+Vacuum term:
+
+{{< katex display >}}
+\delta W_V = \int_{vac} \dd V \frac{B_1 ^2}{\mu_0}
+{{< /katex >}}
+Plasma (free) term:
+
+{{< katex display >}}
+\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\
++ \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0}  - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\
++ \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\
++ \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1  \quad \leftarrow \text{Current-driven (kink)}  \\
+- 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)}
+{{< /katex >}}
+Shear Alfven, fast magnetosonic, and acoustic modes are stabilizing. Current-driven and pressure-driven modes can lead to instability.
+{{< /hint >}}
diff --git a/content/docs/notes/_index.md b/content/notes/_index.md
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+---
+title: "Crews (2018)"
+bookToc: false
+---
+
+# Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities
+
+Shielded potential of a test electron:
+
+{{< katex display >}}
+\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D}
+{{< /katex >}}
+
+where the Debye length is {{< katex >}} \lambda_D = \sqrt{\frac{\epsilon_0 T_e}{ n_e e}} {{< /katex >}}. The mean free path between large-angle collisions is estimated as
+
+{{< katex display >}}
+\lambda_{mfp} \sim \frac{\epsilon_0 T_e ^2}{\phi_e n_e \log ( \Lambda)}
+{{< /katex >}}
+where {{< katex >}}\phi_e = e^2 / 4 \pi \epsilon_0{{< /katex >}} are the constants from the Coulomb force law.
+
+Smooth out the discreteness of particles via spatial average over small volumes:
+
+{{< katex display >}}
+\rho \rightarrow \langle \rho_c \rangle \qquad \vec E \rightarrow \langle \vec E \rangle + \delta \vec E
+{{< /katex >}}
+
+The mean field {{< katex >}}\langle \vec E \rangle{{< /katex >}} is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales {{< katex >}}\lambda_{mfp} / \lambda_D{{< /katex >}}
+
+{{< katex display >}}
+\frac{\lambda_{mfp}}{\lambda_D} \sim \frac{T_e ^{3/2}}{n_e ^{1/2}}
+{{< /katex >}}
+
+Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified.
+
+## Phase space mechanics
+
+To arrive at a kinetic equation governing the collisionless mechanics, consider the one-dimensional motion of a single particle
+
+{{< katex display >}}
+\dot{x} = v \qquad \dot v = F(x)
+{{< /katex >}}
+
+and define the phase space coordinates as {{< katex >}}\vec{\dot r} = \vec F \equiv [ v, F(x) ]{{< /katex >}}. The flux vector is similar to the velocity field of a fluid flow. The streamlines of {{< katex >}}\vec F{{< /katex >}} are the streamlines which a particle will follow if the flux is constant in time. Phase flow is always analogous to that of an incompressible fluid because the flux divergence is zero:
+
+{{< katex display >}}
+\div [v, F(x)] = \pdv{v}{x} + \pdv{F(x)}{v}
+{{< /katex >}}
+
+If the phase fluid density is given by a function {{< katex >}}f(x, v, t){{< /katex >}} where {{< katex >}}t{{< /katex >}} is the time parameter, because any instantiation of a particle can not leave the phase plane, the probability density will be conserved. We can write a conservation law:
+{{< katex display >}}
+\pdv{f(x, v, t)}{t} = - \div ( f (x, v, t) \vec F)
+{{< /katex >}}
+and due to the flow's incompressibility
+{{< katex display >}}
+\div (f \vec F) = f ( \div \vec F) + \vec F \cdot \grad f = \vec F \cdot \grad f \\
+\rightarrow \pdv{f}{t} = - \left[ v, F(x) \right] \cdot \left[ \pdv{f}{x}, \pdv{f}{v} \right] \\
+\rightarrow \pdv{f}{t} + v \pdv{f}{x} + F(x) \pdv{f}{v} = 0
+{{< /katex >}}
+
diff --git a/docs/404.html b/docs/404.html
index 2b463d1b..db59d422 100644
--- a/docs/404.html
+++ b/docs/404.html
@@ -13,8 +13,8 @@
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+<link rel="stylesheet" href="/r/book.min.380d288a2e7eab258a84160c4cb9893132f45d4b08499d9a8114e97c3dc3733e.css" integrity="sha256-OA0oii5&#43;qyWKhBYMTLmJMTL0XUsISZ2agRTpfD3Dcz4=">
+<script defer src="/r/en.search.min.e183efc585b4431392428eef634aace5569ece2f19bfd8169c77c3d190d9ac41.js" integrity="sha256-4YPvxYW0QxOSQo7vY0qs5Vaezi8Zv9gWnHfD0ZDZrEE="></script>
 
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+    }img{filter:invert(87.05%)}}
\ No newline at end of file
diff --git a/docs/categories/index.html b/docs/categories/index.html
index 4859bfd9..f848f978 100644
--- a/docs/categories/index.html
+++ b/docs/categories/index.html
@@ -17,8 +17,8 @@
 
 <link rel="alternate" hreflang="zh" href="https://peppyhare.github.io/r/zh/categories/" title="Categories">
 
-<link rel="stylesheet" href="/r/book.min.df9df75f294a40a9e31f3fa2e59bca6c22879beb4e792e932789fc04009208dd.css" integrity="sha256-3533XylKQKnjHz&#43;i5ZvKbCKHm&#43;tOeS6TJ4n8BACSCN0=">
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+<link rel="stylesheet" href="/r/book.min.380d288a2e7eab258a84160c4cb9893132f45d4b08499d9a8114e97c3dc3733e.css" integrity="sha256-OA0oii5&#43;qyWKhBYMTLmJMTL0XUsISZ2agRTpfD3Dcz4=">
+<script defer src="/r/en.search.min.e183efc585b4431392428eef634aace5569ece2f19bfd8169c77c3d190d9ac41.js" integrity="sha256-4YPvxYW0QxOSQo7vY0qs5Vaezi8Zv9gWnHfD0ZDZrEE="></script>
 
 <script defer src="/r/sw.min.7dd1b982abb2287010a43b99e3233ff99ced25fbe0e72dc2e6ad364145177485.js" integrity="sha256-fdG5gquyKHAQpDuZ4yM/&#43;ZztJfvg5y3C5q02QUUXdIU="></script>
 <link rel="alternate" type="application/rss+xml" href="https://peppyhare.github.io/r/categories/index.xml" title="My Notes" />
@@ -61,6 +61,8 @@ <h2 class="book-brand">
 
   
 
+  
+
 
 
   
@@ -73,20 +75,20 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/example/" class="">Example Site</a>
+    <span>Notes</span>
   
 
           
   <ul>
     
       
-        <li>
+        <li class="book-section-flat" >
           
   
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/example/table-of-contents/" class="">Table of Contents</a>
+    <span>Working</span>
   
 
           
@@ -99,20 +101,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/example/table-of-contents/with-toc/" class="">With ToC</a>
-  
-
-        </li>
-      
-    
-      
-        <li>
-          
-  
-  
-
-  
-    <a href="https://peppyhare.github.io/r/docs/example/table-of-contents/without-toc/" class="">Without ToC</a>
+    <a href="https://peppyhare.github.io/r/notes/working/crews2018/" class="">Crews (2018)</a>
   
 
         </li>
@@ -130,9 +119,9 @@ <h2 class="book-brand">
   
 
   
-    <input type="checkbox" id="section-4e46b01272d410b3a99461d79326ddf4" class="toggle"  />
-    <label for="section-4e46b01272d410b3a99461d79326ddf4" class="flex justify-between">
-      <a  class="">Collapsed</a>
+    <input type="checkbox" id="section-edf338bf880234e7aba93d04934ebde0" class="toggle"  />
+    <label for="section-edf338bf880234e7aba93d04934ebde0" class="flex justify-between">
+      <a  class="">MHD Theory</a>
       <span>▾</span>
     </label>
   
@@ -147,53 +136,12 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/example/collapsed/3rd-level/" class="">3rd Level</a>
-  
-
-          
-  <ul>
-    
-      
-        <li>
-          
-  
-  
-
-  
-    <a href="https://peppyhare.github.io/r/docs/example/collapsed/3rd-level/4th-level/" class="">4th Level</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/" class="">Syllabus</a>
   
 
         </li>
       
     
-  </ul>
-
-        </li>
-      
-    
-  </ul>
-
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-      
-    
-  </ul>
-
-        </li>
-      
-    
-      
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-          
-  
-  
-
-  
-    <span>Notes</span>
-  
-
-          
-  <ul>
-    
       
         <li>
           
@@ -201,30 +149,12 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/notes/crews2018/" class="">Crews (2018)</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/" class="">Plasma Models</a>
   
 
         </li>
       
     
-  </ul>
-
-        </li>
-      
-    
-      
-        <li class="book-section-flat" >
-          
-  
-  
-
-  
-    <span>Shortcodes</span>
-  
-
-          
-  <ul>
-    
       
         <li>
           
@@ -232,7 +162,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/buttons/" class="">Buttons</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/" class="">Plasma Fluid Model</a>
   
 
         </li>
@@ -245,7 +175,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/columns/" class="">Columns</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
   
 
         </li>
@@ -258,7 +188,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/details/" class="">Details</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
   
 
         </li>
@@ -271,7 +201,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/expand/" class="">Expand</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/" class="">Boundary Conditions</a>
   
 
         </li>
@@ -284,7 +214,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/hints/" class="">Hints</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class="">Equilibrium for Fusion</a>
   
 
         </li>
@@ -297,7 +227,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/katex/" class="">Katex</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/" class="">1-D Equilibria</a>
   
 
         </li>
@@ -310,7 +240,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/mermaid/" class="">Mermaid</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/" class="">2D Equilibria</a>
   
 
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@@ -323,15 +253,11 @@ <h2 class="book-brand">
   
 
   
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-    <label for="section-d3fc1bf6d66cd84b896a0af9f40cb1d5" class="flex justify-between">
-      <a href="https://peppyhare.github.io/r/docs/shortcodes/section/" class="">Section</a>
-      <span>▾</span>
-    </label>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/" class="">Equilibrium of 3D Configurations</a>
   
 
-          
-  <ul>
+        </li>
+      
     
       
         <li>
@@ -340,7 +266,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/section/page1/" class="">Page1</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/" class="">MHD Stability</a>
   
 
         </li>
@@ -353,7 +279,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/section/page2/" class="">Page2</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/formulary/" class="">Formulary</a>
   
 
         </li>
@@ -364,19 +290,6 @@ <h2 class="book-brand">
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-      
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-          
-  
-  
-
-  
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/tabs/" class="">Tabs</a>
-  
-
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-      
-    
   </ul>
 
         </li>
@@ -490,6 +403,7 @@ <h2 class="book-brand">
     window.scrollTo(0,document.body.scrollHeight);
 };</script>
 
+
       <footer class="book-footer">
         
   <div class="flex flex-wrap justify-between">
diff --git a/docs/categories/index.xml b/docs/categories/index.xml
index d3527042..55875be0 100644
--- a/docs/categories/index.xml
+++ b/docs/categories/index.xml
@@ -4,10 +4,6 @@
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-    
-	<atom:link href="https://peppyhare.github.io/r/categories/index.xml" rel="self" type="application/rss+xml" />
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index 2b1ba6ff..f1bbd43a 100644
--- a/docs/docs/example/collapsed/3rd-level/4th-level/index.html
+++ b/docs/docs/example/collapsed/3rd-level/4th-level/index.html
@@ -16,7 +16,7 @@
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index 5121ce3b..0b2048f3 100644
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index 7f5aacf0..e070bd5f 100644
--- a/docs/docs/example/collapsed/3rd-level/index.xml
+++ b/docs/docs/example/collapsed/3rd-level/index.xml
@@ -4,11 +4,7 @@
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     <description>Recent content on My Notes</description>
-    <generator>Hugo -- gohugo.io</generator>
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-	<atom:link href="https://peppyhare.github.io/r/docs/example/collapsed/3rd-level/index.xml" rel="self" type="application/rss+xml" />
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@@ -20,4 +16,4 @@ Venasque repulsa Samos qui, exspectatum eram animosque hinc, aut manes, Assyrii.
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index 75ec3f70..e9aeb2ff 100644
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index 280744ec..60472d28 100644
--- a/docs/docs/example/collapsed/index.xml
+++ b/docs/docs/example/collapsed/index.xml
@@ -4,10 +4,6 @@
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-	<atom:link href="https://peppyhare.github.io/r/docs/example/collapsed/index.xml" rel="self" type="application/rss+xml" />
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index d01989d5..b38d0c7f 100644
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@@ -16,7 +16,7 @@
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index 77538f5e..9f096008 100644
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index f9963aec..4583b371 100644
--- a/docs/docs/example/index.xml
+++ b/docs/docs/example/index.xml
@@ -4,11 +4,7 @@
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     <link>https://peppyhare.github.io/r/docs/example/</link>
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@@ -20,4 +16,4 @@
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index 69dd9a3d..612d4606 100644
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index 3fd967c3..203afb74 100644
--- a/docs/docs/example/table-of-contents/index.xml
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@@ -4,11 +4,7 @@
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@@ -30,4 +26,4 @@ Quam iugulum regia simulacra, plus meruit humo pecorumque haesit, ab discedunt d
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index 3c225839..19762f79 100644
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+'use strict';(function(){const indexCfg={cache:true};indexCfg.doc={id:'id',field:['title','content'],store:['title','href','section'],};const index=FlexSearch.create('balance',indexCfg);window.bookSearchIndex=index;index.add({'id':0,'href':'/r/notes/UWAA558/01-syllabus/','title':"Syllabus",'section':"MHD Theory",'content':"Syllabus #  The course topics planned for this section are (in rough order):\nParticle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, \u0026ldquo;Frozen-In\u0026rdquo; Flux)\nIdeal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces\nOne-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q\nTwo-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)\nMHD Stability, General Concepts, Linearized MHD, Exponential (Linear) Stability, Force Operator and Properties, Variational Formulation, Energy Principle, Intuitive Form of delta W, Classification of Instabilities (internal/external, pressure-drive/current-driven, kink/interchange/ballooning)\nStability of One-Dimensional Equilibria, Modal Analysis, Rayleigh-Taylor, Theta-Pinch, Z-Pinch (Kadomtsev Condition), Screw-Pinch (Kruskal-Shafranov Condition, Suydam Criterion), RFP, \u0026ldquo;Straight\u0026rdquo; Tokamak\nStability of Two-Dimensional Equilibria, Tokamak, Mercier Criterion, Elmo Bumpy Torus\nResistive (Tearing) Instabilities, Stability of Non-static Equilibria, Nonlinear Stability Effects\nCourse Motivation #  Plasma phenomena tend to be hard to treat because of the span of relevant scales. You have ions, electrons, and photons interacting through electromagnetic interactions. There is a tremendous variation in mass across species, which leads to a large span of both spatial and temporal scales. The species can interact through both short scale collisions and long range interactions through EM forces. In contrast, in normal gas dynamics you may consider only the short-scale interactions. As a consequence, we can describe dispersive plasma waves.\nFor comparison, remember in gas dynamics, the speed of sound is\n  \\[\\frac{\\omega}{k} = v_s = \\pdv{\\omega}{k} \\qquad \\text{(gas)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Here the phase velocity  \\( \\frac{\\omega}{k} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is equal to the phase velocity  \\( \\pdv{\\omega}{k} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . In a plasma, we can have non-linear dispersion relations in which the phase and group velocity are different.\n \\[\\frac{\\omega}{k} = v(\\omega, k) \\neq \\pdv{\\omega}{k} \\qquad \\text{(plasma)} \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The number of particles we typically deal with in a laboratory plasma is roughly on the order of a mole of particles\n \\[\\text{particles} \\sim O(10^{23})\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  With long-range interactions, we have a combinatorial explosion of interacting particles! It is not possible to track individual particles at such a scale, so we need much simpler plasma descriptions. These are the plasma models we will discuss in the next chapter. When simplifying our models, we need to pay careful attention to the simplifications we are making because in general, inaccurate physics lead to incorrect conclusions.\n"});index.add({'id':1,'href':'/r/notes/UWAA558/02-plasma-models/','title':"Plasma Models",'section':"MHD Theory",'content':"Plasma Models #  Working towards MHD #  Let\u0026rsquo;s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we\u0026rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.\nThe most important question to ask ourselves is \u0026ldquo;when is this model going to be useful?\u0026rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid. This means that we can define a fluid element (some lil' box of plasma) and define the physical properties (mass, density, magnetization, etc.) of the element. We need to make some assumptions about scale in order to do this. In terms of spatial scales, we abstract properties below a discrete scale   \\( a_0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  into the properties of a fluid element\n \\[\\frac{a_0}{L} \\rightarrow 0 \\qquad a_0 = \\text{discrete scale} \\qquad L = \\text{spatial scale of interest}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Length scales smaller than the discrete scale will not be properly captured by the model, so scales like the particle radius will be meaningless in our fluid model.\nPlasma Definition #  A plasma is a quasi-neutral gas of charged and neutral particles which exhibit collective behavior. The particles (electrons, ions, neutrals) interact through EM fields and collisions.\nMathematically you would think that plasmas could be treated as individual particles. Doing so gives an N-body problem with classical interactions through the Lorentz force (Coulomb interaction removed by assumption of quasi-neutrality) and binary collisions. Each particle  \\( i \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  has a well-defined mass  \\( m_i \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and charge  \\( q_i \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  which do not change in time. The governing equations are\n \\[\\dv{\\vec v_i}{t} = \\frac{q_i}{m_i} (\\vec E \u0026#43; \\vec v_i \\cross \\vec B) \u0026#43; \\sum_{j \\neq i} \\left[ \\left. \\dv{\\vec v_{ij}}{t} \\right|_{coll} (\\vec r_i - \\vec r_j) \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\dv{\\vec r_i}{t} = \\vec v_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThe fields E and B are described by the Maxwell equations\n \\[\\pdv{B}{t} = - \\curl E\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\frac{1}{c^2} \\pdv{E}{t} = \\curl B - \\mu_0 \\sum_i q_i v_i \\delta (r - r_i)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\div B = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\epsilon_0 \\div E = \\sum_i q_i \\delta (r - r_i)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nKlimontovich Equation #  Re-writing the force relations as a statement of conservation in phase space, we get Klimontovich equation for species  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\dv{N_\\alpha}{t} = 0 = \\pdv{N_\\alpha}{t} \u0026#43; \\pdv{}{q} \\cdot (\\dot{q} N_\\alpha) \u0026#43; \\pdv{}{p} \\cdot (\\dot{p}N_\\alpha)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The particle phase space is defined by  \\[N_\\alpha(p, q) = \\sum_i \\delta(p - p_i) \\delta(q - q_i)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( q \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  are generalized momentum and position coordinates. The resulting  \\( N(p) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  looks very spiky, with nonzero values only at the exact values inhabited by particles. Unfortunately, that means that only the tools of discrete mathematics are applicable to the distribution, and we\u0026rsquo;re forbidden from our favorite tool (calculus). To make the analysis possible, we can smooth over the discreteness by performing an ensemble average of the Klimontovich equation. This gives us a statistical description using smooth distribution functions:\n \\[f(x, v, t) \\qquad \\frac{(\\text{no. of particles})}{(\\text{unit distance})^3(\\text{unit velocity})^3}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  That is,  \\( f(x, v, t) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the number of particles at position  \\( x \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  with velocity  \\( v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  at time  \\( t \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . We also work with normalized distributions which give the probability of finding a particle.\nBy our definition, we can integrate to get the total number of particles at time  \\( t \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\n \\[\\int \\int \\dd x \\dd v f(x, v, t) = N(t)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The ensemble averaging process works well if the the number of particles is very large\n \\[N \\gg 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  See a kinetic theory text (e.g. Krall and Trivelpiece) for a full description of the ensemble averaging process.\nNow we can write our Klimontovich equation in terms of continuous quantities\n \\[\\pdv{f}{t} \u0026#43; \\left[ \\pdv{}{q} \\cdot (\\dot{q} f) \u0026#43; \\pdv{}{p} \\cdot (\\dot{p}f) \\right] = \\text{(cross terms)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The collision terms now have an infinite number of cross terms. We call this the BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy.\nExpressing as the Boltzmann equation (generally called the Boltzmann-Maxwell equation, since the solution requires solving for the electromagnetic fields of the Maxwell equations)\n \\[\\pdv{f_\\alpha}{t} \u0026#43; \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \u0026#43; \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} = \\left. \\pdv{f_\\alpha}{t} \\right|_{coll}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We leave the collision term in. A lot of the work of kinetic theory is coming up with an applicable form of the collision operator which is appropriate but still simple enough to solve. We often write the collision operator as the product of binary collisions\n \\[ \\left. \\pdv{f_\\alpha}{t} \\right|_{coll} = \\sum_\\beta C_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Notice that the terms  \\( \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  are advection equations, advecting in  \\( \\vec x \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  respectively. If we ignore the collision term (set the RHS to zero) we have the Vlasov equation.\nNote that the fields E and B at any location are generated from the charges and currents of the entire plasma volume, including externally applied fields. As a result, there\u0026rsquo;s an inherent integrating process taking into account the sources across the whole volume that leads to long-range smoothly varying forces. This is in contrast to the collisional effects, which by their nature lead to very short range abrupt forces. It makes sense to make a distinction between the long-range electromagnetic forces and the short range collisional forces.\nBecause the Boltzmann-Maxwell model is inherently 6-dimensional, it is a very challenging model to implement. The B-M model provides a complete description, but it is often too detailed to solve.\nIf we solve the Vlasov equation for two parallel opposite beams, for example, we see that the\nAs it turns out, the integral, centroid, and variance are all that are required to fully describe a Maxwellian distribution\n \\[f_{M, \\alpha} (\\vec v) = n_0 \\left( \\frac{ m_\\alpha }{2 \\pi T_\\alpha} \\right)^{3/2} \\text{exp}\\left[ - \\frac{ \\frac{1}{2} m_\\alpha (v - v_\\alpha)^2}{T_\\alpha} \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  In other words, we can write the Maxwellian distribution as  \\( f_M(n_0, \\vec v_\\alpha, T_\\alpha ; \\vec v) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . We care about Maxwellian distributions so much in plasma physics because it is the solution to the Boltzmann equation for  \\( \\left. \\pdv{f_\\alpha}{t} \\right|_{coll} = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  (Vlasov equation). That\u0026rsquo;s not saying that there are no collisions, it is saying that there are so many collisions that the effect is isotropic and the overall force is zero.\nAnother feature of the Maxwellian distribution is  \\( \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . This is a famous result called the Boltzmann H-theorem, and says that any initial distribution will relax (and very quickly) to a Maxwellian distribution.\nBy replacing our velocity distribution with the associated Maxwell distribution, we arrive at a Plasma Fluid Model\n"});index.add({'id':2,'href':'/r/notes/UWAA558/03-plasma-fluid-model/','title':"Plasma Fluid Model",'section':"MHD Theory",'content':"Plasma Fluid Model #  We take velocity moments of each of the pieces of the kinetic model:\n distribution function,   \\( f_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( \\rightarrow \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  fluid variables Boltzmann Equation  \\( \\rightarrow \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  governing equations describing the evolution of the fluid variables.  Starting with the zeroth moment (integral) of the distribution function:\n \\[\\int f_\\alpha (\\vec x, \\vec v, t) \\dd \\vec v = n_\\alpha(\\vec x, t)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  1st Moment (momentum):\n \\[m_\\alpha \\int \\vec v \\cdot f_\\alpha(\\vec x, \\vec v, t) \\dd \\vec v = \\vec p_\\alpha (\\vec x, t) = m_\\alpha n_\\alpha \\vec v_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  which is to say that the velocity is the 1st moment divided by the zeroth moment\n \\[v_\\alpha = \\frac{\\int \\vec v f_\\alpha}{\\int f_\\alpha}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  2nd Moment:\n \\[\\int \\vec v \\vec v f_\\alpha (\\vec x, \\vec v, t) \\dd \\vec v = \\vec E_\\alpha(\\vec x, t) (\\text{energy tensor})\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can simplify the 2nd moment by taking a reduced 2nd moment. This means that we\u0026rsquo;re going to insert a dot product\n \\[\\int \\vec v \\cdot \\vec v f_\\alpha (\\vec x, \\vec v, t) \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Before moving forward, we want to define a \u0026ldquo;random\u0026rdquo; velocity. Note that\n \\[\\int \\vec v f_\\alpha - \\vec v_\\alpha \\int f_\\alpha = 0 \\rightarrow \\int (\\vec v - \\vec v_\\alpha) f_\\alpha \\dd \\vec v = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can define a random velocity  \\( \\vec w = \\vec v - \\vec v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . It is random in the sense that it is a fluctuation about the mean velocity, and when we integrate it we get zero. We can use this to define the energy tensor using the mean velocity to get a meaningful result. The pressure tensor is the second moment, using the random velocity\n \\[\\vec P_\\alpha = m_\\alpha \\int \\vec w \\vec w f_\\alpha \\dd \\vec v = P_\\alpha \\overline{I} \u0026#43; \\overline{\\Pi}_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where we\u0026rsquo;ve decomposed the pressure into an isotropic value  \\( P_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and what\u0026rsquo;s called the Braginskii stress tensor  \\( \\overline{\\Pi}_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The average isotropic pressure is given by the reduced 2nd moment:\n \\[P_\\alpha = n_\\alpha T_\\alpha = \\frac{1}{3} \\int m_\\alpha \\vec w \\cdot \\vec w f_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where the factor of  \\( 1/3 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  comes from the number of degrees of freedom in our system. It is related to the thermodynamic factor  \\( \\gamma \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( \\gamma = \\frac{DOF \u0026#43; 2}{DOF} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Now we can define the temperature  \\( T_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as\n \\[T_\\alpha (\\vec x, t) = \\frac{1}{DOF} \\frac{\\int m_\\alpha \\vec w \\cdot \\vec w f_\\alpha \\dd \\vec v}{\\int f_\\alpha \\dd \\vec v}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Now that we\u0026rsquo;ve got  \\( n_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and  \\( T_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we have what we need to define a Maxwellian distribution. Higher moments would be required to describe non-Maxwellian distribution functions. For example, the 3rd moment is called the skewness of the distribution. The 4th moment is the kurtosis. So on and so forth. These give a measure of degree of departure from a Maxwellian distribution, in which case it is often more useful to talk about the excess kurtosis, where the excess kurtosis of a Maxwellian is defined to be zero. You can continue to calculate the moment expansion, and in general it requires an infinite number of moments to describe an arbitrary distribution function. Because of the Boltzmann H-theorem and the tendency of plasmas to quickly relax to Maxwellian, we can usually get away with using just the first three moments.\nNow, what are the governing equations? We get these by taking moments of the Boltzmann equation. Let\u0026rsquo;s proceed carefully in sections, so we\u0026rsquo;ll integrate each piece of the BE in terms.\n0th Moment of Boltzmann Equation (Conservation) #   \\[\\int \\pdv{f_\\alpha}{t} \\dd \\vec v \u0026#43; \\int \\vec v \\cdot \\pdv{f_\\alpha}{x} \\dd v \u0026#43; \\int \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\int \\left. \\pdv{f_\\alpha}{t} \\right|_{coll} \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  First, we take  \\( \\int \\pdv{f_\\alpha}{t} \\dd \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Because t and v are both independent variables, with an argument of sufficient smoothness we can reverse the order of integration and differentiation\n \\[\\int \\pdv{f_\\alpha}{t} \\dd \\vec v = \\pdv{}{t}\\int f_\\alpha \\dd v = \\pdv{n_\\alpha}{t}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  For  \\( \\int \\vec v \\cdot \\pdv{f_\\alpha}{x} \\dd v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we can perform an integration by parts\n \\[ \\int \\vec v \\cdot \\pdv{f_\\alpha}{x} \\dd v = \\int \\pdv{}{\\vec x} \\cdot ( \\vec v f_\\alpha) \\dd \\vec v - \\int f_\\alpha \\pdv{}{\\vec x} \\cdot \\vec v \\dd \\vec v \\\\ = \\int \\pdv{}{\\vec x} \\cdot ( \\vec v f_\\alpha) \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int \\vec v f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot (n_\\alpha \\vec v_\\alpha) = \\div (n_\\alpha \\vec v_\\alpha)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Once again, we\u0026rsquo;ve switched the order of integration of  \\( x \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , which we can only do because we have specified that  \\( f \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is a distribution function, and as such meets the criterion of sufficient smoothness.\nFor the last part, we can write it as a surface integral\n \\[\\int \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\oint f_\\alpha \\frac{q_\\alpha}{m_\\alpha} ( \\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\dd S_v - \\int \\frac{q_\\alpha}{m_\\alpha} f_\\alpha \\pdv{}{\\vec v} \\cdot (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  For  \\( f_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to be well-defined, we require  \\( \\lim_{v \\rightarrow \\infty} v^3 f_\\alpha = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  so the surface term vanishes and we\u0026rsquo;re left with\n \\[\\int \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = - \\int \\frac{q_\\alpha}{m_\\alpha} f_\\alpha \\pdv{}{\\vec v} \\cdot (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\dd \\vec v \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Distributing the divergence through,\n \\[\\pdv{}{\\vec v} \\cdot \\vec E = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  because  \\( \\vec E \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is independent of  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\n \\[\\pdv{}{\\vec v} \\cdot ( \\vec v \\cross \\vec B) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  because  \\( \\vec v \\cross \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is always orthogonal to  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nWe\u0026rsquo;ve been working through this in normal vector notation for familiarity, but there is another notation known as Einstein Tensor Notation using the lovely Levi-Civita symbol  \\( \\epsilon_{ijk} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\n \\[\\epsilon_{ijk} = 1 \\qquad \\text{even permutations of i j k} \\\\ = -1 \\qquad \\text{odd permutations of i j k} \\\\ = 0 \\qquad \\text{any repeated indexes}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can write out vector products as products of indices and operators, and any repeated indices are implicitly summed:\n \\[\\vec v \\cross \\vec B = \\epsilon_{ijk} v_j B_k = \\sum_{jk} \\epsilon_{ijk} v_j B_k\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( \\epsilon_{ijk} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is defined to be 1 for even permutations of ijk, -1 for negative permutations if ijk, and 0 for any repeated indices.\nWe can now write the derivative with respect to  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as\n \\[\\pdv{}{\\vec v} \\cdot \\vec v \\cross \\vec B = \\partial _{v_i} \\epsilon_{ijk} v_j B_k \\\\ = \\epsilon_{ijk} (\\partial_{v_i} v_j) B_k\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We see that we\u0026rsquo;re taking the derivative of the j-th component of velocity with respect to the i-th component of velocity, and that\u0026rsquo;s how we can most easily point out that the quantity is zero without relying on properties of 3-vector products.\nWe\u0026rsquo;ll also want to use the divergence in Einstein tensor notation\n \\[\\div \\vec A = \\partial_i A_j \\delta_{ij} = \\partial_i A_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Finally, we come back to the collision term in the zeroth moment of the B-M equation\n \\[\\sum_\\beta \\int C_{\\alpha \\beta} \\dd \\vec v = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can say the collision term is zero by making a physical argument, rather than a mathematical one. We assert that collisions cannot create or destroy particles. This assumption is now baked into our equations going forward, but is not always true! Ionization, recombination, fusion reactions all create/destroy species.\nFinally, we\u0026rsquo;ve got\n \\[\\pdv{n_\\alpha}{t} \u0026#43; \\div ( n_\\alpha \\vec v_\\alpha) = 0 \\qquad \\text{continuity equation}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  So by taking the 0th moment of the Boltzmann Equation, we\u0026rsquo;ve arrived at the continuity equation for  \\( n_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  by introducing  \\( \\vec v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\n1st Moment of Boltzmann Equation #   \\[\\int \\vec v \\pdv{f_\\alpha}{t} \\dd \\vec v \u0026#43; \\int \\vec v \\vec v \\pdv{f_\\alpha}{\\vec x} \\dd \\vec v \u0026#43; \\int \\vec v \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\int \\left. \\vec v \\pdv{f_\\alpha}{t} \\right|_{coll}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Term 1:\n \\[\\int \\vec v \\pdv{f_\\alpha}{t} \\dd \\vec v = \\pdv{}{t} \\int \\vec v f_\\alpha \\dd \\vec v - \\int \\pdv{\\vec v}{t} f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{t} \\left( n_\\alpha \\vec v_\\alpha \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Before we get to the second term, let\u0026rsquo;s have an aside about dyad math. An outer product  \\( \\vec A \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  gives a second-rank tensor. In Cartesian coordinates, it looks like\n \\[\\vec A \\vec B = \\begin{bmatrix} A_x B_x \u0026amp; A_x B_y \u0026amp; A_x B_z \\\\ A_y B_x \u0026amp; A_y B_y \u0026amp; A_y B_z \\\\ A_z B_x \u0026amp; A_z B_y \u0026amp; A_z B_z \\\\ \\end{bmatrix}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  A useful property to do with dot products:\n \\[\\vec A \\vec B \\cdot \\vec C = \\vec A ( \\vec B \\cdot \\vec C ) = (\\vec A \\vec B) \\cdot \\vec C\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[( \\vec V \\cdot \\grad) \\vec B = \\vec V \\cdot ( \\grad \\vec B)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Back to term 2:\n \\[\\int \\vec v \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \\dd \\vec v = \\int \\vec v \\cdot \\pdv{}{\\vec x} ( \\vec v f_\\alpha) \\dd \\vec v - \\int \\vec v f_\\alpha \\pdv{}{\\vec x} \\cdot \\vec v \\dd \\vec v \\\\ = \\int \\pdv{}{\\vec x} \\cdot ( \\vec v \\vec v f_\\alpha) \\dd \\vec v - \\int \\vec v f_\\alpha \\cdot \\pdv{\\vec v}{\\vec x} \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int \\vec v \\vec v f_\\alpha \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Re-expanding in terms of random velocity  \\( \\vec v = \\vec v_\\alpha \u0026#43; \\vec w \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[= \\pdv{}{\\vec x} \\cdot \\int (\\vec v_\\alpha \u0026#43; \\vec w) (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int (\\vec v_\\alpha \\vec v_\\alpha \u0026#43; \\vec v_\\alpha \\vec w \u0026#43; \\vec w \\vec v_\\alpha \u0026#43; \\vec w \\vec w) f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\left[ \\vec v_\\alpha \\vec v_\\alpha \\int f_\\alpha \\dd \\vec v \u0026#43; \\vec v_\\alpha \\int \\vec w f_\\alpha \\dd \\vec v \u0026#43; \\left( \\int \\vec w f_\\alpha \\dd \\vec v \\right) \\vec v_\\alpha \u0026#43; \\int \\vec w \\vec w f_\\alpha f_\\alpha \\dd \\vec v \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The middle two terms are zero by the very definition of  \\( \\vec v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The random velocity  \\( \\vec w \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is defined such that the integral of the random velocity over all phase space is zero.\n \\[= \\pdv{}{\\vec x} \\cdot \\left( \\vec v_\\alpha \\vec v_\\alpha n_\\alpha \\right) \u0026#43; \\pdv{}{\\vec x} \\cdot \\frac{ \\vec P_\\alpha}{m_\\alpha} \\\\ = \\div (n_\\alpha \\vec v_\\alpha \\vec v_\\alpha) \u0026#43; \\frac{1}{m_\\alpha} \\div \\vec P_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Substituting in the pressure tensor,\n \\[= \\div ( n_\\alpha \\vec v_\\alpha \\vec v_\\alpha ) \u0026#43; \\frac{1}{m_\\alpha} \\left( \\grad P_\\alpha \u0026#43; \\div \\vec \\Pi _\\alpha \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Moving on to term 3:\n \\[\\int \\vec v \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\int \\frac{q_\\alpha}{m_\\alpha} \\pdv{}{\\vec v} \\cdot \\left[( \\vec E \u0026#43; \\vec v \\cross \\vec B) f_\\alpha \\right] \\vec v \\dd \\vec v - \\int \\frac{q_\\alpha}{m_\\alpha} f_\\alpha \\pdv{}{\\vec v} \\cdot \\left[ \\vec E \u0026#43; \\vec v \\cross \\vec B \\right] \\vec v \\dd \\vec v \\\\ = \\int \\frac{q_\\alpha}{m_\\alpha} \\pdv{}{\\vec v} \\cdot \\left[ f_\\alpha (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\vec v \\right] \\dd \\vec v - \\int \\frac{q\\alpha}{m\\alpha} f ( \\vec E \u0026#43; \\vec v\\alpha \\cross \\vec B) \\cdot \\pdv{\\vec v}{\\vec v} \\dd \\vec v \\\\ = \\frac{q\\alpha}{m\\alpha} \\oint f\\alpha ( \\vec E \u0026#43; \\vec v\\alpha \\cross \\vec B) \\vec v \\cdot \\dd \\vec S_v - \\frac{q\\alpha}{m\\alpha} \\left[ \\vec E \\int f \\dd \\vec v\\alpha - \\vec B \\cross \\int \\vec v f\\alpha \\dd \\vec v \\right] \\\\ = - \\frac{q\\alpha}{m\\alpha} n\\alpha (\\vec E \u0026#43; v\\alpha \\cross \\vec B)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Finally, the 4th term gives\n \\[ \\int \\left. \\vec v \\pdv{f_\\alpha}{t} \\right|_{coll} = \\sum_\\beta \\int \\vec v C_{\\alpha \\beta} \\dd \\vec v \\\\ = \\int \\vec v C_{\\alpha \\alpha} \\dd \\vec v \u0026#43; \\sum_{\\beta \\neq \\alpha} \\int \\vec v C_{\\alpha \\beta} \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Collisions of like particles  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  do not result in a net change of momentum of species  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , so all we have left is the change in momentum due to collisions of unlike particles\n \\[= \\sum_{\\beta \\neq \\alpha} \\left[ \\vec v_{\\alpha} \\int C_{\\alpha \\beta} \\dd \\vec v \u0026#43; \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  For the same reason as before, we assert that collisions between particles  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  do not lead to the creation or destruction of any species, so the 0th moment  \\( \\int C_{\\alpha \\beta} \\dd \\vec v = 0\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . This leads to the conclusion that only random motion contributes to momentum transfer, not  \\( \\vec v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Viscosity and friction are good examples of similar physical processes where bulk velocity does not transfer momentum, but random motion does.\nThe momentum transfer from  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  must equal transfer from  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Momentum is globally conserved.\n \\[\\int \\vec w C_{\\alpha \\beta} \\dd \\vec v = - \\int \\vec w C_{\\beta \\alpha} \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Now we can finally write out the full momentum equation\n \\[\\pdv{}{t} (n_\\alpha \\vec v_\\alpha ) \u0026#43; \\div (n_\\alpha \\vec v_\\alpha \\vec v_\\alpha) \u0026#43; \\frac{1}{m_\\alpha} \\div \\vec P_\\alpha - \\frac{q_\\alpha}{m_\\alpha} n_\\alpha ( \\vec E \u0026#43; \\vec v _\\alpha \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The collision term is often represented as the momentum transfer vector  \\( \\vec R_{\\alpha \\beta} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  from  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\n \\[\\vec R_{\\alpha \\beta} \\equiv m_\\alpha \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\pdv{}{t} (m_\\alpha n_\\alpha v_\\alpha) \u0026#43; \\div (m_\\alpha n_\\alpha \\vec v_\\alpha \\vec v_\\alpha) \u0026#43; \\grad \\vec P_{\\alpha} \u0026#43; \\div \\vec \\Pi_{\\alpha} - q_\\alpha n_\\alpha (\\vec E \u0026#43; \\vec v _\\alpha \\cross \\vec B) = \\sum_{\\beta} R_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nOnce again, we\u0026rsquo;ve written the above in a conservation law form. The terms that aren\u0026rsquo;t strictly conservation terms are the source term  \\( q_\\alpha n_\\alpha (\\vec E \u0026#43; \\vec v _\\alpha \\cross \\vec B) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and sink term  \\( \\sum_{\\beta} R_{\\alpha \\beta} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nIntroducing the mass density\n \\[\\rho_\\alpha = m_\\alpha n_\\alpha \\qquad \\text{mass density}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\pdv{}{t} (\\rho_\\alpha \\vec v _\\alpha) \u0026#43; \\div (\\rho_\\alpha \\vec v_\\alpha v_\\alpha \u0026#43; \\ldots \\\\ = \\rho_\\alpha \\pdv{\\vec v_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\left( \\pdv{\\rho_\\alpha}{t} \u0026#43; \\div \\rho_\\alpha \\vec v_\\alpha \\right) \u0026#43; \\rho_\\alpha \\vec v_\\alpha \\cdot \\grad \\vec v_\\alpha \u0026#43; \\ldots\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The term in parentheses is just the continuity equation, which is zero, and what\u0026rsquo;s left is a more usual form of the momentum equation\n \\[\\text{Momentum Equation:}\\\\ \\rho_\\alpha \\left(\\pdv{\\vec v_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad \\vec v_\\alpha \\right) \u0026#43; \\grad \\vec P_\\alpha \u0026#43; \\div \\vec \\Pi_\\alpha - q_\\alpha n_\\alpha (\\vec E \u0026#43; \\vec v_\\alpha \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\vec R_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  So by taking the 1st moment of the BE, we arrive at the momentum equation for  \\( \\vec v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and we have introduced  \\( \\vec P_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nMomentum transfer (collisions) #  Looking back at  \\( \\vec R_{\\alpha \\beta} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , it\u0026rsquo;s worth noting that the actual situation is complicated by the potential for multi-body collisions. While it is true that the probability of, e.g. three-body collisions is small, but there are many three-body collisional processes which are extremely important in plasma physics. Three-body recombination is the primary loss term in the ionization balance, for example. For now the simplest model is to only include binary collisions that result in small angel deflections. If we suppose we only have two species with distributions that look like the following:\nthen we expect collisions between species to tend to drag the distributions towards each other. We should expect the collision-based momentum transfer to be proportional to the distance between the distributions, i.e. the total current density  \\( \\vec j \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nFor an ion-electron plasma, the collision momentum transfer vector (to ions from electrons) is\n \\[\\vec R_{ie} = - n_e e \\eta \\vec j\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where\n \\[\\eta = \\frac{m_e \\nu_{ei}}{n e^2} \\qquad \\text{resistivity}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\vec j = n e (\\vec v_i - \\vec v_e) \\qquad \\text {current density for Z = 1}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  2nd Moment of Boltzmann Equation #  Finally we\u0026rsquo;re on to our last expansion term. In general, we would say that the second moment would be\n \\[\\int \\vec v \\vec v \\pdv{f_\\alpha}{t} \\dd \\vec v = \\pdv{}{t} \\int \\vec v \\vec v f_\\alpha \\dd \\vec v = \\pdv{}{t} \\vec E_\\alpha / m_\\alpha \\rightarrow \\pdv{}{t} \\vec P_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Usually we think of the second moment as giving us energy, a scalar, rather than a tensor equation that we have here. Since the pressure tensor is symmetric, the energy equation will contain 6 independent quantities. Together with the continuity and velocity equations, we have 10 quantities in total, so we call this expansion a 10-moment expansion.\nInstead of the full tensor equation, we\u0026rsquo;re actually going to compute the reduced 2nd moment by contracting  \\( \\vec v \\cdot \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\int \\vec v \\cdot \\vec v \\pdv{f_\\alpha}{t} \\dd v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  So, let\u0026rsquo;s get after it:\n \\[\\int v^2 \\pdv{f_\\alpha}{t} \\dd \\vec v \u0026#43; \\int v^2 \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \\dd \\vec v \u0026#43; \\int v^2 \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\left. \\int v^2 \\pdv{f_\\alpha}{t} \\right|_{coll} \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Simplifying\u0026hellip; Term 1:\n \\[\\int v^2 \\pdv{f_\\alpha}{t} \\dd \\vec v = \\int \\pdv{}{t} (v^2 f_\\alpha) \\dd \\vec v - \\int \\pdv{v^2}{t} f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{t} \\int (\\vec v_\\alpha \u0026#43; \\vec w) \\cdot (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{t} \\left[ \\int v_\\alpha ^2 f_\\alpha \\dd \\vec v \u0026#43; 2 \\int \\vec v_\\alpha \\cdot \\vec w f_\\alpha \\dd \\vec v \u0026#43; \\int w^2 f_\\alpha \\dd \\vec v \\right] \\\\ = \\pdv{}{t} \\left[ v_\\alpha ^2 \\int f_\\alpha \\dd \\vec v \u0026#43; 2 \\vec v_\\alpha \\cdot \\int \\vec w f_\\alpha \\dd \\vec v \u0026#43; 3 \\frac{n_\\alpha T_\\alpha}{m_\\alpha} \\right] \\\\ = \\pdv{}{t} \\left[ n_\\alpha v_\\alpha ^2 \u0026#43; 3 \\frac{n_\\alpha T_\\alpha}{m_\\alpha} \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Term 2:\n \\[\\int v^2 \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \\dd \\vec v = \\int \\pdv{}{\\vec x} \\cdot (v^2 \\vec v f_\\alpha) \\dd \\vec v - \\int f_\\alpha \\pdv{}{\\vec x} \\cdot (v^2 \\vec v) \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int ( \\vec v_\\alpha \u0026#43; \\vec w) \\cdot (\\vec v_\\alpha \u0026#43; \\vec w) \\vec w f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int (v_\\alpha ^2 \u0026#43; 2 \\vec v_\\alpha \\cdot \\vec w \u0026#43; w^2) \\vec v f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\left[ \\int v_\\alpha ^2 (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \u0026#43; \\int 2 \\vec v_\\alpha \\cdot \\vec w (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \u0026#43; \\int w^2 (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \\right] \\\\ = \\pdv{}{\\vec x} \\cdot \\left[ v_\\alpha ^2 \\vec v_\\alpha \\int f_\\alpha \\dd \\vec v \u0026#43; v_\\alpha ^2 \\int \\vec w f_\\alpha \\dd \\vec v \u0026#43; 2 \\vec v_\\alpha \\cdot \\left(\\int \\vec w f_\\alpha \\dd \\vec v \\right) \\vec v_\\alpha \\\\ \\qquad \u0026#43; 2 \\vec v_\\alpha \\cdot \\int \\vec w \\vec w f_\\alpha \\dd \\vec v \u0026#43; \\vec v_\\alpha \\int w^2 f_\\alpha \\dd \\vec v \u0026#43; \\int \\vec w w^2 f_\\alpha \\dd \\vec v \\right] \\\\ = \\pdv{}{\\vec x} \\cdot (n_\\alpha v_\\alpha ^2 \\vec v_\\alpha) \u0026#43; \\pdv{}{\\vec x} \\cdot 2 \\vec v_\\alpha \\cdot \\frac{\\vec P_\\alpha}{m_\\alpha} \u0026#43; \\pdv{}{\\vec x} \\cdot \\left(\\frac{3 n_\\alpha T_\\alpha \\vec v_\\alpha}{m_\\alpha} \\right) \u0026#43; \\pdv{}{\\vec x} \\cdot \\left( \\frac{2 \\vec h_\\alpha}{m_\\alpha} \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  In keeping with the progression so far, the very last term would be something like a full 3rd moment  \\( \\int \\vec w \\vec w \\vec w f_\\alpha \\dd \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Instead, we have a \u0026ldquo;contracted 3rd moment\u0026rdquo; of the distribution  \\( \\vec h_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  which is actually the heat flux, or the random energy flux of random energy.\nTerm 3:\n \\[\\int v^2 \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v \\\\ = \\oint f_\\alpha \\frac{q_\\alpha}{m_\\alpha} v^2 ( \\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\dd \\vec S_v - \\int f_\\alpha \\frac{q_\\alpha}{m_\\alpha} \\pdv{}{\\vec v} \\cdot \\left[ v^2 ( \\vec E \u0026#43; \\vec v \\cross \\vec B) \\right] \\dd \\vec v \\\\ = - \\frac{q_\\alpha}{m_\\alpha} \\int f_\\alpha \\left[ v^2 \\left(\\pdv{}{\\vec v} \\cdot \\vec E \\right) \u0026#43; \\left(\\vec E \\cdot \\pdv{}{\\vec v} v^2 \\right) \\right] \\dd \\vec v \\\\ \\qquad - \\frac{q_\\alpha}{m_\\alpha}\\int f_\\alpha \\left[v^2 \\pdv{}{\\vec v} \\cdot (\\vec v \\cross \\vec B) \u0026#43; \\vec v \\cross \\vec B \\cdot \\pdv{}{\\vec v} v^2 \\right] \\dd \\vec v \\\\ = - \\frac{q_\\alpha}{m_\\alpha} \\int f_\\alpha (2 \\vec E \\cdot \\vec v) \\dd \\vec v \\\\ = - \\frac{2 q_\\alpha}{m_\\alpha} n_\\alpha \\vec v_\\alpha \\cdot \\vec E\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We\u0026rsquo;ve ended up with the work done on a fluid by the electric field. As expected, there is no work term associated with the bulk magnetic field.\nTerm 4:\n \\[\\left. \\int v^2 \\pdv{f_\\alpha}{t} \\right|_{coll} \\dd \\vec v = \\int v^2 C_{\\alpha \\alpha} \\dd \\vec v \u0026#43; \\sum _{\\beta \\neq \\alpha} \\int v^2 C_{\\alpha \\beta} \\dd \\vec v \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can make an energy conservation argument to get rid of the first term. The first term goes to 0 since collisions of like particles results in no net energy exchange.\n \\[= \\sum_{\\beta \\neq \\alpha} \\int ( \\vec v_\\alpha \u0026#43; \\vec w) \\cdot(\\vec v_\\alpha \u0026#43; \\vec w) C_{\\alpha \\beta} \\dd \\vec v \\\\ = \\sum_{\\beta \\neq \\alpha}\\left( v_\\alpha ^2 \\int C_{\\alpha \\beta} \\dd \\vec v \u0026#43; 2 \\vec v_\\alpha \\cdot \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v \u0026#43; \\int w^2 C_{\\alpha \\beta} \\dd \\vec v \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Since collisions (we assume) don\u0026rsquo;t create or destroy particles, the first term  \\( v_\\alpha ^2 \\int C_{\\alpha \\beta} \\dd \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  goes away. The second term  \\( \\vec v_\\alpha \\cdot \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we interpret as a frictional force due to the relative motion between  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[= \\sum_{\\beta \\neq \\alpha} 2 \\frac{\\vec v_\\alpha \\cdot \\vec R_{\\alpha \\beta}}{m_\\alpha} \u0026#43; 2 \\frac{Q_{\\alpha \\beta}}{m_\\alpha}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( Q_{\\alpha \\beta} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the heat exchange term (heat generation term in Braginskii). It describes the heat exchange by random collisions of unlike particles, analogous to viscous heating.\n \\[Q_{\\alpha \\beta} = \\frac{1}{2} \\int m_\\alpha w^2 C_{\\alpha \\beta} \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we were to draw distribution functions for species  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  (as shown), even though they have the same centroid they will interact by viscous heating. Species  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  will heat up and  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  will cool\nIf we multiply by  \\( \\frac{1}{2} m_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  then the energy equation becomes\n \\[\\text{Energy Equation: } \\\\ \\qquad \\pdv{}{t} \\left( \\frac{3}{2} n_\\alpha T_\\alpha \u0026#43; \\frac{1}{2} \\rho_\\alpha v_\\alpha ^2 \\right) \u0026#43; \\\\ \\div \\left(\\frac{3}{2} n_\\alpha T_\\alpha \u0026#43; \\frac{1}{2} \\rho_\\alpha v_\\alpha ^2 \\right) \\vec v_\\alpha \u0026#43; \\div (\\vec v_\\alpha \\cdot \\vec P_\\alpha) \u0026#43; \\div \\vec h_\\alpha - q_\\alpha n_\\alpha \\vec v_\\alpha \\cdot \\vec E \\\\ = \\sum_{\\beta \\neq \\alpha} (v_\\alpha \\cdot \\vec R_{\\alpha \\beta} \u0026#43; Q_{\\alpha \\beta})\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Notice that the total energy appears\n \\[E_\\alpha = \\frac{3}{2} n_\\alpha T_\\alpha \u0026#43; \\frac{1}{2} \\rho_\\alpha v_\\alpha ^2\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The factor of  \\( \\frac{3}{2} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as usual comes from  \\( \\gamma \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{3}{2} = \\frac{1}{\\gamma - 1} = \\frac{1}{5/3 - 1}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where\n \\[\\gamma = \\frac{DOF \u0026#43; 2}{DOF} = \\frac{5}{3}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can remove the kinetic portion of the energy equation by subtracting the product of the momentum equation with  \\( \\vec v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\n \\[\\rho_\\alpha \\vec v_\\alpha \\cdot \\pdv{v_\\alpha}{t} = \\frac{1}{2} \\pdv{}{t} (\\rho_\\alpha v_\\alpha ^2)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\pdv{}{t} \\left( \\frac{3}{2} n_\\alpha T_\\alpha \\right) \u0026#43; \\div \\left(\\frac{3}{2} n_\\alpha T_\\alpha \\vec v_\\alpha \\right) \u0026#43; \\div \\left( \\vec v_\\alpha \\cdot \\vec P_\\alpha \\right) - \\vec v_\\alpha \\cdot (\\div \\vec P_\\alpha) \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\frac{3}{2} n_\\alpha \\left( \\pdv{T_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\div T_\\alpha \\right) \u0026#43; \\frac{3}{2} T_\\alpha \\left[ \\pdv{n_\\alpha}{t} \u0026#43; \\div (n_\\alpha \\vec v_\\alpha) \\right] \u0026#43; \\vec P_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\frac{3}{2} n_\\alpha \\left( \\pdv{T_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\div T_\\alpha \\right) \u0026#43; \\vec P_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nWhere  \\( \\vec P_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is a tensor contraction in two indices. It is a generalization of the dot product, i.e.  \\[\\vec P_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha = \\delta_{ik} \\delta_{jl} P_{ij} \\partial_k v_l \\\\ = P_{xx} \\partial_x v_x \u0026#43; P_{xy} \\partial_y v_x \u0026#43; \\ldots\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{3}{2} n_\\alpha \\left( \\pdv{T_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad T_\\alpha \\right) \u0026#43; P_\\alpha \\div \\vec v_\\alpha \u0026#43; \\vec \\Pi_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Now the continuity, momentum, and energy equations describe the evolution of each fluid  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The only velocity information we have retained in taking moments is the centroid  \\( \\vec v_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and the width  \\( T_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . This is the only information we will include in the 5-moment model (5N-moment plasma fluid model). The higher moments (describing skewness, kurtosis) are not evolved with the 5-moment model. There are fluid models that do evolve those moments, but we won\u0026rsquo;t touch on those here.\nClosure Problem #  We still need to \u0026ldquo;close\u0026rdquo; the fluid model (i.e. solving the Closure Problem). As we have been finding out, each additional moment of the Boltzmann Equation introduces the next-higher moment. In calculating the 0th moment, we introduce the 1st moment, etc. We need to address the closure problem by relating higher moment variables to lower moment variables. Applying a closure scheme (relating variables that are not evolved directly) is usually equivalent to making a statement on heat flow.\nIn our model, we are not evolving the heat flux  \\( \\vec h_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , so we need to relate it to one of the other variables that we do evolve directly. We usually do that by writing down a conductivity relation\n \\[\\vec h_\\alpha = - \\kappa \\grad T_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for some conductivity  \\( \\kappa \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nWe\u0026rsquo;re also only evolving the scalar pressure, so we need to relate the stress tensor  \\( \\vec \\Pi_\\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to the other variables. We can do that with a viscosity relation\n \\[\\vec \\Pi_\\alpha = \\nu \\grad \\vec v_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for some viscosity  \\( \\nu \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . We call the introduced quantities  \\( \\kappa \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\nu \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  the transport coefficients. They are usually derived by performing an expansion of the equations we\u0026rsquo;ve already discussed. They are what lead to \u0026ldquo;nondimensional\u0026rdquo; numbers that characterize the flow in our system.\nWe can also achieve closure by constraining portions of the system. In an isothermal system\n \\[\\text{isothermal: } \\quad T = \\text{const.}, p \\propto n\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{adiabatic: } \\quad p \\propto n^\\gamma\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{force-free, cold: } \\quad p = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThe expressions that lead us to a closure scheme are the equations of state. They are not time-dependent, they simply relate fluid parameters to other properties of the system.\nMaxwell\u0026rsquo;s equations to couple the electromagnetic terms to the fluid variables.  \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\epsilon_0 \\pdv{\\vec E}{t} = \\frac{1}{\\mu_0} \\curl \\vec B - \\sum_{alpha} q_\\alpha n_\\alpha \\vec v_\\alpha \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\epsilon_0 \\div \\vec E = \\sum_\\alpha q_\\alpha n_\\alpha \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\div \\vec B = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThe fluid variables provide the source terms for Ampere\u0026rsquo;s law and Gauss' law, and we\u0026rsquo;ve already seen how the electromagnetic fields appear in the momentum equations as source terms for the fluid forces.\nStopping after the 2nd moment (5N-Moment Plasma Fluid Model) #  What assumptions have we made by stopping here? It\u0026rsquo;s important to know what they are and to recognize what they mean for the fluid model.\n Each species  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is well-represented by a Maxwellian with a small perturbation. The pressure tensor is not strictly diagonal and the heat flux is not zero, so the small perturbations are what lead to the transport coefficients. This is due to the process by which we obtain the transport coefficients, called the Chapman-Enskog expansion. Kinetic effects (stream instabilities, counter-flow instabilities, etc.) are not captured.  The variables of the 5-moment model that we do evolve are  \\[n, v_x, v_y, v_z, T\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nWe can take additional moments. The 10N-moment equation evolves, in addition to the quantities in the 5N-moment model, we evolve all of the independent terms of the pressure tensor  \\[n, v_x, v_y, v_z, P_{xx}, P_{xy}, P_{xz} , P_{yy} , P_{yz} , P_{zz} \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThe 13-N moment model has everything from the 10N-moment model plus the heat flux tensor  \\( h_x, h_y, h_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n"});index.add({'id':3,'href':'/r/notes/UWAA558/04-two-fluid-plasma-model/','title':"Two-Fluid Plasma Model",'section':"MHD Theory",'content':"Two-Fluid Plasma Model (ions-electrons) #  Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let\u0026rsquo;s start with a cold plasma approximation (   \\( p = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ) and neglect collisions. The momentum equation reduces to\n \\[m_\\alpha \\left( \\pdv{\\vec v_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad \\vec v_\\alpha \\right) - q_\\alpha (\\vec E \u0026#43; \\vec v_\\alpha \\cross \\vec B) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  From here on out we can avoid some clutter (and wrist strain) by dropping the  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  subscripts and acknowledging that we have sets of equations for ions and electrons. Apply a perturbation to an equilibrium  \\( g = g_0 \u0026#43; g_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[m \\left(\\pdv{\\vec v_0}{t} \u0026#43; \\vec v_0 \\cdot \\grad \\vec v_0 \u0026#43; \\pdv{\\vec v_0}{t} \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_0 \u0026#43; \\vec v_0 \\cdot \\grad \\vec v_1 \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_1 \\right) \\\\ \\qquad - q (E_0 \u0026#43; E_1) - q (\\vec v_0 \\cross \\vec B_0 \u0026#43; \\vec v_1 \\cross \\vec B_0 \u0026#43; \\vec v_0 \\cross \\vec B_1 \u0026#43; \\vec v_1 \\cross \\vec B_1) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can drop some terms because equilibrium has to satisfy the original equation. We can balance all of the subscript-0 terms and sum them to get zero.\n \\[m \\left( \\pdv{\\vec v_1}{t} \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_0 \u0026#43; \\vec v_0 \\cdot \\grad \\vec v_1 \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_1 \\right) \u0026#43; q (- E_1 - \\vec v_1 \\cross \\vec B_0 - \\vec v_0 \\cross \\vec B_1 - \\vec v_1 \\cross \\vec B_1)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Let\u0026rsquo;s now make the assumption that the perturbation is small, that is  \\( g_1 \\ll g_0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . That means that nonlinear products of perturbation terms are negligible (linearization process).\n \\[m \\left( \\pdv{\\vec v_1}{t} \u0026#43; \\vec v_0 \\cdot \\grad \\vec v_1 \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_0 \\right) - q \\vec E_1 - q (\\vec v_1 \\cross \\vec B_0 \u0026#43; \\vec v_0 \\cross \\vec B_1) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Now, assume that the equilibrium is a static equilibrium, that is  \\( \\vec v_0 = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . If we decompose into components that are parallel and perpendicular to the equilibrium magnetic field  \\( \\vec B_0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , then\n \\[\\pdv{v_{1, \\parallel}}{t} - \\frac{q}{m} E_{1, \\parallel} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\pdv{\\vec v_{1, \\perp}}{t} - \\frac{q}{m} \\left(\\vec E_{1, \\perp} \u0026#43; B_0 \\vec v_{1, \\perp} \\cross \\vu z \\right) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThe parallel component  \\( E_{1, \\parallel} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  will lead us to the ordinary wave (O-wave). Consideration of the more general case with perpendicular components will lead to the X-wave.\nThe plasma velocity is related to the fields through the current density (Maxwell equations). Faraday\u0026rsquo;s law gives\n \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow \\curl \\pdv{\\vec B}{t} = - \\curl \\curl \\vec E = - \\grad (\\div \\vec E) \u0026#43; \\nabla ^2 \\vec E \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nAmpere\u0026rsquo;s law gives\n \\[\\epsilon_0 \\pdv{\\vec E}{t} = \\frac{1}{\\mu_0} \\curl \\vec B - \\sum_{\\alpha} q_\\alpha n_\\alpha \\vec v_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\epsilon_0 \\pdv{^2 \\vec E }{t ^2} = \\frac{1}{\\mu_0} \\curl \\pdv{\\vec B}{t} \\\\ = \\frac{1}{\\mu_0} \\left[ \\nabla ^2 \\vec E - \\grad (\\div \\vec E) \\right] - \\sum_\\alpha q_\\alpha \\pdv{}{t} (n_\\alpha \\vec v_\\alpha) \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nSince this is a linear system, assume that the perturbed quantities have a wave-like structure. That is, the perturbed quantities  \\( g_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  are proportional to  \\( e^{i(\\omega t \u0026#43; \\vec k \\cdot \\vec r)} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . This lets us transform the spatial and temporal derivatives into factors of  \\( \\omega \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\vec k \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[- \\epsilon_0 \\omega ^2 \\vec E_1 = - \\frac{1}{\\mu_0} \\left[k^2 \\vec E_1 - \\vec k (\\vec k \\cdot \\vec E_1) \\right] \u0026#43; i \\omega e n_0 \\vec v_1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Let\u0026rsquo;s now consider only high frequency oscillations, assuming that only the electrons respond and the ions remain stationary. There\u0026rsquo;s nothing particularly complicated about including the ion response, this just lets us drop the  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  subscripts and focus on a single set of equations.\n \\[- i \\omega \\frac{e n_0}{\\epsilon_0} \\vec v_1 = (\\omega ^2 - c^2 k^2) \\vec E_1 \u0026#43; c^2 \\vec k (\\vec k \\cdot \\vec E_1)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Now let\u0026rsquo;s apply the perturbed form to the linearized momentum equation\n \\[\\pdv{v_{1, \\parallel}}{t} - \\frac{q}{m} E_{1, \\parallel} = 0 \\\\ \\rightarrow i \\omega v_{1, \\parallel} = - \\frac{e}{m} E_{1, \\parallel} \\\\ \\rightarrow i \\omega \\frac{e n_0}{\\epsilon_0} v_{1, \\parallel} = \\frac{e^2 n_0}{\\epsilon_0 m} E_{1, \\parallel}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Combine the momentum equation and the Maxwell equations to eliminate  \\( \\vec E_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\vec v_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{e^2 n_0}{\\epsilon_0 m} E_{1, \\parallel} = ( \\omega ^2 - c^2 k^2 ) E_{1, \\parallel} \u0026#43; c^2 k_{\\parallel} \\vec (k \\cdot \\vec E_1)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Consider different possibilities for the  \\( \\vec k \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  vector. If it is along the magnetic field  \\( \\vec k = k_{\\parallel} \\vu{e}_\\parallel \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  (longitudinal wave) then\n \\[\\frac{e^2 n_0}{\\epsilon_0 m} = \\omega ^2 - c^2 k^2 \u0026#43; c^2 k^2 = \\omega_{pe}^2\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  For  \\( \\vec k = k_{\\perp} \\vu e_\\perp \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  (transverse wave) then we get the dispersion relation for the O-wave\n \\[\\text{dispersion relation for O-waves:} \\qquad \\omega^2 - c^2 k^2 = \\omega_p ^2\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The electric field is in the same direction as the magnetic field  \\( (\\vec E_1 = \\vec E_{1, \\parallel}) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , which means the O-wave is linearly polarized. At large  \\( k \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we just have regular light waves, but as we turn the frequency downwards we see a cut-off at the plasma frequency:\nIt turns out that the dispersion relation for the X-wave has the same cut-off, but also has another branch with a resonance\nThe two-fluid plasma model is highly reduced from the full kinetic model, but it is still too complete to be useful when studying gross plasma behavior. Further reductions of the model are possible by making asymptotic assumptions:\nLow-frequency Asymptotic Assumption #   Eliminate high frequency, short wavelength phenomena by using pre-Maxwell field equations. Formally, this is  \\( \\epsilon_0 \\rightarrow 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .  The direct consequences of the low-frequency approximation are\n \\[c^2 = \\frac{1}{\\epsilon_0 \\mu_0} \\qquad c \\rightarrow \\infty\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\omega_p ^2 = \\frac{n e^2}{\\epsilon_0 m} \\qquad \\omega_p \\rightarrow \\infty\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\lambda_D = \\frac{v_T}{\\omega_p} \\rightarrow 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThis means that all phenomena will have  \\( \\omega \\ll \\omega_p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , limiting the frequencies we can resolve to the ion plasma frequency. The characteristic speeds will be limited by the speed of light\n \\[\\frac{\\omega}{k} \\ll c\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and all characteristic lengths will be much greater than the Debye length\n \\[x_0 \\gg \\lambda_D\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Looking at Gauss' law,  \\[\\epsilon_0 \\div \\vec E = \\sum_\\alpha q_\\alpha n_\\alpha \\rightarrow \\sum_\\alpha q_\\alpha n_\\alpha = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nso we now have charge neutrality everywhere in the domain. For H plasma, locally we have  \\( n_e = n_i \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  everywhere.\nLooking at Ampere\u0026rsquo;s law,\n \\[\\epsilon_0 \\pdv{\\vec E}{t} = \\frac{1}{\\mu_0} \\curl \\vec B - \\sum_\\alpha q_\\alpha n_\\alpha \\vec v_\\alpha = 0 \\\\ \\rightarrow \\vec j = \\frac{1}{\\mu_0} \\curl \\vec B\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Things we do not get from this approximation are  \\( \\vec E = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  or  \\( \\pdv{\\vec E}{t} = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . It does mean that plasma dynamics occur on a sufficiently large spatial scale that charge separation is small, and they occur on a sufficiently long temporal scale that electrons respond quickly.\nTiny electron asymptotic assumption #  2nd approximation: neglect electron inertia in the momentum equation. Formally, we let the electron mass  \\( m_e \\rightarrow 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\omega_{pe}^2 = \\frac{n e^2}{\\epsilon_0 m_e} \\rightarrow \\infty\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\omega_{c, e} = \\frac{e B}{m_e} \\rightarrow \\infty\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[v_{T, e} \\rightarrow \\infty\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The Larmor radius goes to zero  \\[r_{l, e} = \\frac{v_{T, e}}{\\omega_{c, e}} \\rightarrow 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Importantly, as the gyroradius  \\( r_{l, e} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  goes to 0 (because the thermal velocity goes as  \\( \\sqrt{m_e} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and the cyclotron frequency goes as  \\( m_e \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ), this means that the electrons are tied to the magnetic field. The skin depth is also small.  \\[\\delta_e = \\frac{c}{\\omega_{p, e}} \\rightarrow 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nSo all phenomena that we capture must have  \\( \\omega \\ll \\omega_{p, e} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\omega \\ll \\omega_{c, e} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and  \\( x_0 \\gg r_{L, e} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( x_0 \\gg \\delta_e \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  .\nThe electron momentum equation becomes\n \\[\\grad P_e \u0026#43; \\div \\vec \\Pi _e \u0026#43; e n_e (\\vec E \u0026#43; \\vec v \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\vec R_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The momentum equation is now a state equation, not an evolution equation. It now simply relates the dynamical variables to each other at any point in time.\nNow, note that along magnetic field lines electrons can travel long distances at very fast (finite) speeds which can produce low frequency, long wavelength phenomena. Neglecting electron inertia implies that electrons respond instantaneously, meaning we cannot capture these modes. An example of such a phenomena is drift waves.\nThe characteristic speeds  \\( c \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( v_{T, e} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  have disappeared from the model. Remaining is  \\( v_{T, i} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . This means that the ion dynamics dictate the plasma evolution.\n"});index.add({'id':4,'href':'/r/notes/UWAA558/05-mhd-model/','title':"Magnetohydrodynamic (MHD) Model",'section':"MHD Theory",'content':"Magnetohydrodynamic (MHD) Model #  Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD).\nFirst, define the MHD variables:\n  \\[\\text{mass density:} \\qquad \\rho = n_i m_i \u0026#43; n_i m_e\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{fluid velocity:} \\qquad \\vec v = \\frac{n_i m_i \\vec v_i \u0026#43; n_e m_e \\vec v_e}{n_i m_i \u0026#43; n_e m_e}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{current density:} \\qquad \\vec j = q_i n_i \\vec v_i - e n_e \\vec v_e \\rightarrow \\vec v_e = \\frac{q_i n_i \\vec v_i}{e n_e} - \\vec j / e n_e\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{total pressure:} \\qquad p = p_i \u0026#43; p_e = n_e T_e \u0026#43; n_i T_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{total temperature:} \\qquad T = \\frac{n_i T_i \u0026#43; n_e T_e}{(n_i \u0026#43; n_e)/2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Now let\u0026rsquo;s begin applying asymptotic approximations. For the mass density, applying the first approx (charge neutrality) we have\n \\[\\rho \\approx n (m_i \u0026#43; m_e)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Using approx 2 (vanishing electron mass)\n \\[\\rho \\approx n m_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\[n = n_i \\qquad n_e = Z n\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThe center-of-mass velocity (charge neutrality) gives\n \\[\\vec v \\approx \\frac{m_i \\vec v_i \u0026#43; m_e Z \\vec v_e}{m_i \u0026#43; Z m_e}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  with small electron mass approximation:\n \\[\\vec v \\approx \\vec v_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The current density is (charge neutrality approx)\n \\[\\vec j \\approx Z e n (\\vec v_i - \\vec v_e) \\qquad \\vec v_e = \\vec v_i - \\vec j / Z e n\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The pressure and total temperature are (with charge neutrality)\n \\[P \\approx n (T_i \u0026#43; Z T_e)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[T \\approx \\frac{T_i \u0026#43; Z T_e}{(1 \u0026#43; Z)/2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  MHD Momentum Equation #  Now we combine the two-fluid equations with these asymptotic approximations to obtain the governing equations for the MHD variables. Multiplying the ion continuity equation by the ion mass gives\n \\[\\pdv{}{t} (m_i n_i) \u0026#43; \\div (m_i n_i \\vec v_i) = 0 \\\\ \\rightarrow \\pdv{\\rho}{t} \u0026#43; \\div (\\rho \\vec v) = 0 \\quad \\text{(MHD continuity eq.)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we multiply the two-fluid continuity equations by the charge and sum them, we get\n \\[\\pdv{}{t}(q_i n_i - e n_e) \u0026#43; \\div (q_i n_i \\vec v_i - e n_e \\vec v_e) = 0 \\\\ \\rightarrow \\div \\vec j = 0 \\quad \\text{(no accumulation of charge)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we add the electron and ion momentum equations and apply the small electron mass approximation, and recognizing that  \\( \\vec R_{ei} = - \\vec R_{ie} = n e \\eta \\vec j \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\rho_i \\left(\\pdv{\\vec i}{t} \u0026#43; \\vec v_i \\cdot \\grad \\vec v_i \\right) \u0026#43; \\grad (P_i \u0026#43; P_e) \u0026#43; \\div (\\vec \\Pi _i \u0026#43; \\vec \\Pi_e) \\\\ \\qquad - (q_i n_i - e n_e) \\vec E - (q_i n_i \\vec v_i - e n_e \\vec v_e) \\cross \\vec B = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{MHD Momentum Equation} \\\\ \\rightarrow \\rho \\left(\\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = - \\div (\\vec \\Pi _i \u0026#43; \\vec \\Pi _e)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n!!! info \u0026ldquo;MHD Momentum Equation\u0026rdquo;\n \\[ \\rho \\left(\\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = - \\div (\\vec \\Pi _i \u0026#43; \\vec \\Pi _e) \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   If we take the electron momentum equation, apply the small electron mass asymptotic approximation and introduce the current density, then we have\nMHD Ohm\u0026rsquo;s Law #   \\[e n_e (\\vec E \u0026#43; \\vec v_i \\cross \\vec B - \\vec j \\cross \\vec B / Z e n) = - \\grad P_e - \\div \\vec \\Pi_e \u0026#43; \\vec R_{ei}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Substituting  \\( \\vec R_{ei} = e n_e \\eta \\vec j \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{MHD Generalized Ohms Law}\\\\ \\vec E \u0026#43; \\vec v \\cross \\vec B = \\eta \\vec j \u0026#43; \\frac{1}{Z e n} \\left(\\vec j \\cross \\vec B - \\grad p_e - \\div \\vec \\Pi_e \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n!!! info \u0026ldquo;MHD Generalized Ohms Law\u0026rdquo;\n \\[ \\vec E \u0026#43; \\vec v \\cross \\vec B = \\eta \\vec j \u0026#43; \\frac{1}{Z e n} \\left(\\vec j \\cross \\vec B - \\grad p_e - \\div \\vec \\Pi_e \\right) \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   MHD Energy Equation #  The energy equation is found by adding the ion and electron energies, but first we need to manipulate them into a common form. The ion energy equation is\n \\[\\frac{1}{\\gamma - 1} \\left[ \\pdv{}{t} (n_i T_i) \u0026#43; \\div (n_i T_i \\vec v_i) \\right] \u0026#43; n_i T_i \\div \\vec v_i = RHS_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( RHS_i \\equiv - \\vec \\Pi_i \\cdot \\cdot \\grad \\vec v_i - \\div \\vec h_i \u0026#43; Q_{ie} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{1}{\\gamma - 1} \\left( \\pdv{p_i}{t} \u0026#43; \\vec v_i \\cdot \\grad p_i \\right) \u0026#43; \\frac{\\gamma}{\\gamma - 1} p_i (\\div \\vec v_i) = RHS_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Multiply by  \\( \\frac{\\gamma - 1}{n_i ^\\gamma} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and define the total derivative with respect to the ion velocity as\n \\[\\dv{}{t} \\equiv \\pdv{}{t} \u0026#43; \\vec v_i \\cdot \\grad\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow \\dv{}{t} \\left( \\frac{p_i}{n_i ^\\gamma} \\right) - p_i \\dv{}{t} \\left( \\frac{1}{n_i ^\\gamma} \\right) \u0026#43; \\frac{\\gamma p_i}{n_i ^\\gamma} \\div \\vec v_i = \\frac{\\gamma - 1}{n_i ^\\gamma} RHS_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Simplify further by recognizing that carrying out the total derivative gives\n \\[\\dv{}{t} \\left( \\frac{p_i}{n_i ^\\gamma} \\right) - \\frac{\\gamma p_i}{n_i ^{\\gamma \u0026#43; 1}} \\dv{}{t} n_i \u0026#43; \\frac{\\gamma p_i}{n_i ^\\gamma} \\div \\vec v_i = \\frac{\\gamma - 1}{n_i ^\\gamma} RHS_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The continuity equation says  \\[\\dv{}{t} n_i = - n_i \\div \\vec v_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Multiply by  \\( m_i ^{- \\gamma} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and use  \\( \\rho = n m_i \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and the resulting ion energy equation is\n \\[\\dv{}{t} \\left( \\frac{p_i}{\\rho ^\\gamma} \\right) = \\frac{\\gamma - 1}{\\rho ^\\gamma} RHS_i\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We want to get the electron energy equation in the same form. The steps are very similar:\n \\[\\dv{}{t} \\left( \\frac{p_e}{n_e ^\\gamma} \\right) \u0026#43; \\vec v_e \\cdot \\grad \\left( \\frac{p_e}{n_e ^\\gamma} \\right) = \\frac{\\gamma - 1}{n_e ^\\gamma} RHS_e\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Since we\u0026rsquo;ve defined our center-of-mass reference frame to be that of the ions, we can not use the same total derivative\n \\[\\pdv{}{t} \u0026#43; \\vec v_e \\cdot \\grad \\neq \\dv{}{t}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We now apply the charge neutrality approximation\n \\[n_e = Z n \\qquad \\vec v_e = \\vec v_i - \\frac{1}{Z e n} \\vec j\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow \\dv{}{t} \\left( \\frac{p_e}{Z^\\gamma n^\\gamma} \\right) \u0026#43; \\vec v_i \\cdot \\grad \\left( \\frac{p_e}{Z^\\gamma n^\\gamma} \\right) = \\frac{1}{Z e n} \\vec j \\cdot \\grad \\frac{p_e}{Z^\\gamma n^\\gamma} \u0026#43; \\frac{\\gamma - 1}{Z^\\gamma n^\\gamma} RHS_e\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Multiply by  \\( (Z / m_i)^\\gamma \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and the result is\n \\[\\dv{}{t} \\frac{p_e}{\\rho ^\\gamma} = \\frac{1}{Z e n} \\vec j \\cdot \\grad \\frac{p_e}{\\rho^\\gamma} \u0026#43; \\frac{\\gamma - 1}{\\rho^\\gamma} RHS_e\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Finally we can add the ion energy equation to the electron energy equation to get\n \\[\\dv{}{t} \\left( \\frac{p}{\\rho^\\gamma} \\right) = \\\\ \\frac{\\gamma - 1}{\\rho^\\gamma} \\left[ Q_{ie} \u0026#43; Q_{ei} - \\div (\\vec h_i \u0026#43; \\vec h_e) - \\vec \\pi_i \\cdot \\cdot \\grad \\vec v_i - \\vec \\Pi_e \\cdot \\cdot \\grad \\vec v_e \\right] \u0026#43; \\frac{\\vec j}{Z e n} \\cdot \\grad \\frac{p_e}{\\rho^\\gamma}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  !!! info \u0026ldquo;MHD Energy Equation\u0026rdquo;\n \\[ \\dv{}{t} \\left( \\frac{p}{\\rho^\\gamma} \\right) = \\frac{\\gamma - 1}{\\rho^\\gamma} \\left[ Q_{ie} \u0026#43; Q_{ei} - \\div (\\vec h_i \u0026#43; \\vec h_e) - \\vec \\pi_i \\cdot \\cdot \\grad \\vec v_i - \\vec \\Pi_e \\cdot \\cdot \\grad \\vec v_e \\right] \u0026#43; \\frac{\\vec j}{Z e n} \\cdot \\grad \\frac{p_e}{\\rho^\\gamma} \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   Obviously, we\u0026rsquo;ve retained a number of terms that are specific to the behavior of the electrons. It is possible to incorporate the electron behavior by using a single-fluid MHD model with two temperatures  \\( T_i \\neq T_e \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . One can imagine a hierarchy of models, in which the most simplified is the single-fluid MHD model in which you evolve  \\( \\rho \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and  \\( T \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Moving up a level, you have a MHD model with two temperatures in which you evolve  \\( \\rho \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( T_i \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and  \\( T_e \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Upwards from there you move back into the realm of multi-fluid models.\nNow to relate the fields back to source terms. The low-frequency Maxwell\u0026rsquo;s equations are\n \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\vec j = \\frac{1}{\\mu_0} \\curl \\vec B\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\div \\vec B = \\div \\vec E = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nLike in the multi-fluid plasma model, we still need to close the system by expressing some of our variables using equations of state ( \\( \\vec h \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\vec \\Pi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ).\nTo simplify further, we can make some assumptions about heat flow\n \\[\\text{isothermal} \\qquad \\rightarrow p \\propto n \\qquad T = \\text{const.} \\qquad \\gamma = 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{adiabatic} \\qquad \\rightarrow p \\propto n^\\gamma\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\text{cols plasma / force-free} \\qquad \\rightarrow p = \\text{const.} \\qquad \\gamma = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nIdeal MHD Model #  The extended MHD equations are simpler than the two-fluid model, but they can still be quite complicated. We can often still get useful analysis from further reductions. The ideal MHD model is such a reduction that we can get by dropping (with justification) several terms from the extended model. We justify the simplifications by comparing the magnitude of the neglected terms to the terms that are retained.\nRecall the characteristic speed is  \\( v_{T, i} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . If we say that the characteristic length plasma length is  \\( L \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , then we can define characteristic time  \\( \\tau = L / v_{T, i} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nThe derivation of the two-fluid plasma model assumed a Maxwellian velocity distribution. We need the velocity distribution to thermalize, reach local thermodynamic equilibrium, and become Maxwellian. This means that we need many collisions, in fact so many collisions occurring frequently enough that we can ignore collisional effects. There must then be many collisions during the characteristic time  \\( \\tau \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nFor ions to be thermalized,\n \\[\\frac{\\tau_{ii}}{\\tau} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  And similarly for electrons\n \\[\\frac{\\tau_{e}}{\\tau} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The continuity equation remains unchanged from the extended MHD model\n!!! info \u0026ldquo;Ideal MHD Continuity Equation\u0026rdquo;\n \\[ \\pdv{\\rho}{t} \u0026#43; \\div (\\rho \\vec v) = 0 \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   The momentum equation is\n \\[\\rho \\left( \\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = - \\div (\\vec \\Pi_e \u0026#43; \\vec \\Pi_e)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Drop the anisotropic pressure\n!!! info \u0026ldquo;Ideal MHD Momentum Equation\u0026rdquo;\n \\[ \\rho \\left( \\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = 0 \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   The generalized Ohm\u0026rsquo;s law is\n \\[\\vec E \u0026#43; \\vec v \\cross \\vec B = \\eta \\vec j \u0026#43; \\frac{1}{Z e n} \\left( \\vec j \\cross B - \\grad p_e - \\div \\vec \\Pi_e \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We\u0026rsquo;re going to drop the entire right hand side\n!!! info \u0026ldquo;Ideal MHD Generalized Ohm\u0026rsquo;s Law\u0026rdquo;\n \\[ \\vec E \u0026#43; \\vec v \\cross \\vec B = 0 \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   For the energy equation we have\n \\[\\dv{}{t} \\left( \\frac{p}{\\rho^\\gamma} \\right) = \\frac{\\gamma - 1}{\\rho^\\gamma} [Q_{ie} \u0026#43; Q_{ei} - \\div (\\vec h_i \u0026#43; \\vec h_e) - \\vec \\Pi_i \\cdot \\cdot \\grad \\vec v_i - \\vec \\Pi_e \\cdot \\cdot \\grad \\vec v_e] \u0026#43; \\frac{\\vec j}{Z e n} \\cdot \\grad \\frac{p_e}{\\rho^\\gamma}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We neglect the entire right-hand side\n!!! info \u0026ldquo;Ideal MHD Energy Equation\u0026rdquo;\n \\[ \\pdv{\\rho}{t} \u0026#43; \\div (p \\vec v) = (1 - \\gamma) p \\div \\vec v \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   Collision/Pressure terms #  If we assume that the ions and electrons are in thermal equilibrium \\( T_i = T_e \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , we can relate the collision times\n \\[\\tau_{ee} : \\tau_{ii} : \\tau_{ei} = 1 : \\left( \\frac{m_i}{m_e} \\right) ^{1/2} : \\frac{m_i}{m_e}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The collision times are specifically collisional relaxation times of the Boltzmann equation\n \\[\\left. \\pdv{f}{t} \\right|_{coll} = \\frac{f - f_{\\text{Maxwellian}}}{\\tau_{\\alpha \\beta}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  For electrons, the thermalization condition is much stricter for the ions\n \\[\\frac{\\tau_{ee}}{\\tau} = \\left( \\frac{m_e}{m_i} \\right)^{1/2} \\frac{\\tau_{ii}}{\\tau} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Neglect the anisotropic pressure tensor in the momentum and generalized Ohm\u0026rsquo;s law,  \\( \\div \\vec \\Pi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .  \\( \\vec \\Pi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is primarily the shear stress tensor. The ion thermal speed gives us a characteristic velocity for the plasma, so we use it to characterize the shear stress\n \\[\\vec \\Pi_{i, max} \\sim 2 \\mu \\left( \\pdv{u}{x} - \\frac{1}{3} \\div \\vec v \\right) \\sim \\mu \\frac{v_{T, i}}{L}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Standard treatments of the viscosity (Braginskii, etc.) show that viscosity scales with the number density, temperature, and collision time\n \\[\\mu \\sim n T_i \\tau_{ii} \\sim p_i \\tau_{ii}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\Pi_{i, max} \\sim p_i \\frac{\\tau_{ii} v_{T, i}}{L}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The specific term we want to get rid of is  \\( \\div \\vec \\Pi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , so let\u0026rsquo;s compare it to a term we want to keep  \\( \\grad P \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{\\div \\vec \\Pi}{\\grad p} \\sim \\frac{p_i \\tau_{ii} v_{T, i} / L^2}{p_i / L} \\sim \\frac{\\tau_{ii}}{\\tau}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  So, to neglect the anisotropic pressure term in the momentum equation, once again we require\n \\[\\frac{\\tau_{ii}}{\\tau} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  In other words, as long as the plasma is collision-dominated, we can drop the ion anisotropic pressure term. What about associated the electron term? If you can assume  \\( T_i \\approx T_e \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Then  \\( p_i \\approx p_e \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for a neutral plasma, and\n \\[\\frac{\\div \\vec \\Pi_e}{\\grad p} \\sim \\frac{p_i \\tau_{ee} v_{T, i} / L^2}{p_i / L} \\sim \\frac{\\tau_{ee}}{\\tau} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Magnetic terms #  In the generalized Ohm\u0026rsquo;s law, the diamagnetic drift term is\n \\[\\frac{\\grad p_e}{Z e n} \\sim \\frac{n T_e / L}{Z en} \\sim \\frac{T_i / L}{Ze} \\sim \\frac{m_i v_{T, i}^2}{L Z e}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Compare  \\( \\grad p_e / Z e n \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to a term that we\u0026rsquo;re going to keep, which is the dynamo term  \\( \\vec v \\cross \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{ \\grad p_e / Z en}{|\\vec v \\cross \\vec B|} \\sim \\frac{m_i v_{T, i}^2}{LZe}{v_{T, i} B} \\sim \\frac{m_i}{ZeB}\\frac{v_{T, i}}{L} = \\frac{v_{T, i}}{\\omega_{c, i}} \\frac{1}{L} \\sim \\frac{r_{L, i}}{L} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  So to neglect the diamagnetic drift term, we need the plasma to be well-magnetized. This means the Larmor radius must be much less than the plasma characteristic length  \\( r_{L, i} \\ll L \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nNow what can we do with the Hall term  \\( \\frac{\\vec j \\cross \\vec B}{Zen} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . For a static plasma (or one with subsonic flows):\n \\[\\rho (\\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v) - \\grad p - \\vec j \\cross \\vec B = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  so by \u0026ldquo;subsonic\u0026rdquo; we mean that the static pressure is much larger than the dynamic pressure and we can discard the  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  terms.\n \\[\\vec j \\cross \\vec B \\approx \\grad p\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Comparing the Hall term to the dynamo term  \\( \\vec v \\cross \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  also gives the same requirement\n \\[\\frac{r_{L, i}}{L} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  How well do some real plasmas hold up to the requirements of ideal MHD? Consider field nulls in a Z-pinch or an FRC, and weakly magnetized plasmas (such as those in a Hall thruster). Clearly, the magnetization requirement does not hold up at all points in space, and ideal MHD does not necessarily apply across the whole domain.\nResistivity #  We neglect resistivity and the resistive electric field  \\( \\eta \\vec j \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  in the generalized Ohm\u0026rsquo;s law\n \\[\\eta \\vec j \\sim \\eta \\frac{\\grad p}{B} \\sim \\frac{ \\eta n T}{LB}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\eta = \\frac{m_e \\nu_{ei}}{n e^2} = \\frac{m_e}{n e^2 \\tau_{ei}} = \\frac{m_e ^2/ m_i}{n e^2 \\tau_{ee}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow \\eta \\vec j \\sim \\frac{m_e ^2}{e^2 L B} \\frac{v_{T, i} ^2}{\\tau_{ee}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  How does it compare to  \\( \\vec v \\cross \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{\\eta \\vec j}{|\\vec v \\cross \\vec B|} \\sim \\left(\\frac{m_e}{m_i} \\right)^2 \\frac{m_i}{e ^2 B^2} \\frac{v_{T, i} ^2}{L^2} \\frac{L}{v_{T, i} \\tau_{ee}} \\sim \\left( \\frac{m_e}{m_i}\\right)^2 \\frac{v_{T, i} ^2}{\\omega_{c, i} ^2} \\frac{1}{L^2} \\sim \\left( \\frac{m_e}{m_i} \\right) ^{3/2} \\left( \\frac{r_{L, i}}{L} \\right) ^{2} \\frac{\\tau}{\\tau_{ii}} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  This now places a lower limit on  \\( \\tau_{ii} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ; to neglect resistivity  \\( \\tau_{ii} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  cannot be too low\n \\[\\left( \\frac{m_e}{m_i} \\right) ^{3/2} \\left( \\frac{r_{L, i}}{L} \\right) ^{2} \\ll \\frac{\\tau_{ii}}{\\tau} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Heating sources #  Neglect the collisional heating sources  \\( Q_{ei} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( Q_{ie} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  in the energy equation. We do that by assuming that they are equal and opposite, which only happens when the temperatures are equal. In other words, we are again assuming local thermodynamic equilibrium between electrons and ions.\n \\[1 \\gg \\frac{\\tau_{ei}}{\\tau} = \\left( \\frac{m_i}{m_e} \\right)^{1/2} \\frac{\\tau_{ii}}{\\tau} \\rightarrow \\frac{\\tau_{ii}}{\\tau} \\ll \\left( \\frac{m_e}{m_i} \\right) ^{1/2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  This is a much more restrictive condition than ion collisionality. Alternatively, we could track two temperature independently.\nWe also neglect the heat flux terms  \\( \\div (\\vec h_i \u0026#43; \\vec h_e) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  in the energy equation. Consider parallel (to the magnetic field) heat conduction which dominates since  \\( \\kappa_\\perp \\ll \\kappa_\\parallel \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , so\n \\[\\div \\vec h \\approx \\grad_\\parallel \\cdot (\\kappa_\\parallel \\grad_\\parallel T)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\grad_\\parallel \\cdot \\left[ ( \\kappa_{\\parallel, i} \u0026#43; \\kappa_{\\parallel, e}) \\grad_\\parallel T \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\kappa_{\\parallel, e} \\sim \\frac{n T_e}{m_e} \\tau_{ee} \\qquad \\text{ and } \\qquad \\kappa_{\\parallel, i} \\sim \\frac{n T_i}{m_i} \\tau_{ii}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\frac{\\kappa_{\\parallel, e}}{\\kappa_{\\parallel, i}} \\sim \\frac{\\tau_{ee}}{\\tau_{ii}} \\frac{m_i}{m_e} \\sim \\left( \\frac{m_i}{m_e} \\right) ^{1/2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Compare the thermal conductivity to the rate of pressure change  \\( \\pdv{p}{t} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{\\grad_\\parallel \\cdot (\\kappa_{\\parallel, e} \\grad_\\parallel T)}{\\pdv{p}{t}} \\sim \\frac{n T^2 \\tau_{ee}/m_e L^2}{nT/\\tau} \\sim \\left( \\frac{m_i}{m_e} \\right)^{1/2} \\frac{\\tau_{ii}}{\\tau} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We\u0026rsquo;re back to the same requirement that we have ion-electron local thermodynamic equilibrium.\nAnisotropic pressure terms in energy equation #  Neglect the anisotropic pressure terms in the energy equation. That is,  \\( \\vec \\Pi_i \\cdot \\cdot \\grad \\vec v_i \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\vec \\Pi_e \\cdot \\cdot \\grad \\vec v_e \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Skipping ahead, the result is\n \\[\\frac{\\tau_{ii}}{\\tau} \\ll 1 \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\left( \\frac{m_e}{m_i} \\right)^{1/2} \\frac{\\tau_{ii}}{\\tau} \\frac{r_{L, i}}{L} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nElectron convection term #  Lastly, we neglect the electron convection term in the energy equation  \\( \\frac{\\vec j}{Zen} \\cdot \\grad \\frac{P_e}{\\rho^\\gamma} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The result is\n \\[\\frac{r_{L, i}}{L} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Conservation Law Form of MHD #   \\[\\pdv{}{t} q \u0026#43; \\div \\vec f = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can express momentum in conservation law form:\n \\[\\pdv{}{t} (\\rho \\vec v) \u0026#43; \\div \\left[ \\rho \\vec v \\vec v - \\frac{ \\vec B \\vec B}{\\mu_0} \u0026#43; (p \u0026#43; \\frac{B^2}{2 \\mu_0})\\vec 1 \\right] = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The conservation law form for the magnetic field looks like\n \\[\\pdv{ \\vec B}{t} \u0026#43; \\div \\left[ \\vec v \\vec B - \\vec B \\vec v \\right ] = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  And of course the energy equation for\n \\[E = \\frac{1}{\\gamma - 1} p \u0026#43; \\frac{1}{2} \\rho v^2 \u0026#43; \\frac{ B^2}{2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\pdv{E}{t} \u0026#43; \\div \\left[ \\left( E \u0026#43; p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) - \\left( \\frac{ \\vec B \\cdot \\vec v}{\\mu_0} \\right) \\vec B \\right] = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Conservation law forms are particularly useful when considering equilibrium steady-state force balance. This means that in steady-state equilibrium we have\n \\[\\div \\left[ \\rho \\vec v \\vec v - \\frac{ \\vec B \\vec B}{\\mu_0} \u0026#43; \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\vec 1 \\right] = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can use this relationship and integrate over various volumes to determine the relationship between the various force balance terms\nExamples of equilibrium plasma confinement #  For a plasma in static equilibrium  \\( \\vec v = 0 \\rightarrow \\vec j \\cross \\vec B = \\grad p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[ \\div \\vec T = 0 = \\div \\left( \\right) \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\vec j \\cross \\vec B = \\frac{1}{\\mu_0} (\\curl \\vec B) \\cross \\vec B = \\frac{1}{\\mu_0} (\\vec B \\cdot \\grad \\vec B - \\frac{1}{2} \\grad B^2) \\\\ = \\frac{1}{\\mu_0} ( B^2 \\vu e_B \\cdot \\grad \\vu e_B \u0026#43; \\frac{1}{2} \\vu e_B \\vu e_B \\cdot \\grad B^2 - \\frac{\\grad B^2}{2} )\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( \\vu e_B = \\vec B / B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\( \\vu e_B \\cdot \\grad \\vu e_B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the curvature of  \\( \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Write it like a curvature\n \\[\\vec K = - \\vu r / R_c\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\vu e_B \\vu e_B \\cdot \\grad B^2\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is gradient of  \\( B^2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  that is parallel to B. Multiply by  \\( e_B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  gives the component of gradient along  \\( e_B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The difference between that and  \\( \\grad B^2 / 2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  gives you the perpendicular gradient\n \\[\\vec j \\cross \\vec B = \\frac{1}{\\mu_0} (B^2 \\vec \\kappa - \\frac{1}{2} \\grad _\\perp B^2) = \\grad p = \\grad_\\perp p\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  identify  \\( B^2 \\vec \\kappa \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is magnetic tension resulting from having a bent magnetic field line.  \\( \\frac{1}{2} \\grad_\\perp B^2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is magnetic pressure. They have to balance the plasma pressure at equilibrium.\nFor example, consider a cylindrical plasma that\u0026rsquo;s in equilibrium with a helical magnetic field  \\[\\vec B = B_\\theta (r) \\vu \\theta \u0026#43; B_z (r) \\vu z\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nHow is plasma pressure profile determined by the different components of the magnetic field? If we want to maximize the amount of pressure we confine, what should be maximized/minimized?\n \\[\\frac{B^2}{\\mu_0} \\vu \\kappa = \\grad _\\perp ( p \u0026#43; \\frac{B^2}{2 \\mu_0} ) \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is straight and has no curvature, so the only magnetic tension comes from  \\( B_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , so the magnetic tension from  \\( B_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  must balance the total pressure.\nThe role of  \\( B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is displacing plasma pressure. The utility in defining  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as\n \\[\\beta = \\frac{\\text{plasma pressure}}{\\text{magnetic pressure}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Conditions of Ideal MHD Validity #  The conditions for ideal MHD to be valid are\n High Ion Collisionality:  \\( \\frac{\\tau_{ii}}{\\tau} \\ll 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Small ion Larmor radius:  \\( \\frac{r_{L, i}}{L} \\ll 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Low resistivity:  \\( \\left(\\frac{m_e}{m_i} \\right)^{3/2} \\left( \\frac{r_{L, i}}{L} \\right)^2 \\ll 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   For a given plasma in force balance, we can relate the plasma pressure to the magnetic pressure\n \\[\\beta = \\frac{n T}{B^2 / 2 \\mu_0} = 4 \\times 10^{-16} \\frac{n_{cm^{-3}} T_{keV}}{B^2 _{T}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Ion collision time is (Spitzer collisionality)\n \\[\\tau_{ii} = 2.09 \\times 10^{7} \\frac{T_{eV} ^{3/2} \\mu ^{1/2}}{\\ln \\Lambda n_{cm ^{-3}}} \\left[\\text{s}\\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( \\mu \\equiv m_i / m_p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nPutting the conditions for ideal MHD in terms of  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\tau_{ii} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,\n High collisionality:  \\[\\frac{\\tau_{ii}}{\\tau} = 2.14 \\times 10^{12} \\frac{T^2}{n L} \\ll 1 \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow T_{eV} \\ll 6.8 \\times 10^{-7} L_{cm} ^{1/2} n_{cm^{-3}} ^{1/2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Small gyroradius:  \\[\\frac{r_{L, i}}{L} = 2.3 \\times 10^7 \\frac{1}{Z L} \\sqrt{ \\frac{\\mu B}{n}} \\ll 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow n_{cm^{-3}} \\gg 5.3 \\times 10^{14} \\frac{\\mu B_{T}}{Z^2 L_{cm} ^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Low resistivity:  \\[\\left( \\frac{m_e}{m_i} \\right) ^{3/2} \\left( \\frac{r_{L, i}}{L} \\right) ^2 \\frac{\\tau}{\\tau_{ii}} = 5.65 \\frac{\\mu B}{Z^2 L T^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow T_{eV} \\gg 2.4 \\frac{\\mu ^{3/2} \\beta^{1/2}}{Z L _{cm} ^{1.2}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   Perpendicular MHD #  For most configurations for magnetic fusion confinement, we are able to satisfy 2. and 3. but often have densities much too low to meet the high collisionality constraint. However, in practice ideal MHD does accurately model macroscopic behavior of many plasmas. At the same time, magnetized / fusion plasmas are often largely collisionless. We can understand why by re-writing the collisionality requirement as\n \\[1 \\gg \\frac{\\tau_{ii}}{\\tau} = \\frac{\\tau_{ii} v_{T, i}}{\\tau v_{T, i}} \\sim \\frac{\\lambda}{L}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( \\lambda \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the mean free path. The ratio  \\( \\lambda / L \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is also known as the Knudsen number. In magnetized plasmas, the mean free path is often very long, but the path between collisions can cover a great distance only by following magnetic field lines. Motion  \\( \\perp \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to the magnetic field is constrained by  \\( r_{L, i} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . This suggests an approach wherein we divide the plasma model into a 1-D kinetic model to be solved along the magnetic field and a 2-D MHD model  \\( \\perp \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to the magnetic field.\nWe consider diffusivity (terms like viscosity, conductivity) parallel and perpendicular to  \\( \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[k_\\parallel \\sim \\frac{\\lambda ^2}{\\tau_{ii}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[k_\\perp \\sim \\frac{r_{L, i} ^2}{\\tau_{ii}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\frac{k_\\parallel}{k_\\perp} \\sim \\left( \\frac{\\lambda}{r_{L, i}} \\right)^2 \\sim (\\omega_{c, i} \\tau_{ii} )^2\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThis changes one of our conditions for validity. Specifically, we can write\n \\[\\frac{r_{L, i}}{L} \\frac{1}{ \\omega_{c, i} \\tau_{i i}} = 254 \\frac{\\mu \\beta}{Z^2 L_{cm} T_{eV} ^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  or\n \\[T_{eV} \\gg 16 \\frac{\\mu ^{1/2} \\beta^{1/2}}{Z L^{1/2}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  This is now only slightly more restrictive than the low-resistivity condition. In fact, most of the plasmas that we looked at before (large tokamaks \u0026amp; toruses, propulsion systems) which had temperatures above the validity range now fall comfortably within the region of validity. The perpendicular MHD model is equivalent to a collisionless model, giving a much wider applicability than the collisional MHD model.\nWhat about the parallel component? Our collisionality condition still isn\u0026rsquo;t valid parallel to the field. Basically, we ignore the parallel component, which is the same as assuming that  \\( \\rho \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( T \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  are constant along magnetic field lines, with  \\( \\div \\vec v = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  . Let\u0026rsquo;s write out the expressions for collision-less MHD and for ideal MHD and compare:\n    Collisionless MHD Ideal MHD     Continuity  \\( \\pdv{\\rho}{t} = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( \\pdv{\\rho}{t} = - \\rho \\div \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };    Momentum  \\( \\rho \\dv{\\vec v_\\perp}{t} = \\vec j \\cross \\vec B - \\grad_\\perp p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( \\rho \\dv{\\vec v}{t} = \\vec j \\cross \\vec B - \\grad p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };    Parallel constraint  \\( \\vec B \\cdot \\grad \\frac{v_\\parallel}{B} = - \\div \\vec v _\\perp \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };     Energy  \\( \\dv{p}{t} = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( \\dv{p}{t} = - \\gamma p \\div \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };     Collisionless MHD reproduces many of the effects of ideal MHD but has a wider region of validity. Corollary: ideal MHD is accurate beyond its region of validity, unless results lead to parallel gradients. For example, we know that MHD is not valid when representing confinement of a plasma confined in a magnetic mirror, which is an inherently kinetic phenomenon. But ideal MHD can generate parallel gradients within its region of validity, and we need to be careful. Ideal MHD does not require different models  \\( \\parallel \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\perp \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to the magnetic field, and is therefore preferred. We will continue to use ideal MHD outside of its region of validity.\n"});index.add({'id':5,'href':'/r/notes/UWAA558/06-boundary-conditions/','title':"Boundary Conditions",'section':"MHD Theory",'content':"Boundary Conditions #  Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region.\nPerfectly Conducting Wall #  For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:\n  \\[\\left. \\vec E_t \\right|_{wall} = 0 \\quad \\rightarrow \\quad \\left. \\vu n \\cross \\vec E \\right| _{wall} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Applying Faraday\u0026rsquo;s law at the wall,\n \\[\\left. \\vu n \\cdot \\pdv{\\vec B}{t} \\right|_{wall} = \\left. - \\vu n \\cdot \\curl \\vec E \\right|_{wall} = \\left. \\div (\\vu n \\cross \\vec E) \\right| _{wall} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\pdv{}{t} \\vu n \\cdot \\vec B |_{wall} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If initially there is no normal magnetic field, then  \\[\\vu n \\cdot \\vec B|_{wall} = 0 \\quad \\text{if initially true}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nAnd of course, for an impermeable wall,\n \\[\\vu n \\cdot \\vec v |_{wall} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Is this a sufficient set of boundary conditions? Think back to the governing equations in conservation form  \\[\\pdv{}{t} \\vec Q \u0026#43; \\div \\vec F = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The boundary conditions come into play when defining  \\( \\vec F \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  at the boundary. In particular, we need to know what  \\( \\dd \\vec S \\cdot \\vec F |_{wall} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is. In our governing equations, this will involve  \\( \\vec E \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nInsulating Boundary #  As a slight modification, an insulating boundary can have a tangential electric field. Consider a simple geometry of parallel electrodes with an insulating wall between them.\nFrom Ohm\u0026rsquo;s law  \\[\\vec E \u0026#43; \\vec v \\cross \\vec B = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  so the only way an electric field tangential to the wall can exist is if  \\( \\vu n \\cdot \\vec v \\neq 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nFor either a perfectly conducting or an insulating boundary, the other variables are arbitrary:  \\( \\rho \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  ,  \\( p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\vec v_t \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\vec B_t \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nVacuum Region #  The plasma (radius  \\( R_p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ) is supported by a region of vacuum out to a perfectly conducting wall at some radius  \\( R_w \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . We assume that there is no plasma in the vacuum region. The governing equations in vacuum are just Maxwell\u0026rsquo;s equations\n \\[\\curl \\vec B_{vac} = 0 \\qquad \\text{and} \\qquad \\div \\vec B_{vac} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  At the wall,  \\[\\vu n \\cross \\vec E |_{wall} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\left. \\vu n \\cdot \\pdv{\\vec B}{t} \\right|_{wall} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nWhat happens at the plasma-vacuum interface? We need to specify jump conditions and continuity conditions. Let\u0026rsquo;s use square brackets to signify a jump:\n \\[\\left[ X \\right] = \\left. X \\right|_{R_p \u0026#43; dr} - \\left. X \\right|_{R_p - dr}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The normal magnetic field has to be continuous.\n \\[[\\vu n \\cdot \\vec B]_{R_p} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The tangential magnetic field jump is given by the surface current density at the jump\n \\[\\left[ \\vu n \\cross \\vec B \\right] _{R_p} = \\mu_0 \\vec K \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Integrating  \\( \\grad_\\perp (p \u0026#43; \\frac{B^2}{2 \\mu_0}) = \\frac{B^2}{\\mu_0} \\vec \\kappa \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  over a differential volume across the surface gives\n \\[\\left[ p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right] _{R_p} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The plasma shape is determined self-consistently by the wall shape and surface current. This is a free-boundary problem. Another option is to specify the plasma shape, and then determine the required wall shape. This is a fixed-boundary problem.\nThe most realistic case includes externally applied magnetic fields coming from source coils, perhaps computed by Biot-Savart law. The vacuum magnetic field is then  \\( \\vec B_{vac} = \\vec B_{ext} \u0026#43; \\vec B_{plasma} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The crazy coil shapes in the stellarator design come from the 3D geometry computations solving this problem.\nConservation of Magnetic Flux (\u0026ldquo;Frozen-In\u0026rdquo; Flux) #  Locally,  \\( \\vec E \u0026#43; \\vec v \\cross \\vec B = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  with Faraday\u0026rsquo;s law  \\[\\pdv{B}{t} = - \\curl \\vec E = - B \\div \\vec v \u0026#43; \\vec B \\cdot \\grad \\vec v - \\vec v \\cdot \\grad B\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  From the continuity equation,  \\[\\pdv{\\rho}{t} \u0026#43; \\vec v \\cdot \\grad \\rho = - \\rho \\div \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Combining we find that  \\[\\dv{\\vec B}{t} = \\frac{\\vec B}{\\rho} \\dv{\\rho}{t} \u0026#43; \\vec B \\cdot \\grad \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow \\dv{}{t} \\left( \\frac{\\vec B}{\\rho} \\right) = \\frac{\\vec B}{\\rho} \\cdot \\grad \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThis says that the field and plasma density move together. Locally, if the magnetic field increases then mass density increases, such that the ratio  \\( \\vec B / \\rho \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  remains constant. In the direction parallel to the magnetic field we have a term that involves field line twisting, which is a bit more complicated, but in the perpendicular direction  \\[\\dv{}{t} \\left( \\frac{\\vec B}{\\rho} \\right) _\\perp = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nIf we consider globally the magnetic flux through a moving surface S at velocity  \\( \\vec u \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The magnetic flux penetrating the surface is  \\[\\Psi = \\int \\vec B \\cdot \\dd \\vec S\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  or  \\[\\dv{\\Psi}{t} = \\int \\dv{\\vec B}{t} \\cdot \\vu n \\dd S\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[= \\int \\pdv{\\vec B}{t}\\cdot \\vu n \\dd S \u0026#43; \\oint \\vec B \\cross \\vec u \\dd \\vec l\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Using Faraday\u0026rsquo;s law\n \\[\\dv{\\Psi}{t} = \\int - \\curl \\vec E \\cdot \\vu n \\dd S \u0026#43; \\oint \\vec B \\cross \\vec u \\cdot \\dd \\vec l\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[= \\oint (- \\vec E \u0026#43; \\vec B \\cross \\vec u) \\cdot \\dd \\vec l\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Using the electric field from Ohm\u0026rsquo;s law  \\[\\dv{\\Psi}{t} = \\oint(\\vec v - \\vec u) \\cross \\vec B \\cdot \\dd \\vec l\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  This tells us that if the surface moves with the plasma  \\( \\vec u = \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  then  \\[\\dv{\\Psi}{t} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  the flux through the surface is constant, and the flux is a constant of the topology. This is a direct consequence of ideal MHD. If we add even a small amount of resistivity, we dramatically alter the results in a process called \u0026ldquo;tearing\u0026rdquo; where the magnetic field \u0026ldquo;tears\u0026rdquo; and reconnects with itself.\n"});index.add({'id':6,'href':'/r/notes/UWAA558/07-equilibrium-for-fusion/','title':"Equilibrium for Fusion",'section':"MHD Theory",'content':"Equilibrium for Fusion (   \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ) #  For a fusion device we would like to determine a magnetic configuration that confines plasma while it fuses. At fusion temperatures, the power required to maintain the equilibrium will be substantial. For a device to be useful, the power required to sustain the equilibrium must be less than the power released from fusion. Important loss terms for a confined plasma are transport (thermal conduction primarily) and radiation terms. The scaling factors are  \\( P_{Brem} \\sim n^2 T^{1/2} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( P_{cycl} \\sim n^2 T^2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for radiation, and  \\( P_L \\sim \\frac{3nT}{\\tau_E} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for thermal losses.\nWe know that the fusion source term will primarily come from the DT fusion reaction\n \\[\\text{D} \u0026#43; \\text{T} \\rightarrow \\text{He}^4 (3.5\\, MeV) \u0026#43; \\text{n} (14.1\\, MeV)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The primary fusion reaction releases an  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -particle and a high-energy neutron. The concept of ignition is that the neutron leaves the plasma, and the  \\( \\alpha \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  (with energy  \\( E_\\alpha = 3.5 MeV) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  remains to heat the plasma.\n \\[P_\\alpha = \\frac{1}{4} n^2 \\langle \\sigma v \\rangle E_\\alpha \\qquad \\text{(assuming} \\quad n_D = n_T = n/2 \\text{)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[P_\\alpha \u0026gt; P_L \\quad \\rightarrow \\quad n \\tau_E \u0026gt; \\frac{12 T}{E_\\alpha \\langle \\sigma v \\rangle}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  To sustain fusion, we set the fusion heating term above the thermal loss term. The reaction cross-section  \\( \\sigma \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  can be maximized to give the Lawson criterion\n \\[n \\tau_E \u0026gt; 10^{14} s / cm^3\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The Lawson criterion only applies at fusion temperatures, but it is a useful parameter even outside of ignition since it gives a ratio of fusion power to lost power\n    \\( T_i (keV) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( \\langle \\sigma v \\rangle (cm^3 / s) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Required  \\( n \\tau_E (s / cm^3) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };      1  \\( 7 \\cdot 10^{-21} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( 5 \\cdot 10^{17} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };    5  \\( 1.4 \\cdot 10^{-17} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( 1.2 \\cdot 10^{15} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };    20  \\( 4.3 \\cdot 10^{-16} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( 1.6 \\cdot 10^{14} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };    60  \\( 8.7 \\cdot 10^{-16} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( 2.4 \\cdot 10^{14} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };     We can see that the required  \\( n \\tau_E \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  actually has a minimum around  \\( 20 keV \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  (at least, as far as the data in the table goes). Even though the maximum cross-section is at a much higher temperature, what we\u0026rsquo;re really concerned with is the ratio of the fusion source term to the thermal loss term, which is linear in temperature.\nMHD equilibrium does not place a limit on the density  \\( n \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Instead, it places a limit on  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  in order to achieve equilibrium force-balance  \\( (\\beta = 1) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\beta = \\frac{n (T_e \u0026#43; T_i)}{B^2 / 2 \\mu_0} \\rightarrow n = \\frac{ \\beta B^2}{4 \\mu_0 T}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  In this form, we can more clearly see what our options are to achieve MHD equilibrium. Some devices (large-scale tokamaks) are able to achieve the requisite confinement time at a low  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  by making use of very strong magnetic fields. Other devices are able to make use of more modest magnetic fields by working at a higher  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nTherefore,\n \\[\\tau_E \u0026gt; \\frac{1}{\\beta B^2} \\frac{48 \\mu_0}{E_\\alpha} \\frac{T^2}{\\langle \\sigma v \\rangle}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The term  \\( \\frac{T^2}{\\langle \\sigma v \\rangle} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  has a minimum at  \\( 10-20 keV \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . At 15 keV and a magnetic field of  \\( 5T \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  (many actual components cannot reasonably exceed such magnetic fields) then\n \\[\\tau_E \u0026gt; \\frac{0.1}{\\beta} \\text{s}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  For a large-scale toroidal device with  \\( \\beta = 1\\% \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , the confinement time  \\( \\tau_E \u0026gt; 10s \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . If we consider a common diffusivity (how fast energy will leave due to thermal conductivity)  \\( D_E \\approx 1 m^2 / s\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , so for a characteristic radius  \\( a \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\tau_E \\approx \\frac{a^2}{4 D_E} \\rightarrow a \u0026gt; 6.3 \\text{m}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  This gives you a sense of why low- \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  devices need to be so large. Instead, if we consider  \\( \\beta \\sim 50\\% \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\tau_E \u0026gt; 0.2 \\text{s} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and\n \\[\\beta \\sim 50\\% \\rightarrow a \u0026gt; 0.9 \\text{m}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  When you consider that the cost of a device (to first order) scales with the volume of the device, achieving a high  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is very important for fusion equilibrium. However, when we consider MHD stability we are generally forced into lower  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to avoid destructive instabilities. Configuration optimization is the process of balancing this trade-off.\nVirial Theorem #  Application of the virial theorem to energy balance for the stress tensor  \\( \\vec T \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  tells us that MHD equilibria must be supported by externally supplied currents. Many times you\u0026rsquo;ll hear of theoretical designs for compact toroid devices which can maintain stability under their own currents, but they are the MHD stability equivalent of a perpetual motion machine. A compact toroid cannot exist unsupported.\nWriting static equilibrium:\n \\[\\div \\left[ - \\frac{ \\vec B \\vec B}{\\mu_0} \u0026#43; \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0}\\right) \\vec I \\right] = \\div \\vec T = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we define the direction of the magnetic field to be  \\( \\vu e _B = \\vu z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  then\n \\[\\vec T = p_\\perp ( \\vec I - \\vu e_B \\vu e_B ) \u0026#43; p_\\parallel \\vu e_B \\vu e_B \\\\ = \\begin{bmatrix} p_\\perp \u0026amp; 0 \u0026amp; 0 \\\\ 0 \u0026amp; p_\\perp \u0026amp; 0 \\\\ 0 \u0026amp; 0 \u0026amp; p_\\parallel \\end{bmatrix}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where\n \\[p_\\perp = p \u0026#43; \\frac{B^2}{2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\[p_\\parallel = p - \\frac{B^2}{2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nA gradient vector identity gives\n \\[\\div (\\vec r \\cdot \\vec T) = \\vec r \\cdot ( \\div \\vec T) \u0026#43; \\vec T \\cdot \\cdot \\grad \\vec r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Integrating this expression over a volume and assuming that the volume contains a confined MHD equilibrium that is self-contained and self-supported:\n \\[\\int_V \\div ( \\vec r \\cdot \\vec T) \\dd V = \\int_V (\\vec r \\cdot \\overbrace{\\cancel{(\\div \\vec T)}}^{\\text{MHD equil.}} \u0026#43; \\vec T \\cdot \\cdot \\grad \\vec r) \\dd V\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\grad \\vec r = \\vec I\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  so  \\[\\vec T \\cdot \\cdot \\grad \\vec r = p_\\perp \u0026#43; p_\\perp \u0026#43; p_\\parallel \\\\ = 3p \u0026#43; \\frac{B^2}{2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\int_V (3p \u0026#43; \\frac{B^2}{2 \\mu_0} ) \\dd V = \\int _V \\div ( \\vec r \\cdot \\vec T) \\dd V = \\oint _S (\\vec r \\cdot \\vec T) \\cdot \\vu n \\dd S \\\\ = \\oint _S \\left[ \\vec r \\cdot \\vec I p_\\perp \u0026#43; \\vec r \\cdot \\vu e_B \\vu e_B (p_\\parallel - p_\\perp) \\right]\\cdot \\vu n \\dd S \\\\ =\\oint \\left[ \\left( \\cancel{p} \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\vu r \\cdot \\vu n - \\frac{B^2}{\\mu_0} (\\vec r \\cdot \\vu e_B)(\\vu e_B \\cdot \\vu n) \\right] \\dd S\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Beyond where the plasma is contained, the pressure does not contribute  \\( p = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . If all current sources are contained in the configuration, the magnetic field  \\( \\sim 1/r^3 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for a dipole,  \\( \\sim 1/r^4 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for a quadrupole, etc. Therefore the right-hand side will fall off like\n \\[RHS \\propto \\oint_S B^2 r \\dd S \\propto \\left( \\frac{1}{r^3} \\right) ^2 r r^2 \\propto \\frac{1}{r^3} \\text{(dipole)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  so  \\( RHS \\rightarrow 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as  \\( r \\rightarrow \\infty \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . But what about the left-hand side? Both of the terms in the volume integral are positive definite, so the LHS must be positive finite and the equality can\u0026rsquo;t possibly hold. The assumption that the plasma is self-contained must be invalid. This tells us that we must have external currents.\nMagnetic Flux Surfaces #  The vast difference in thermal conductivity parallel and perpendicular to the magnetic field in a plasma confinement configuration leads to an avoidance of any open field lines. Magnetic equilibria are generally toroidal to eliminate end losses from open configurations. In general fusion confinement devices, magnetic field lines lie on a set of closed nested toroidal surfaces. This means that we can no longer describe any equilibria in a solely 1D geometry. The minor radius is no longer the only important scale length.\nFrom  \\( \\vec j \\cross \\vec B = \\grad p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , we know that the pressure gradient is perpendicular to  \\( \\vec j \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and therefore both  \\( \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\vec j \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  lie on surfaces of uniform pressure. We call these toroidal surfaces either magnetic surfaces or flux surfaces. We can use these surfaces to build a 1-dimensional description.\nAs a brief aside, some geometrical vocabulary will be useful when describing toroidal geometry. A toroid is any surface of revolution with a hole in the middle. A torus is the particular case of a toroid in which the revolved figure is a circle.\nWe will define our global toroidal coordinate system to consist of the major axis  \\( (z) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , the distance from the major axis  \\( (R) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and the azimuthal angle around the major axis  \\( (\\phi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nWe will also make use of a poloidal coordinate system measured by minor radius (distance from the minor axis)  \\( r \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and the poloidal angle from the minor axis  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . We will generally refer to a point on the torus relative to the major axis  \\( (R, z, \\phi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , or relative to the minor axis  \\( (r, \\theta, \\phi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , or in spherical coordinates  \\( (R, \\theta, \\phi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The major radius  \\( R_0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the distance from the major axis to the minor axis. The minor radius  \\( a \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the characteristic distance from the minor axis to the exterior of the revolved figure. Usually we will find symmetry under  \\( \\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nThe aspect ratio of a torus is the ratio of the major radius to the minor radius.\n \\[A = \\frac{R_0}{a}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  When we talk about a \u0026ldquo;toroidal surface,\u0026rdquo; we mean a cross-section of the toroidal rotation. When we talk about a \u0026ldquo;poloidal surface\u0026rdquo; we mean a surface which is coplanar with the minor axis:\nThe poloidal flux is determined by the size of the poloidal surface and the poloidal magnetic field:\n \\[\\Psi _p = \\int_{S_p} \\vec B \\cdot \\dd \\vec S\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and the toroidal flux is determined by the size of a toroidal surface and the toroidal magnetic field:\n \\[\\Psi_t = \\int_{S_t} \\vec B \\cdot \\dd \\vec S\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Considering the poloidal flux, we can see that if we expand the size of the surface towards the minor radius, the flux will increase until eventually we come to a point where the flux begins to decrease. The position of this maximum is called the magnetic axis, which does not necessarily correspond to the minor axis. In fact, it is generally displaced from the minor axis.\nTo refer back to something more familiar, we\u0026rsquo;ll define the same terms for a cylindrical geometry  \\((r, \\theta, z) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . An axial surface corresponds with a toroidal surface, and an azimuthal surface corresponds with a poloidal surface:\nIf we consider the trajectory of a single field line, what sorts of surfaces will it trace out? What surface will contain the field line? As it turns out, there are three options:\n Rational surface - the field line closes on itself, and it does so after a finite number of revolutions. One way to quickly visualize such a surface is to draw a Poincaré puncture plot. Choose a toroidal plane and plot a point wherever the field line punctures the surface. A Poincaré puncture plot of a rational surface contains a finite number of points and no continuous curves. Ergodic surface - the field line completely covers an entire surface, which is to say the field line punctures any toroidal surface an infinite number of times. In other words, it never closes on itself and defines an irrational curve. Stochastic region - In this case, there is no definite surface and the field line fills a volume.  Generally rational surfaces and ergodic surfaces are largely equivalent, but by introducing a small amount of resistivity a rational surface can lead to magnetic islands. One can imagine the addition of resistivity equivalent to allowing a small degree of motion of the magnetic field lines. In an ergodic surface, a flux surface is defined by a single (irrational) field line. If it moves toward itself in one location it will necessarily move away from itself in another location. But in a rational surface, different field lines can lie on the same constant pressure surface and will tend to move towards each other. By concentrating into magnetic islands, the flux surfaces are now more closely spaced, and the pressure gradient increases (a bad thing!)\nSurface quantities: Since pressure, current (not current density!), and flux (not field!) are constant along a flux surface, it is convenient to use flux  \\( \\Psi_p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as a coordinate. A particular poloidal flux itself uniquely determines a poloidal surface with constant pressure and current. The flux surface quantities are  \\( p, \\, \\Psi_p, \\, \\Psi_t, \\, I_p = \\int_{S_p} \\vec j \\cdot \\dd \\vec S, \\, I_t = \\int_{S_t} \\vec j \\cdot \\dd \\vec S\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nSurface quantities are not independent. The poloidal current  \\( I_p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  affects the toroidal field  \\( B_t \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and toroidal flux  \\( \\Psi_t \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The toroidal current affects the poloidal field  \\( B_p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and poloidal flux  \\( \\Psi_p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\n \\[B_t = \\vec B \\cdot \\vu \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\vec B_p = B_\\theta \\vu \\theta \u0026#43; B_z \\vu z \\\\ = B_r \\vu R \u0026#43; B_z \\vu z\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Toroidal Force Balance #  Toroidal equilibria solves the end losses of linear equilibrium, but generates a new force which must be balanced. This is a result of the virial theorem.\nPoloidal Fields (Tire Tube Pressure Force) #  If you think of a flexible bike tire being inflated, as the pressure within the inner tube increases, the major radius will increase! Why is this? The pressure within the tire is isotropic. As we pump up the tire, the pressure increase causes a force imbalance with the atmospheric pressure that causes the tube to expand. To simplify, we can consider a square tube with inner \u0026ldquo;radius\u0026rdquo;  \\( a \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The radial force scales with the pressure over the outer and inner surfaces\n \\[F_r = p S_{outer} - p S_{inner} \\\\ \\sim p (R_0 \u0026#43; a) a - p (R_0 - a) a \\sim 2 a^2 p\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  !!! error \u0026ldquo;Missing content here on how to balance radial forces for purely poloidal fields (axisymmetric) with a conducting wall\u0026rdquo;\nAlternatively, we can use vertical external field coils to increase the field strength near the outer wall. They will also tend to center the plasma between the coils, so they have the advantage of preventing the plasma from drifting upwards or downwards\nToroidal Fields #  Now let\u0026rsquo;s consider driving a toroidal field in the plasma using a central coil on the major axis. Driving a toroidal field  \\( B_\\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  will also create a poloidal current  \\( j_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . We know that the driven field will decay as  \\( 1/R \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  through the plasma. In actuality, the relationship is not perfectly  \\( 1/R \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  because of the generated poloidal current, which will modify the field within the plasma. Depending on the orientation of  \\( j_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , it could either increase or decrease the poloidal field within the plasma. These are two different operations by which the plasma interacts with the magnetic field. When the induced plasma current tends to increase the field, we have a paramagnetic effect. In the opposite case (decrease) there is a diamagnetic effect. Generally, plasmas are diamagnetic, but there are certain situations where they become paramagnetic.\nHere again, we have a larger field on the in-bore side than on the out-bore side, so there will be a force imbalance tending to push the plasma towards larger radii. We might consider surrounding the plasma with a conducting wall, as we did previously, but we run into a difficulty determining a current distribution in the wall which would balance the radial effect. Any current distribution must be circular; because of the geometry of a torus such a current distribution would have the same force contribution on both the inside and outside edges. There is no way to stabilize the distribution with purely toroidal fields.\n"});index.add({'id':7,'href':'/r/notes/UWAA558/08-1d-equilibria/','title':"1-D Equilibria",'section':"MHD Theory",'content':"1-Dimensional Equilibria #  The   \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch #  In a  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  pinch, we have an applied axial field generated by a driven azimuthal current distribution. The way these usually work is that you begin with a plasma generated by some pre-ionization process and zero field. Then you crank up the current to drive an azimuthal current in the plasma (in the opposite direction as the external current).\n \\[\\vec j \\cross \\vec B = \\grad p \\quad \\rightarrow \\quad j_\\theta B_z = \\dv{p}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[j_\\theta = - \\frac{1}{\\mu_0} \\dv{B_z}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\dv{p}{r} = - \\frac{1}{\\mu_0} B_z \\dv{B_z}{r} = - \\dv{}{r} \\left( \\frac{B_z ^2}{2 \\mu_0} \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[p \u0026#43; \\frac{B_z ^2}{2 \\mu_0} = \\text{constant} = \\frac{B_0 ^2}{2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nAt equilibrium, the magnetic pressure balances the plasma pressure. If we say that the pressure is\n \\[p = p_0 e^{- r^2 / a^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  with  \\( p_0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  the pressure on-axis, then we can solve for the axial field\n \\[B_z = B_0 (1 - B_0 e^{-r^2/a^2})^{1/2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can define the peak  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to be the ratio of the on-axis pressure to the maximum magnetic field\n \\[\\beta_0 = \\frac{p_0}{B_0 ^2 / 2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  By definition, the peak  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  will always be  \\( \\leq 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . We can define  \\( \\langle \\beta \\rangle \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\langle \\beta \\rangle = \\frac{ \\langle p \\rangle }{B_a ^2 / 2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( B_a \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is a characteristic field value, typically taken to be at the plasma edge.\n \\[\\langle \\beta \\rangle = \\frac{2 \\mu_0}{B_0 ^2 \\pi a^2} \\int _0 ^a 2 \\pi r p \\, \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[= \\frac{2}{a^2} \\int_0 ^a \\frac{ rp}{B_0 ^2/2 \\mu_0} \\dd r = \\frac{2}{a^2} \\int_0 ^a \\left( 1 - \\frac{B_z ^2}{B_0 ^2} \\right) r \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nIn this form, we can see that because  \\( B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  will everywhere be less than  \\( B_0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , we can increase  \\( \\langle \\beta \\rangle \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  by driving  \\( B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as low as possible. In this particular example,  \\( \\langle \\beta \\rangle / \\beta_0 = 63\\% \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nZ-pinch #  In the case of a Z-pinch, we only have an applied axial current.\n \\[\\vec j = j_z (r) \\vu z\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  For force balance\n \\[\\vec j \\cross \\vec B = \\grad p \\quad \\rightarrow \\quad - j_z B_\\theta = \\dv{p}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[j_z = \\frac{1}{\\mu_0} \\frac{1}{r} \\dv{}{r} ( r B_\\theta) \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[- \\dv{p}{r} = \\frac{B_\\theta}{\\mu_0 r} \\dv{}{r} ( r B_\\theta)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we find it convenient we can separate this into a magnetic pressure term\n \\[- \\dv{}{r} \\left( p \u0026#43; \\frac{B_\\theta ^2}{2 \\mu_0} \\right) = \\frac{B_\\theta ^2}{\\mu_0 r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Bennett Profile #  An example of an achievable distribution is the Bennett profile, which has a diffuse form\n \\[B_\\theta = \\frac{\\mu_0 I}{2 \\pi} \\frac{r}{r^2 \u0026#43; a^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[j_z = \\frac{I}{\\pi} \\frac{a^2}{(r^2 \u0026#43; a^2) ^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[p = \\frac{\\mu_0 I^2}{8 \\pi ^2} \\frac{a^2}{(r^2 \u0026#43; a^2)^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Interestingly,  \\( j \\propto p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . For a uniform temperature,  \\( j \\propto n \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Since current density is the product of  \\( \\vec v \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( n \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , this says that we have a uniform drift velocity and all particles are drifting with the same velocity at all points along the profile. If we consider what the equilibrium profile looks like for a Bennett profile:\nSo for  \\( r \u0026lt; a \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we have magnetic tension and pressure which balance the plasma pressure. For  \\( r \\geq a \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we have magnetic tension which balances both plasma pressure and magnetic pressure.\nThe Z-pinch  \\( \\langle \\beta \\rangle \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\langle \\beta \\rangle \\equiv \\frac{ \\langle p \\rangle}{B_a ^2 / 2 \\mu_0} \\\\ = \\frac{2 \\mu_0}{B_a ^2 \\pi a^2} \\int_0 ^a 2 \\pi r p \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we multiply the force balance by  \\( r^2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and integrate\n \\[\\int_0 ^a r^2 \\dv{p}{r} \\dd r \u0026#43; \\frac{1}{\\mu_0} \\int_0 ^a r B_\\theta \\dv{}{r} (r B_\\theta) \\dd r = 0 \\\\ 0 = \\left[ r^2 p \\right] _0 ^a - \\int_0 ^a p \\dd (r^2) \u0026#43; \\left[ \\frac{(r B_\\theta)^2}{2 \\mu_0} \\right] _0 ^a\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we have a discrete pinch such that  \\( p(a) = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  then the first term vanishes.\n \\[\\int_0 ^a 2 r p \\dd r = \\frac{(a B_a)^2}{2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we substitute our definition of  \\( \\langle \\beta \\rangle \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , we find  \\( \\langle \\beta \\rangle = 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . For a diffuse pinch in which  \\( p(a) \\neq 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we end up with  \\( \\langle \\beta \\rangle \\leq 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and we have a wall-supported plasma. Ideal confinement ( \\( \\langle \\beta \\rangle = 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ) is a very nice property and is what makes the Z-pinch configuration so interesting.\nStability Considerations #  Instability results if there exists a plasma displacement that leads to a lower energy state. There are several ways to provide stability in the context of MHD. The two most common are magnetic shear and magnetic well.\nMagnetic Shear #  In ideal MHD, magnetic field lines can not break or tear. Let\u0026rsquo;s consider some flux surface containing field lines  \\( \\vec B_3 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Behind it, we have another flux surface containing field lines  \\( B_2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  which are not parallel to  \\( \\vec B_3 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and the same for  \\( \\vec B_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nBecause the field lines are a different angles to each other, these flux surfaces can not interpenetrate. In other words, if the flux surface pressures are  \\( P_1 \u0026gt; P_2 \u0026gt; P_3 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , we can maintain the pressure gradient and prevent the flux surfaces from moving each other. What prevents the surfaces from achieving a lower energy state is the magnetic shear between flux surfaces.\nWithout shear, the surfaces can interpenetrate and exchange positions. In the case of a toroidal geometry, magnetic shear is defined by the rotational transform  \\( \\iota \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , or by the safety factor\n \\[q = \\frac{2 \\pi}{\\iota} \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Generally speaking,  \\( q \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is generally referenced for tokamaks and  \\( \\iota \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is referenced for stellarators. Another way of picturing the safety factor in a toroidal geometry is\n \\[q \\equiv \\frac{\\text{no. of windings long way}}{\\text{no. of windings short way}} \\\\ = \\frac{ \\dv{\\psi_t}{V}}{\\dv{\\phi_p}{V}} = \\dv{\\phi_t}{\\phi_p} \\\\ = \\frac{n}{m} = \\frac{\\text{toroidal transits}}{\\text{poloidal transits}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  In a cylindrical (1D) geometry it is just\n \\[q = \\frac{\\text{longitudinal transits}}{\\text{azimuthal transits}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Let\u0026rsquo;s calculate the safety factor for a toroidal geometry:\n \\[\\dv{\\phi_p}{V} = \\frac{ B_\\theta 2 \\pi R \\dd r}{2 \\pi R_0 2 \\pi r \\dd r} \\\\ = \\frac{ B_\\theta}{2 \\pi r} \\frac{R}{R_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\dv{\\phi_t}{V} = \\frac{B_\\phi 2 \\pi r \\dd r}{2 \\pi R_0 2 \\pi r \\dd r} \\\\ = \\frac{B_\\phi}{2 \\pi R_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[q = \\frac{r B_\\phi}{R B_\\theta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  In a cylindrical geometry the analysis is even simpler\n \\[q = 2 \\pi \\frac{r B_z}{L B_\\theta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  As a note, it would appear that  \\( q \\rightarrow 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  at the magnetic axis as  \\( r \\rightarrow 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , but in general  \\( B_\\theta \\rightarrow 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as well, and the safety factor is generally bounded at  \\( r \\rightarrow 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\( q \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is a flux surface quantity.\nWe care about magnetic shear. How does that relate to the safety factor? Magnetic shear is defined as\n \\[s \\equiv 2 \\frac{\\dd q / q}{\\dd V /V} = 2 \\dv{\\ln (q)}{\\ln(V)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Even a uniform  \\( B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  or  \\( B_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  produces a finite magnetic shear because of the way that  \\( r \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( B_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  change. The safety factor is often considered synonymous with magnetic shear, and often we don\u0026rsquo;t even compute  \\( s \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nShear is generally a stabilizing effect. Interchange between flux surfaces can be prevented/inhibited by shear, or by making it energetically unfavorable. Shear tends to stabilize current-driven instabilities.\nMagnetic Well #  As before, we can consider two adjacent flux surfaces  \\( B_1, P_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( B_2, P_2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . If  \\( B_2 \u0026gt; B_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( P_2 \u0026gt; P_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , the interchange is energetically favorable. But if  \\( B_2 \u0026gt; B_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( P_2 \u0026lt; P_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  then the interchange may be unfavorable without any magnetic shear.\nConsider a plasma confined by an externally applied magnetic field generated by a coil  \\( I \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nOn the left side, the magnetic field gradient is in the same direction as the plasma pressure gradient, which is a destabilizing configuration. Flux surfaces are able to interchange easily, and the magnetic field is described as having bad curvature. On the right side, the gradients are in the same direction and the magnetic field has a good curvature.\nWe can define the \u0026ldquo;wellness\u0026rdquo;  \\( W \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  as\n \\[W \\equiv \\frac{ \\text{total pressure change relative to mag. pressure}}{\\text{relative volume change}} \\\\ = \\frac{\\dd \\langle p \u0026#43; B^2/2 \\mu_0 \\rangle / \\langle B^2/2 \\mu_0 \\rangle}{\\dd V / V}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where the angle brackets indicate a quantity integrated along a field line  \\[\\langle Q \\rangle \\equiv \\frac{\\int_0 ^L \\frac{ Q \\dd l}{|B|}}{\\int_0 ^l \\frac{\\dd l}{|B|}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nFor a stabilizing effect, the wellness must be greater than 0. This means that the magnetic pressure must increase faster than the pressure decreases to prevent pressure-driven instabilities.\nSince  \\( W \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is evaluated along a field line, it is also a surface quantity.\nApplication to 1D Equilibria #   \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch: Since  \\( B_\\theta = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( q \\rightarrow \\infty \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , which really just means  \\( q \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is not well defined for a  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch. If we consider some small  \\( \\delta B_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , we get a very large  \\( q \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . From a magnetic shear perspective, a  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch has very large values of shear and very good stability properties.\nThe wellness is\n \\[W = \\frac{V}{\\langle B^2 \\rangle} \\dv{}{V} \\langle 2 \\mu_0 p \u0026#43; B^2 \\rangle \\\\ = \\frac{\\pi r^2 L}{B_z ^2} \\frac{1}{2 \\pi r L} \\dv{}{r} (2 \\mu_0 p \u0026#43; B_z ^2) \\\\ = \\frac{\\mu_0 r}{B_z ^2} \\dv{}{r} \\left( p \u0026#43; \\frac{B_z ^2}{2 \\mu_0} \\right) = 0 \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  so a  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch has neutral magnetic well.\nVacuum case:\n \\[W = \\frac{\\mu_0 r}{ B_z ^2} \\dv{}{r} \\left( \\frac{ B_z ^2}{2 \\mu_0} \\right) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  So vacuum magnetic fields also have neutral wellness. This leads to a general result sometimes referred to as \u0026ldquo;a plasma cannot dig its own well.\u0026rdquo; In other words, by introducing plasma to a magnetic configuration, it cannot make the configuration more stable than it was. Plasmas make stability more challenging, not less.\nZ-pinch:\nSince  \\( B_z = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( q = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and there is no magnetic shear. Even for a small value of  \\( \\delta B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  you still get a small  \\( q \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The magnetic well properties of a Z-pinch are\n \\[W = \\frac{V}{\\langle B^2 \\rangle} \\dv{}{V} \\langle 2 \\mu_0 p \u0026#43; B^2 \\rangle \\\\ = \\frac{\\mu_0 r}{B_\\theta ^2} \\dv{}{r} \\left( p \u0026#43; \\frac{B_\\theta ^2}{2 \\mu_0} \\right) \\\\ = \\frac{\\mu_0 r}{B_\\theta ^2} \\left( - \\frac{ B_\\theta ^2}{ \\mu_0 r} \\right) = -1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Recall that  \\( W \u0026gt; 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for stability, so the Z-pinch has negative magnetic well and provides no pressure stability.\nIn summary,\n Both  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; - and Z-pinch have high  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch is stable Z-pinch is unstable End losses in a  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  pinch enormous since  \\( k_\\parallel \\gg k_\\perp \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   Screw Pinch #  A natural extension is to combine a moderate toroidal field and a moderate poloidal field to produce a screw pinch configuration.\n \\[\\curl \\vec B = \\mu_0 \\vec j\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\rightarrow j_\\theta = - \\frac{1}{\\mu_0 } \\dv{B_z}{r} \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[j_z = \\frac{1}{\\mu_0 r} \\dv{}{r} (r B_\\theta)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nFor static MHD equilibrium\n \\[\\vec j \\cross \\vec B = \\grad p\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\dv{}{r} \\left( p \u0026#43; \\frac{ B_\\theta ^2 \u0026#43; B_z ^2}{2 \\mu_0} \\right) = - \\frac{B_\\theta ^2}{\\mu_0 r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nWe can define a toroidal  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( B_0 = B_z (a) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\beta_t = \\frac{\\langle p \\rangle }{B_0 ^2 / 2 \\mu_0} \\\\ = \\frac{2 \\mu_0}{ B_0 ^2} \\frac{1}{\\pi a^2} \\int _0 ^a 2 \\pi r p \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nand in the poloidal direction with  \\( B_{\\theta, a} = B_\\theta (a) = \\frac{\\mu_0 I}{2 \\pi a} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\beta_p = \\frac{\\langle p \\rangle}{B_{\\theta, a} ^2 / 2 \\mu_0} \\\\ = \\frac{8 \\pi ^2 a^2}{ \\mu_0 I_0 ^2} \\left( \\frac{1}{\\pi a^2} \\int _0 ^a 2 \\pi r p \\dd r \\right) \\\\ = \\frac{16 \\pi ^2}{\\mu_0 I_0 ^2} \\int_0 ^a r p \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  To proceed we can multiply the force balance by  \\( r^2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and integrate\n \\[\\underbrace{\\int_0 ^a r^2 \\pdv{p}{r} \\dd r}_{(1)} \u0026#43; \\underbrace{\\int _0 ^a \\pdv{}{r} \\left( \\frac{ B_\\theta ^2 \u0026#43; B_z ^2 }{2 \\mu_0} \\right) \\dd r}_{(2)} \u0026#43; \\underbrace{\\int_0 ^a r^2 \\frac{B_\\theta ^2}{\\mu_0 r} \\dd r}_{(3)} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[(1) = \\int_0 ^a r^2 \\dd p = \\left. r^2 p \\right|_0 ^a - \\int_0 ^a p \\dd (r^2) = - \\int_0 ^a 2 r p \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[(2) = \\int_0 ^a r^2 \\dd \\left( \\frac{B_\\theta ^2 \u0026#43; B_z ^2}{2 \\mu_0 } \\right) \\\\ = \\left. r^2 \\left( \\frac{ B_\\theta ^2 \u0026#43; B_z ^2}{2 \\mu_0 } \\right) \\right|_0 ^a - \\int_0 ^a r \\left( \\frac{ B_\\theta ^2 \u0026#43; B_z ^2}{\\mu_0} \\right) \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[(3) = \\int_0 ^a r \\frac{B_\\theta ^2}{\\mu_0} \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Combining we have\n \\[- \\int_0 ^a 2 r p \\dd r \u0026#43; \\frac{a^2 B_{\\theta, a}^2}{2 \\mu_0} \u0026#43; \\overbrace{\\frac{a^2 B_0 ^2}{2 \\mu_0}}^{B_0 = B_z(r = a)} - \\int_0 ^a r \\frac{B_z ^2}{\\mu_0} \\dd r = 0 \\\\ - \\int_0 ^a 2 r p \\dd r \u0026#43; \\frac{ \\mu_0 I_0 ^2}{8 \\pi ^2} \u0026#43; \\int_0 ^a r \\left( \\frac{ B_\\theta ^2 - B_z ^2}{\\mu_0} \\right) \\dd r = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Dividing  \\( \\int_0 ^a 2 r p \\dd r \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  gives\n \\[\\left[ \\frac{16 \\pi^2}{\\mu_0 I^2} \\int_0 ^a r p \\dd r \\right] ^{-1} \u0026#43; \\left[\\frac{4 \\mu_0}{B_0 ^2 a^2} \\int_0 ^a r p \\dd r \\right] ^{-1} \\frac{2}{a^2} \\int_0 ^a \\left(1 - \\frac{B_z ^2}{B_0 ^2} \\right) r \\dd r = 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  or\n \\[\\frac{1}{\\beta_p} \u0026#43; \\frac{\\alpha_t}{\\beta_t} = 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\[\\alpha_t = \\frac{2}{a^2} \\int_0 ^a \\left( 1 - \\frac{B_z^2}{B_0 ^2} \\right) r \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the diamagnetism.\n \\[\\beta _p = \\left( 1 - \\frac{\\alpha_t}{\\beta_t} \\right) ^{-1}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we have a diamagnetic current, then  \\( \\alpha \u0026gt; 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . This maximizes confinement, since we have confinement in the azimuthal field, as well as the axial field. The limit where you have a skin current such that  \\( B_z = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  inside the plasma results in the best confinement and  \\( \\alpha _t = 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nLooking at the safety factor,\n \\[q = \\frac{2 \\pi r B_z}{L B_\\theta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we look at the edge  \\( r = a \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\[q_a = \\frac{2 \\pi a B_0}{L B_{\\theta, a}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nAs it turns out, this value of the edge safety factor is critically important, and for stability we require that  \\( q_a \u0026gt; 1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nThe magnetic shear of a screw pinch is\n \\[s = 2 \\frac{V}{q} \\dv{q}{V} \\\\ V = \\pi r^2 L \\\\ \\dd V = 2 \\pi L r \\dd r = \\frac{2 V}{r} \\dd r \\\\ s = \\frac{r}{q} \\dv{q}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The shear can be adjusted by changing the applied axial field.\nThe magnetic well is\n \\[W = \\frac{V}{B^2} \\dv{(2 \\mu_0 p \u0026#43; B^2)}{V} \\\\ = \\frac{\\mu_0 r}{B^2} \\dv{}{r} \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\\\ = \\frac{\\mu_0 r}{B^2} \\left( - \\frac{B_\\theta ^2}{\\mu_0 r} \\right) \\\\ = - \\frac{B_\\theta ^2}{B^2} \\\\ = - \\frac{B_\\theta ^2}{B_\\theta ^2 \u0026#43; B_z ^2} \\\\ = - \\left( 1 \u0026#43; \\frac{B_z ^2}{B_\\theta ^2} \\right) ^{-1}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  So the well is always less than zero, but adding  \\( B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  improves the well.\nBy combining the properties of  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch and Z-pinch, we are able to sacrifice some  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to achieve better stability properties. Of course, we have not addressed the end losses in any way; to do that, we need to connect the ends.\n"});index.add({'id':8,'href':'/r/notes/UWAA558/09-2d-equilibria/','title':"2D Equilibria",'section':"MHD Theory",'content':"2D Equilibria #  Let\u0026rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:\n  \\[\\vec j \\cross \\vec B = j_\\theta B_z - j_z B_\\theta \\\\ = \\grad p = \\dv{p}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we move to an axisymmetric 2-dimensional torus, replacing our cylindrical coordinate system with a toroidal one\n \\[\\vec j \\cross \\vec B = \\vec j_\\theta \\cross \\vec B_\\phi \u0026#43; \\vec j_\\phi \\cross \\vec B_\\theta = \\grad p\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Eventually, the toroidal force balance will lead to the Grad-Shafranov Equation, which tells us how we can solve for a general equilibrium that solves  \\( \\vec j \\cross \\vec B = \\grad p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nLet\u0026rsquo;s consider how we might achieve such a configuration. A toroidal magnetic field can be achieved by driving current through a poloidal coil. A more complicated problem is how to drive toroidal current. In general this is done by means of a transformer, where the plasma itself is the secondary circuit. Driving a time-varying current through the primary induces a toroidal current through the plasma. This is called a transformer drive for current.\nGrad-Shafranov equation #  Computing  \\( j_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( B_\\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  can be computationally difficult in a toroidal geometry, so let\u0026rsquo;s do some work towards simplifying our force balance expression. The toroidal magnetic vector potential is defined as\n \\[\\vec B_\\theta = \\curl \\vec A_\\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If we integrate  \\( B_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  over a poloidal surface, Stokes' theorem gives\n \\[\\int _{S_p} \\curl \\vec A_\\phi \\cdot \\dd \\vec S = \\oint \\vec A_\\phi \\cdot \\dd \\vec l \\\\ = \\int _{S_p} B_\\theta \\cdot \\dd \\vec S = \\Psi _p \\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If the equilibrium is axisymmetric,  \\( A_\\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  must be uniform along  \\( \\dd l \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , so\n \\[A_\\phi \\vu \\phi \\cdot \\oint \\dd \\vec l = A_\\phi 2 \\pi R = \\Psi _p \\\\ \\rightarrow A_\\phi = \\frac{ \\Psi_p}{R} \\vu \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where we absorb the factor of  \\( 2 \\pi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  into the poloidal flux  \\( \\Psi _p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . After some manipulation, we can relate  \\( B_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to the poloidal flux\n \\[\\vec B_\\theta = \\curl \\vec A_\\phi = - \\frac{ \\vu R}{R} \\pdv{\\Psi}{z} \u0026#43; \\frac{\\vu z}{R} \\pdv{\\Psi}{R}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\mu_0 j_\\phi \\cross B_\\theta = (\\curl \\vec B_\\theta) \\cross \\vec B_\\theta \\\\ = \\curl \\vec B_\\theta \\cross \\left( \\grad \\Psi \\cross \\frac{\\vu \\phi}{R} \\right) \\\\ = - \\left[ \\pdv{}{R} \\left( \\frac{1}{R} \\pdv{\\Psi}{R} \\right) \u0026#43; \\frac{1}{R} \\pdv{\\Psi ^2}{z^2} \\right] \\cdot \\left[\\frac{ \\grad \\Psi}{R} ( \\vu \\phi \\cdot \\vu \\phi) - \\frac{ \\phi}{R} \\cancel{(\\grad \\Psi \\cdot \\vu \\phi)} \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  That gives the first component of  \\( \\grad p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , now let\u0026rsquo;s do the other one\n \\[\\mu_0 \\vec j_\\theta \\cross \\vec B_\\phi = ( \\curl \\vec B_\\phi) \\cross \\vec B_\\phi \\\\ = \\left[ - \\vu R \\pdv{B_\\phi}{z} \u0026#43; \\vu z \\frac{1}{R} \\pdv{}{R} ( R B_\\phi) \\right] \\cross \\vec B_\\phi \\\\ = - \\frac{B_\\phi}{R} \\left[ \\vu R \\pdv{}{R} (R B_\\phi) \u0026#43; \\vu z \\pdv{}{z} (R B_\\phi) \\right] \\\\ = - \\frac{B_\\phi}{R} \\grad (R B_\\phi)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Finally, since pressure is a flux surface quantity we can write\n \\[\\grad p = \\dv{p}{\\Psi} \\grad \\Psi = p\u0026#39; \\grad \\Psi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The toroidal force balance now looks like\n \\[\\mu_0 p\u0026#39; \\grad \\Psi = - \\frac{1}{R} \\left( \\pdv{}{R} \\frac{1}{R} \\pdv{\\Psi}{R} \u0026#43; \\frac{1}{R} \\pdv{^2 \\Psi}{z^2} \\right) \\grad \\Psi - \\frac{B_\\phi}{R} \\grad(R B_\\phi)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We notice that the only vector quantities here are  \\( \\grad \\Psi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\grad (R B_\\phi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , so  \\( \\grad (R B_\\phi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  must be parallel to  \\( \\grad \\Psi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and is a flux surface quantity. We can define our new flux surface quantity as\n \\[F(\\Phi) \\equiv R B_\\phi = \\frac{\\mu_0 I_\\theta}{2 \\pi} = \\frac{\\mu_0}{2 \\pi} \\int_{S_p} \\vec j_\\theta \\cdot \\dd \\vec S\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\grad F = \\dv{F}{\\Psi} \\grad \\Psi = F\u0026#39; \\grad \\Psi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Now each term in the toroidal force balance has a factor of  \\( \\grad \\Psi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  attached. Let\u0026rsquo;s multiply through by  \\( R^2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and factor out the gradient to arrive at the Grad-Shafranov equation:\n \\[R^2 \\mu_0 p\u0026#39; = - \\Delta ^\\star \\Psi - F F\u0026#39;\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\[\\Delta ^\\star \\equiv R \\pdv{}{R} \\frac{1}{R} \\pdv{}{R} \u0026#43; \\pdv{^2}{z^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nTo solve the Grad-Shafranov equation, you solve for  \\( \\Psi(R, z) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , which determines  \\( p(\\Psi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( F(\\Psi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , which directly gives you  \\( p(R, z) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( F(R, z) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and completely defines the equilibrium.\nYou can solve for the other terms as well. Since  \\( \\vec B_\\theta = \\frac{\\grad \\Psi}{R} \\cross \\vu \\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\vec j_\\phi = - \\frac{1}{\\mu_0 R} \\Delta ^\\star \\Psi \\vu \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and since  \\( \\vec B_\\phi = \\frac{F}{R} \\vu \\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\vec j_\\theta = - \\frac{1}{\\mu_0 R} \\grad (R B_\\phi) \\cross \\vu \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  For the G-S equation to be solvable, you need to specify the equilibrium by specifying  \\( p(\\Phi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( F(\\Phi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . In practice, this is usually done by making experimental measurements to determine  \\( p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( F \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . A common code that does this is called EFIT, which takes the boundary conditions of the magnetic field and measurements of temperature, density to perform a least-squares fit to solve the G-S equation.\nIn general, the Grad-Shafranov equation leads to a matrix equation\n \\[\\overline \\vec A \\vec \\Psi \u0026#43; \\vec f(\\Psi) = \\vec g\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Depending on the conditions we place on  \\( \\Psi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ,  \\( \\vec f(\\Psi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  can be a nonlinear function.\nSolutions to the Grad-Shafranov equation #  In the limit that  \\( \\vec j \\parallel \\vec B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , then  \\( \\vec j \\cross \\vec B = 0 = \\grad p \\rightarrow p\u0026#39; = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . These are called force-free states. In the G-S equation, the pressure term vanishes and we\u0026rsquo;re left with\n \\[\\Delta ^\\star \\Psi \u0026#43; F F\u0026#39; = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Spheromaks and RFPs are examples of nearly force-free states in which the current is nearly parallel to the magnetic field. Notice that in completely force-free states,  \\( \\langle \\beta \\rangle = 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nAnother interesting limit is the case where  \\( F F\u0026#39; \\gg \\Delta ^\\star \\Psi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . Now we have  \\[\\grad p \\approx \\vec j_\\theta \\cross \\vec B_\\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  which looks like a  \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch which has been connected at the ends. Remember from the previous section that we can not maintain radial force balance with purely toroidal fields, so the toroidal current is not zero (hence the  \\( \\approx \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ) but is just high enough to maintain radial force balance. This sort of configuration is called a high- \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  tokamak.\nThe other limit is  \\( F F\u0026#39; \\ll \\Delta ^\\star \\Psi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\grad p \\approx \\vec j_\\phi \\cross \\vec B_\\theta\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  which looks like an end-connected z-pinch. This configuration is usually called an Ohmically heated Tokamak, and the majority of currently operating tokamaks operate this way. As we know, a purely poloidal field has very bad stability properties, so  \\( \\vec B_\\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  needs to be added to provide stability. The toroidal  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is very small  \\[\\beta _t \\ll 1 \\qquad \\beta _p \\approx 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nStability Considerations #  The same stability factors exist in 2D equilibria that we found for 1D equilibria:\nMagnetic shear - the safety factor  \\( q(\\Psi) = \\frac{\\Delta \\phi}{\\Delta \\theta} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  for  \\( \\Delta \\theta = 2 \\pi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . We can calculate  \\( q \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  more easily by integrating along a flux surface in the poloidal plane:\n \\[q(\\Psi) = \\frac{1}{2 \\pi} \\int_0 ^{\\Delta \\phi} \\dd \\phi \\\\ = \\frac{1}{2 \\pi} \\int _0 ^{2 \\pi} \\dv{\\phi}{\\theta} \\dd \\theta \\\\ = \\frac{1}{2 \\pi} \\int_0 ^{2 \\pi} \\dd \\theta \\left. \\frac{r B_\\phi}{R B_\\theta} \\right|_{\\text{along flux surf.}} \\\\ = \\frac{F(\\Psi)}{2 \\pi} \\oint \\frac{ \\dd l_p}{R^2 B_\\theta} \\qquad \\dd l_p = r \\dd \\theta\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Magnetic well: similarly we can get the magnetic well factor by integrating around a flux surface in the poloidal plane\n \\[\\langle Q \\rangle = \\frac{\\oint \\frac{Q \\dd l_p}{B_\\theta}}{\\oint \\frac{ \\dd l_p}{B_\\theta}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Shafranov Shift #  Remember that when we had an equilibrium which had a toroidal current and a corresponding poloidal magnetic field, and a poloidal magnetic field, then radial force balance will tend to shift the configuration outwards away from the major axis and a conducting wall or external coil will be required to maintain the equilibrium. The radial force balance is really achieved by  \\( \\vec j_\\phi \\cross B_p \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nAs we move towards the magnetic axis,  \\( B_p \\rightarrow 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  by definition. With less poloidal field to balance the radial force imbalance, there is more radial expansion. This means that inner portion of the plasma (inner flux surfaces) must shift radially further to achieve radial force balance.\nThe shift increases with plasma pressure. This effect is further enhanced if we have low poloidal fields, for example in the high- \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  tokamak configurations.\nLow aspect ratios also enhance the effect. Recall that the radial force imbalance increases with smaller aspect ratio, leading to a larger shift.\n"});index.add({'id':9,'href':'/r/notes/UWAA558/10-equilibrium-of-3d-configurations/','title':"Equilibrium of 3D Configurations",'section':"MHD Theory",'content':"Equilibrium of 3D Configurations #  In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated.\nSome general features of 3D equilibria are:\n No net toroidal current. This means that they tend to be steady-state configurations. Radial confinement is accomplished by toroidal fields, as in the end-connected   \\( \\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; -pinch. As we saw, toroidal fields cannot provide radial confinement in a purely axisymmetric configuration, but radial variation with  \\( \\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  can provide confinement. Toroidal effect (radial force balance) is generated by helical magnetic fields. You can do an expansion of the magnetic field into a toroidal component, and a helical component that traces out a twisted shape as you move around the torus. These twisted shapes are what lead to radial confinement.  ELMO Bumpy Torus (EBT) #  In contrast to most other 3D configurations, even though the EBT is a 3D equilibrium, it has no helical windings.\nSince there are no helical windings, we have to provide radial stability in another way. In the EBT configuration, you also drive hot poloidal electron rings (driven by electron cyclotron resonance) to provide both stability and heating.\nStellarator #  The stellarator configuration is composed of a number of helical current lines (generated by helical coils with alternating currents), and a net toroidal field driven by poloidal coils. The direction of the currents alternate, for a total of  \\( 2l \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  current lines.\nThe result is a net magnetic field with a ratio such that  \\( B_{\\text{helical}} \\gg B_\\phi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nStellarators raise some very complicated engineering challenges both in the design and construction of the complicated geometry. It is also very difficult to maintain no net current within the plasma, especially during start-up. As you add plasma, you raise from zero  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  to a finite  \\( \\beta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , introducing things like bootstrap currents that need to be balanced.\nTorsatron #  Similar to a stellarator, the torsatron does not have alternating currents. All of the helical current lines are in the same direction. There are also no toroidal field coils.\nThe engineering is slightly simpler, but it is slightly less efficient at generating the helical magnetic field.\nThe flux surfaces in stellarators and torsatrons have geometrical cross-sections depending on the number  \\( l \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  of helical current lines. About the current lines, the flux surfaces are nearly circular. The flux surfaces within the plasma volume are determined by the separatrix of the helical coil fields.\n"});index.add({'id':10,'href':'/r/notes/UWAA558/11-mhd-stability/','title':"MHD Stability",'section':"MHD Theory",'content':"MHD Stability #  Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear. We can define the initial deviation from equilibrium to be a linear phenomenon. As usual, we perform this linearization by letting   \\( Q(r, t) = Q_0 \u0026#43; Q_1(r, t) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  with  \\( Q_1 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  being a small first-order perturbation. Since the equilibrium is both time and space independent, the general form of the perturbation is\n \\[Q_1(r, t) = \\vu Q_1 e^{-i (\\omega t - \\vec k \\cdot \\vec r)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\grad p_0 = \\vec j_0 \\cross \\vec B_0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[p = p_0 \u0026#43; p_1 \\qquad \\rho = \\rho_0 \u0026#43; \\rho_1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\vec j = \\vec j_0 \u0026#43; \\vec j_1 \\qquad \\vec B = \\vec B_0 \u0026#43; \\vec B_1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nand for a static equilibrium  \\[\\vec v = \\vec v_1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nIn our momentum equations of the perturbed quantities, we assume that the static equilibrium holds, so most of the equilibrium terms drop out. We can define a velocity displacement  \\( \\vec \\xi = \\int_0 ^t \\vec v_1 \\dd t \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . As we integrate the field and pressure in time,\n \\[\\pdv{B_1}{t} = \\curl (\\vec v_1 \\cross \\vec B_0)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\int \\pdv{B_1}{t} = \\vec B_1 = \\curl \\int \\vec v_1 \\cross \\vec B_0 \\dd t \\\\ = \\curl (\\vec \\xi \\cross \\vec B_0)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nIf we do the same for the pressure equation, we get\n \\[p_1 = - \\vec \\xi \\cdot \\grad p_0 - \\Gamma p_0 \\div \\vec \\xi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( \\Gamma \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the ratio of specific heats, to avoid confusion with typical perturbation growth rate  \\( \\gamma \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; .\nIf we combine all of these together, substituting into the momentum equation, we can express the perturbation entirely in terms of  \\( \\vec \\xi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and the equilibrium properties:\n \\[\\rho_0 \\pdv{ ^2 \\vec \\xi }{t^2} = \\grad (\\vec \\xi \\cdot \\grad p_0 \u0026#43; \\Gamma p_0 \\div \\vec \\xi)\\\\ \u0026#43; \\frac{1}{\\mu_0} \\left[(\\curl \\vec B_0) \\cross \\curl (\\vec \\xi \\cross \\vec B_0) \\right] \\\\ \u0026#43; \\frac{1}{\\mu_0} \\left[ \\curl \\curl ( \\vec \\xi \\cross \\vec B_0) \\cross \\vec B_0 \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We define the right-hand-side as the linearized forcing function of our equilibrium  \\[\\rho_0 \\pdv{ ^2 \\vec \\xi }{t^2} = \\vec F(\\vec \\xi _i , p_0, \\vec B_0)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nFor a linear force function, we can also write it in terms of a spring constant tensor\n \\[\\rho_0 \\pdv{ ^2 \\vec \\xi }{t^2} = \\vec F(\\vec \\xi) = - \\overline \\vec K \\cdot \\vec \\xi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  We can determine the stability behavior of a configuration by specifying an initial condition\n \\[\\vec \\xi (t = 0) = 0 \\qquad \\text{and} \\qquad \\left. \\pdv{\\xi}{t} \\right| _{t = 0} = f(\\vec r)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and boundary conditions. A boundary condition may be a rigid wall  \\[\\vec \\xi \\cdot \\vu n |_{wall} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nOne way we can tell whether a given solution is unstable is to assume a variation of the form  \\[\\vec \\xi \\propto e^{-i \\omega t}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If  \\( \\omega^2 \u0026gt; 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , the displacement will oscillate in time without growth, and if  \\( \\omega^2 \u0026lt; 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  then the displacement will grow. In other words, if  \\( \\omega \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is real, then the mode is stable, and if  \\( \\omega \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is imaginary then the mode is unstable. The eigenvalue equation to be solved is\n \\[- \\omega ^2 \\rho_0 \\vec \\xi = \\vec F(\\vec \\xi)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  which we can write as a matrix equation  \\[\\overline \\vec A\\, \\overline X = \\lambda \\overline X\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\frac{1}{\\rho_0} \\vec F (\\vec \\xi) = - \\omega ^2 \\vec \\xi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nFor any arbitrary linear forcing function, we might get an infinite number of eigenvalues. How do we know which ones to look at? It turns out that the linearized force function  \\( \\vec F(\\vec \\xi) \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  has the property of being self-adjoint, so  \\[\\int \\vec \\eta \\cdot \\vec F(\\vec \\xi) \\dd V = \\int \\xi \\cdot \\vec F( \\vec \\eta) \\dd V\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( \\vec \\eta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( \\vec \\xi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  are arbitrary displacements that satisfy the same boundary conditions. If  \\( \\vec F \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is self-adjoint, then the system is Hermitian, which guarantees that we get real eigenvalues ( \\( \\omega^2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ) , orthogonal eigenfunctions, and most importantly we are guaranteed to have an ordered spectrum of eigenvalues. That is to say  \\( \\omega_0 ^2 \u0026lt; \\omega _1 ^2 \u0026lt; \\omega _2 ^2 \u0026lt; \\ldots \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . This means that the eigenvalue of the lowest mode is guaranteed to be the most negative, and therefore dictates the stability of the system. If the lowest eigenvalue is negative, then the system is necessarily unstable, and if the lowest eigenvalue is positive, then we are guaranteed that all modes are stable.\nBecause  \\( \\vec F \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is self-adjoint, we can make use of the energy principle to write the variation in the sum of the kinetic and potential energy as:\n \\[0 = \\dv{}{t} \\left[ \\frac{1}{2} \\int \\rho_0 \\left( \\pdv{\\vec \\xi}{t} \\right) ^2 \\dd V - \\frac{1}{2} \\int \\vec \\xi \\cdot \\vec F ( \\vec \\xi) \\dd V \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The kinetic energy term will always be positive, so we can formulate the stability based on the potential energy, often called a  \\( \\delta W \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  approach  \\[\\delta W = - \\frac{1}{2} \\int \\vec \\xi \\cdot \\vec F ( \\vec \\xi) \\dd V\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the change in potential energy due to a displacement  \\( \\xi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . If the potential energy decreases due to a displacement  \\( \\xi \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , then the kinetic energy must necessarily increase, so  \\( \\delta W \u0026lt; 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  indicates instability.\nWe can write the change in kinetic energy for our normal mode decomposition as  \\[\\delta T = \\frac{1}{2} \\int \\rho _0 \\left( \\pdv{\\xi}{t} \\right) ^2 \\dd V = - \\frac{1}{2} \\omega ^2 \\int \\rho _0 \\vec \\xi ^ \\star \\cdot \\vec \\xi \\dd V \\\\ = - \\delta W = \\frac{1}{2} \\int \\vec \\xi ^\\star \\cdot \\vec F(\\vec \\xi) \\dd V\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\omega^2 = \\frac{- \\int \\xi ^\\star \\cdot \\vec F \\dd V}{\\int \\rho_0 \\xi ^\\star \\cdot \\xi \\dd V} = \\frac{\\delta W}{\\frac{1}{2} \\int \\rho_0 \\xi ^\\star \\cdot \\xi \\dd V}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nThe denominator is strictly positive, so the sign of  \\( \\omega^2 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is determined by the sign of  \\( \\delta W \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\delta W \u0026lt; 0 \\rightarrow \\omega^2 \u0026lt; 0 \\rightarrow \\text{unstable} \\\\ \\delta W \u0026gt; 0 \\rightarrow \\omega^2 \u0026gt; 0 \\rightarrow \\text{stable}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Analyzing the form of  \\( \\delta W \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  (within the plasma volume)\n \\[\\delta W = \\frac{1}{2} \\int_{plasma} \\dd V \\Gamma p_0(\\div \\vec \\xi) ^2 \u0026#43; \\vec \\xi \\cdot \\grad p_0 (\\div \\vec \\xi) \\qquad \\qquad \\\\ \\qquad \\qquad \u0026#43; \\frac{1}{\\mu_0} \\left[ \\curl ( \\vec \\xi \\cross \\vec B_0) \\right]^2 \\\\ \\qquad \\qquad - \\frac{1}{\\mu_0} \\left[\\vec \\xi \\cross ( \\curl \\vec B_0) \\right] \\cdot \\left[ \\curl ( \\vec \\xi \\cross \\vec B_0) \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Generally speaking, the plasma volume does not extend to infinity, and we care very much about the boundary. The total  \\( \\delta W \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the sum of that in the plasma volume  \\( \\delta W_F \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , the surface  \\( \\delta W_S \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , and the vacuum region  \\( \\delta W_V \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The vacuum term looks like\n \\[\\delta W_V = \\frac{1}{2} \\int _{vac} \\dd V \\frac{ (\\curl ( \\vec \\xi \\cross \\vec B_0))^2}{\\mu_0} = \\int_{vac} \\dd V \\frac{\\vec B_1 ^2}{\\mu_0} \u0026gt; 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  so the vacuum term is always positive, and has a stabilizing influence. The surface contribution offsets this\n \\[\\delta W_S = \\frac{1}{2} \\oint \\dd S ( \\vu n \\cdot \\vec \\xi) ^2 \\left[ \\left[ \\grad \\left( p_0 \u0026#43; \\frac{B_0^2}{2 \\mu_0} \\right) \\right] \\right] \\cdot \\vu n\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Instabilities can be characterized as:\n Internal/fixed boundary  \\( \\delta W = \\delta W_F \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  External/free boundary  \\( \\delta W = \\delta W_F \u0026#43; \\delta W_S \u0026#43; \\delta W_V \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   The plasma portion can be re-written slightly as\n \\[\\delta W_F = \\frac{1}{2} \\int \\dd V \\frac{ |B_{1, \\perp}|^2}{\\mu_0} \\quad \\leftarrow \\text{Shear Alfven} \\\\ \u0026#43; \\mu_0 \\left| \\frac{B_{1, \\parallel}}{\\mu_0} - \\frac{B_0 \\xi \\cdot \\grad p_0}{B_0} ^2 \\right|^2 \\quad \\leftarrow \\text{Fast magnetosonic} \\\\ \u0026#43; \\Gamma p_0 |\\div \\xi|^2 \\quad \\leftarrow \\text{Acoustic}\\\\ \u0026#43; \\frac{\\vec j_0 \\cdot \\vec B_0}{B_0 ^2} (\\vec B_0 \\cross \\vec \\xi) \\cdot \\vec B_1 \\quad \\leftarrow \\text{Current-driven (kink)} \\\\ - 2 ( \\vec \\xi \\cdot \\grad p_0)(\\vec \\xi \\cdot \\vec \\kappa) \\quad \\leftarrow \\text{pressure-driven (interchange/balooning)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\( \\vec \\kappa \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the curvature vector  \\( \\vu e_B \\cdot \\grad \\vu e_B \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . If we look at each of these terms, the first three terms are all going to be stabilizing effects, which means that all instability is going to come from the last two terms, the current-driven instability term and the pressure-driven instability term.\nGoing back to the screw pinch,  \\[\\dv{p}{r} = j_\\theta B_z - j_z B_\\theta\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  we have current in the same direction as magnetic field ( \\( j_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  with  \\( B_\\theta \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and  \\( j_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  with  \\( B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; ), so kink instabilities are possible. We also have a pressure gradient, so interchange instabilities are also possible.\nAs a concrete example, look at the pressure driven instability term in a Z-pinch.  \\[\\kappa = - \\frac{ \\vu r}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\vec \\xi = \\xi _r \\vu r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };   \\[\\delta W_{F, pressure} = \\int \\dd V \\xi _r \\dv{p_0}{r} \\frac{\\xi_r}{r} \\\\ = \\int \\dd V \\frac{2 \\xi _r ^2}{r} \\dv{p_0}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \nIn a Z-pinch, it is always the case that  \\( \\dv{p_0}{r} \u0026lt; 0 \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . As shown by Kadomtsev (1965) it turns out that these modes can be stabilized by adding  \\( B_z \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; , but this also introduces kink modes.\nGoing back to our stabilizing quantities of wellness and shear, current-driven instabilities are generally managed through shear, and pressure-driven instabilities are stabilized by well.\n"});index.add({'id':11,'href':'/r/notes/UWAA558/formulary/','title':"Formulary",'section':"MHD Theory",'content':"   \\( \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Formulary #  \u0026ldquo;Kinetic Description\u0026rdquo;\n \\[\\dv{\\vec v}{t} = \\frac{q_i}{m_i} (\\vec E \u0026#43; \\vec v_i \\cross \\vec B) \u0026#43; \\sum_{j \\neq i} \\left[ \\left. \\dv{\\vec v_{ij}}{t} \\right|_{coll} \\delta(\\vec r_i - \\vec r_j) \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\frac{1}{c^2} \\pdv{\\vec E}{t} = \\curl \\vec B - \\mu_0 \\sum_i q_i \\vec v_i \\delta (\\vec r - \\vec r_i)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\div \\vec{B} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\div \\vec E = \\frac{1}{\\epsilon_0} \\sum_i q_i \\delta (\\vec r - \\vec r_i)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \nKlimontovich equation:\n \\[\\dv{N}{t} = 0 = \\pdv{N}{t} \u0026#43; \\pdv{}{q_i} \\cdot (\\dot{q_i} N) \\\\ N \\equiv \\sum_i \\delta (p - p_i) \\delta(q - q_i)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \u0026ldquo;Plasma Fluid Description\u0026rdquo;\nBoltzmann Equation\n \\[\\pdv{f_\\alpha}{t} \u0026#43; \\vec v \\cdot \\pdv{f_\\alpha}{t} \u0026#43; \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} = \\left. \\pdv{f_\\alpha}{t} \\right|_{coll} = \\sum_{\\beta \\neq \\alpha} C_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \nMaxwellian distribution:\n \\[f_\\alpha (\\vec v) = n_\\alpha \\left( \\frac{m_\\alpha}{2 \\pi T} \\right)^{3/2} e^{- \\frac{m_\\alpha(\\vec v - \\vec v_\\alpha)^2}{2T}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Moments of fluid model (moments of distribution \\( \\rightarrow \\) moments of Boltzmann equation:\n \\[\\text{Continuity:} \\qquad n_\\alpha = \\int f_\\alpha \\dd \\vec v \\\\ \\quad \\rightarrow \\pdv{n_\\alpha}{t} \u0026#43; \\div (n_\\alpha \\vec v_\\alpha) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\text{Momentum:} \\qquad n_\\alpha \\vec v_\\alpha = \\int \\vec v f_\\alpha \\dd v \\\\ \\quad \\rightarrow \\quad \\pdv{}{t} (n_\\alpha \\vec v_\\alpha ) \u0026#43; \\div (n_\\alpha \\vec v_\\alpha \\vec v_\\alpha) \u0026#43; \\frac{1}{m_\\alpha} \\div \\vec P_\\alpha - \\frac{q_\\alpha}{m_\\alpha} n_\\alpha ( \\vec E \u0026#43; \\vec v _\\alpha \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\rightarrow \\rho_\\alpha \\left(\\pdv{\\vec v_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad \\vec v_\\alpha \\right) \u0026#43; \\grad \\vec P_\\alpha \u0026#43; \\div \\vec \\Pi_\\alpha - q_\\alpha n_\\alpha (\\vec E \u0026#43; \\vec v_\\alpha \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\vec R_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\text{Energy:} \\qquad \\int \\vec v \\vec v \\pdv{f_\\alpha}{t} \\dd \\vec v = \\pdv{}{t} \\int \\vec v \\vec v f_\\alpha \\dd \\vec v = \\pdv{}{t} \\vec E_\\alpha / m_\\alpha \\rightarrow \\pdv{}{t} \\vec P_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\rightarrow \\quad \\frac{3}{2} n_\\alpha \\left( \\pdv{T_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad T_\\alpha \\right) \u0026#43; P_\\alpha \\div \\vec v_\\alpha \u0026#43; \\vec \\Pi_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Closure relations\n \\[\\vec h_\\alpha = - \\kappa \\grad T_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\overline \\Pi_ \\alpha = \\nu \\grad \\vec v_\\alpha\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n Ideal MHD #  \u0026ldquo;Ideal MHD\u0026rdquo;\nContinuity:\n \\[\\pdv{\\rho}{t} \u0026#43; \\div (\\rho \\vec v) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Momentum:\n \\[\\rho \\left( \\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Generalized Ohm\u0026rsquo;s Law\n \\[\\vec E \u0026#43; \\vec v \\cross \\vec B = \\frac{1}{Zen}\\cancel{(\\vec j \\cross \\vec B - \\grad p_e)} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Energy\n \\[\\dv{}{t} \\left( \\frac{p}{\\rho^\\gamma} \\right) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \u0026ldquo;Lawson Criterion\u0026rdquo;\n \\[n \\tau_E \u0026gt; 10^4 s/m^3\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \u0026ldquo;Conservation Law Form of Ideal MHD\u0026rdquo;\nContinuity:\n \\[\\pdv{\\rho}{t} \u0026#43; \\div (\\rho \\vec v) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Momentum:\n \\[\\pdv{(\\rho \\vec v)}{t} \u0026#43; \\div \\left[ \\rho \\vec v \\vec v - \\frac{\\vec B \\vec B}{\\mu_0} \u0026#43; \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\overline{I} \\right] = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Energy:\n \\[\\pdv{\\epsilon}{t} \u0026#43; \\div \\left[ \\left( \\epsilon \u0026#43; p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\vec v - (\\vec B \\cdot \\vec v) \\frac{\\vec B}{\\mu_0} \\right] = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\pdv{\\vec B}{t} \u0026#43; \\div ( \\vec v \\vec B - \\vec B \\vec v) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  where\n \\[\\epsilon = \\frac{1}{\\gamma - 1} p \u0026#43; \\frac{1}{2} \\rho v^2 \u0026#43; \\frac{B^2}{2\\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \nStatic Equilibrium:\n \\[\\vec j \\cross \\vec B = \\grad p\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\frac{B^2}{\\mu_0} \\vec K = \\grad_\\perp (p \u0026#43; \\frac{B^2}{2 \\mu_0})\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\vec K \\equiv \\frac{\\vec B}{|B|} \\cdot \\grad \\frac{ \\vec B}{|B|}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \nConservation of flux:\n \\[\\vec E \u0026#43; \\vec v \\cross \\vec B = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\rightarrow \\dv{}{t} \\left( \\frac{\\vec B}{\\rho} \\right) = \\frac{\\vec B}{\\rho} \\cdot \\grad \\vec v\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n 1D Equilibria #  \u0026quot;\\( \\theta \\)-pinch\u0026quot;\n \\[B_\\theta = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[j_\\theta B_z = \\dv{p}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[j_\\theta = - \\frac{1}{\\mu_0} \\dv{B_z}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\rightarrow p \u0026#43; \\frac{B_z ^2}{2 \\mu_0} = \\frac{B_0 ^2}{2 \\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\langle \\beta \\rangle = \\frac{2}{a^2} \\int_0 ^a \\frac{r p}{B_0 ^2 / 2 \\mu_0} \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[q = \\infty\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[W = \\frac{\\mu_0 r}{B_z ^2} \\dv{}{r} \\left( p \u0026#43; \\frac{B_z ^2}{2 \\mu_0} \\right) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \u0026ldquo;Z-pinch\u0026rdquo;\n \\[B_z =0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\grad p = \\dv{p}{r} = - j_z B_\\theta\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[- \\dv{}{r} \\left( p \u0026#43; \\frac{B_\\theta ^2}{2 \\mu_0} \\right) = \\frac{B_\\theta ^2}{\\mu_0 r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\langle \\beta \\rangle = \\frac{2 \\mu_0}{B_0 ^2 \\pi a^2} \\int _0 ^a 2 \\pi r p \\dd r = 1 \\quad \\text{ if } \\quad p(a) = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[q = S = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[W = 1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \u0026ldquo;Screw pinch\u0026rdquo;\n \\[\\dv{}{r} \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) = - \\frac{B_\\theta ^2}{\\mu_0 r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\beta_t = \\frac{2 \\mu_0}{B_z (a) ^2} \\left( \\frac{1}{\\pi a^2} \\int_0 ^a 2 \\pi r p \\dd r \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\beta_p = \\left( 1 - \\frac{\\alpha_t}{\\beta _t} \\right)^{-1} \\qquad \\alpha_t \\equiv \\frac{2}{a^2} \\int_0 ^a \\left(1 - \\frac{B_z ^2}{B_0 ^2} \\right) r \\dd r\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[q = \\frac{2 \\pi r B_z}{L B_\\theta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[q_a = \\frac{4 \\pi ^2 a^2 B_0}{\\mu_0 I_a}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[S = \\frac{r}{q} \\dv{q}{r}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[W = - \\frac{B_\\theta ^2}{B_\\theta ^2 \u0026#43; B_z ^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \u0026ldquo;Stability\u0026rdquo;\nShear:\n \\[S = 2 \\frac{ dq / q}{dV / V} = 2 \\frac{d \\ln q}{d \\ln V}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[q = \\frac{\\text{\\# long windings}}{\\text{\\# short windings}} = \\dv{\\psi_t}{\\psi_p}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Shear for toroid\n \\[q = \\frac{r B_\\phi}{R B_\\theta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Shear for cylinder\n \\[q = 2 \\pi \\frac{r B_z}{L B_\\theta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Well\n \\[W = \\frac{ d \\langle p \u0026#43; B^2 / 2 \\mu_0 \\rangle / \\langle B^2 / 2 \\mu_0 \\rangle}{dV / V}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  For stabilization, \\( B^2 / 2 \\mu_0 \\) should increase faster than \\( p \\) decreases\n 2D Equilibria #  \u0026ldquo;Grad-Shafranov Equation: Static toroidal equilibrium\u0026rdquo;\n \\[\\grad p = \\vec j_\\theta \\cross \\vec B_\\phi \u0026#43; \\vec j_\\phi \\cross \\vec B_\\theta\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[A_\\phi = \\frac{\\phi}{R} \\vu \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\phi = \\frac{\\phi_p}{2 \\pi}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\vec B_\\theta = - \\frac{\\vu R}{R} \\pdv{\\psi}{z} \u0026#43; \\frac{ \\vu z}{R} \\pdv{\\psi}{R} = \\frac{ \\grad \\psi}{R} \\cross \\vu \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[F \\equiv R B_\\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\Delta ^\\star \\equiv R \\pdv{}{R} \\frac{1}{R} \\pdv{}{R} \u0026#43; \\pdv{^2}{z^2}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\Delta ^\\star \\psi = \\pdv{^2 \\psi}{z^2} \u0026#43; \\pdv{^2 \\psi}{R^2} - \\frac{1}{R} \\pdv{\\psi}{R}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\vec j_\\phi = - \\frac{1}{\\mu_0 R} \\Delta ^\\star \\psi \\vu \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\vec j_\\theta = \\frac{1}{\\mu_0 R} \\grad (F) \\cross \\vu \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[R^2 \\mu_0 \\pdv{p}{\\psi} = - \\Delta ^\\star \\psi - F \\pdv{F}{\\psi}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[q(\\psi) = \\frac{F(\\psi)}{2 \\pi} \\oint_{p} \\frac{r \\dd \\theta}{R^2 B_\\theta}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Limits:\n \\[\\text{Force-free:} \\qquad \\vec j \\parallel \\vec B\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\rightarrow \\Delta ^\\star \\psi \u0026#43; F F\u0026#39; = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\text{Connected $\\theta$ pinch:} \\qquad FF\u0026#39; \\gg \\Delta ^\\star \\psi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\rightarrow \\grad p \\approx \\vec j_\\theta \\cross \\vec B_\\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\text{Connected Z-pinch:} \\qquad FF\u0026#39; \\ll \\Delta ^\\star \\phi\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\rightarrow \\grad p \\approx j_\\phi \\cross B_\\theta\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n MHD Stability #  \u0026ldquo;Linear stability\u0026rdquo;\n \\[\\pdv{\\rho_1}{t} = - \\vec v_1 \\grad \\rho_0 - \\rho_0 \\div \\vec v_1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\pdv{\\vec B_1}{t} = \\curl ( \\vec v_1 \\cross \\vec B_0)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\rho_0 \\pdv{\\vec v_1}{t} = - \\grad p_1 \u0026#43; \\vec j_0 \\cross \\vec B_1 - \\vec j_1 \\cross \\vec B_0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\pdv{p_1}{t} = - \\vec v_1 \\cdot \\grad p_0 - \\gamma p_0 \\div \\vec v_1\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  For linear perturbation \\( \\vec \\xi = \\int_0 ^t \\vec v_1 \\dd t \\) the momentum equation becomes\n \\[\\rho_0 \\pdv{^2 \\xi}{t^2} = \\vec F(\\xi)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  where\n \\[F(\\xi) = \\grad (\\xi \\cdot \\grad p_0 \u0026#43; \\gamma p_0 \\div \\xi) \u0026#43; \\frac{1}{\\mu_0} \\left[ ( \\curl \\vec B_0) \\cross \\curl (\\xi \\cross \\vec B_0) \u0026#43; \\curl \\curl (\\xi \\cross \\vec B_0) \\cross \\vec B_0 \\right]\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \nEigenvalues of \\( \\frac{1}{\\rho_0} \\vec F (\\xi) = \\omega^2 \\xi \\) are real and ordered. Only need to check \\( n=0 \\) to determine stability/instability of configuration.\n \u0026quot;\\( \\delta W \\) Approach\u0026quot;\n\\( \\delta W = \\) change in potential energy due to a displacement \\( \\xi \\)\n \\[\\delta W \u0026lt; 0 \\rightarrow \\text{instability}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }; \n \\[\\delta W = - \\frac{1}{2} \\int \\xi \\cdot F(\\xi) \\dd V = \\delta W_F \u0026#43; \\delta W_S \u0026#43; \\delta W_V\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Surface term:\n \\[\\delta W_s = \\frac{1}{2} \\oint \\dd S (\\vu n \\cdot \\xi) ^2 \\left( \\vu n \\cdot \\grad p_0 \u0026#43; \\left[ \\vu n \\cdot \\grad \\frac{B_0 ^2}{2 \\mu_0} \\right]_{jump} \\right)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Vacuum term:\n \\[\\delta W_V = \\int_{vac} \\dd V \\frac{B_1 ^2}{\\mu_0}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Plasma (free) term:\n \\[\\delta W_F = \\frac{1}{2} \\int \\dd V \\frac{ |B_{1, \\perp}|^2}{\\mu_0} \\quad \\leftarrow \\text{Shear Alfven} \\\\ \u0026#43; \\mu_0 \\left| \\frac{B_{1, \\parallel}}{\\mu_0} - \\frac{B_0 \\xi \\cdot \\grad p_0}{B_0} ^2 \\right|^2 \\quad \\leftarrow \\text{Fast magnetosonic} \\\\ \u0026#43; \\Gamma p_0 |\\div \\xi|^2 \\quad \\leftarrow \\text{Acoustic}\\\\ \u0026#43; \\frac{\\vec j_0 \\cdot \\vec B_0}{B_0 ^2} (\\vec B_0 \\cross \\vec \\xi) \\cdot \\vec B_1 \\quad \\leftarrow \\text{Current-driven (kink)} \\\\ - 2 ( \\vec \\xi \\cdot \\grad p_0)(\\vec \\xi \\cdot \\vec \\kappa) \\quad \\leftarrow \\text{pressure-driven (interchange/balooning)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  };  Shear Alfven, fast magnetosonic, and acoustic modes are stabilizing. Current-driven and pressure-driven modes can lead to instability.\n "});index.add({'id':12,'href':'/r/notes/working/crews2018/','title':"Crews (2018)",'section':"Working",'content':"Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities #  Shielded potential of a test electron:\n  \\[\\phi(r) = \\frac{-e}{4 \\pi \\epsilon_0 r} e ^{- r / \\lambda_D}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where the Debye length is  \\( \\lambda_D = \\sqrt{\\frac{\\epsilon_0 T_e}{ n_e e}} \\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The mean free path between large-angle collisions is estimated as\n \\[\\lambda_{mfp} \\sim \\frac{\\epsilon_0 T_e ^2}{\\phi_e n_e \\log ( \\Lambda)}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\(\\phi_e = e^2 / 4 \\pi \\epsilon_0\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  are the constants from the Coulomb force law.\nSmooth out the discreteness of particles via spatial average over small volumes:\n \\[\\rho \\rightarrow \\langle \\rho_c \\rangle \\qquad \\vec E \\rightarrow \\langle \\vec E \\rangle \u0026#43; \\delta \\vec E\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  The mean field  \\(\\langle \\vec E \\rangle\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales  \\(\\lambda_{mfp} / \\lambda_D\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n \\[\\frac{\\lambda_{mfp}}{\\lambda_D} \\sim \\frac{T_e ^{3/2}}{n_e ^{1/2}}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified.\nPhase space mechanics #  To arrive at a kinetic equation governing the collisionless mechanics, consider the one-dimensional motion of a single particle\n \\[\\dot{x} = v \\qquad \\dot v = F(x)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and define the phase space coordinates as  \\(\\vec{\\dot r} = \\vec F \\equiv [ v, F(x) ]\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; . The flux vector is similar to the velocity field of a fluid flow. The streamlines of  \\(\\vec F\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  are the streamlines which a particle will follow if the flux is constant in time. Phase flow is always analogous to that of an incompressible fluid because the flux divergence is zero:\n \\[\\div [v, F(x)] = \\pdv{v}{x} \u0026#43; \\pdv{F(x)}{v}\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  If the phase fluid density is given by a function  \\(f(x, v, t)\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  where  \\(t\\)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  is the time parameter, because any instantiation of a particle can not leave the phase plane, the probability density will be conserved. We can write a conservation law:  \\[\\pdv{f(x, v, t)}{t} = - \\div ( f (x, v, t) \\vec F)\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); };  and due to the flow\u0026rsquo;s incompressibility  \\[\\div (f \\vec F) = f ( \\div \\vec F) \u0026#43; \\vec F \\cdot \\grad f = \\vec F \\cdot \\grad f \\\\ \\rightarrow \\pdv{f}{t} = - \\left[ v, F(x) \\right] \\cdot \\left[ \\pdv{f}{x}, \\pdv{f}{v} \\right] \\\\ \\rightarrow \\pdv{f}{t} \u0026#43; v \\pdv{f}{x} \u0026#43; F(x) \\pdv{f}{v} = 0\\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }; \n"});})();
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+'use strict';(function(){const indexCfg={cache:true};indexCfg.doc={id:'id',field:['title','content'],store:['title','href','section'],};const index=FlexSearch.create('balance',indexCfg);window.bookSearchIndex=index;index.add({'id':0,'href':'/r/notes/UWAA558/01-syllabus/','title':"Syllabus",'section':"MHD Theory",'content':"Syllabus #  The course topics planned for this section are (in rough order):\nParticle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, \u0026ldquo;Frozen-In\u0026rdquo; Flux)\nIdeal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces\nOne-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q\nTwo-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)\nMHD Stability, General Concepts, Linearized MHD, Exponential (Linear) Stability, Force Operator and Properties, Variational Formulation, Energy Principle, Intuitive Form of delta W, Classification of Instabilities (internal/external, pressure-drive/current-driven, kink/interchange/ballooning)\nStability of One-Dimensional Equilibria, Modal Analysis, Rayleigh-Taylor, Theta-Pinch, Z-Pinch (Kadomtsev Condition), Screw-Pinch (Kruskal-Shafranov Condition, Suydam Criterion), RFP, \u0026ldquo;Straight\u0026rdquo; Tokamak\nStability of Two-Dimensional Equilibria, Tokamak, Mercier Criterion, Elmo Bumpy Torus\nResistive (Tearing) Instabilities, Stability of Non-static Equilibria, Nonlinear Stability Effects\nCourse Motivation #  Plasma phenomena tend to be hard to treat because of the span of relevant scales. You have ions, electrons, and photons interacting through electromagnetic interactions. There is a tremendous variation in mass across species, which leads to a large span of both spatial and temporal scales. The species can interact through both short scale collisions and long range interactions through EM forces. In contrast, in normal gas dynamics you may consider only the short-scale interactions. As a consequence, we can describe dispersive plasma waves.\nFor comparison, remember in gas dynamics, the speed of sound is\n  document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\[\\frac{\\omega}{k} = v_s = \\pdv{\\omega}{k} \\qquad \\text{(gas)}\\]  Here the phase velocity  \\( \\frac{\\omega}{k} \\)  is equal to the phase velocity  \\( \\pdv{\\omega}{k} \\) . In a plasma, we can have non-linear dispersion relations in which the phase and group velocity are different.\n \\[\\frac{\\omega}{k} = v(\\omega, k) \\neq \\pdv{\\omega}{k} \\qquad \\text{(plasma)} \\]  The number of particles we typically deal with in a laboratory plasma is roughly on the order of a mole of particles\n \\[\\text{particles} \\sim O(10^{23})\\]  With long-range interactions, we have a combinatorial explosion of interacting particles! It is not possible to track individual particles at such a scale, so we need much simpler plasma descriptions. These are the plasma models we will discuss in the next chapter. When simplifying our models, we need to pay careful attention to the simplifications we are making because in general, inaccurate physics lead to incorrect conclusions.\n"});index.add({'id':1,'href':'/r/notes/UWAA558/02-plasma-models/','title':"Plasma Models",'section':"MHD Theory",'content':"Plasma Models #  Working towards MHD #  Let\u0026rsquo;s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we\u0026rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.\nThe most important question to ask ourselves is \u0026ldquo;when is this model going to be useful?\u0026rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid. This means that we can define a fluid element (some lil' box of plasma) and define the physical properties (mass, density, magnetization, etc.) of the element. We need to make some assumptions about scale in order to do this. In terms of spatial scales, we abstract properties below a discrete scale   document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\( a_0 \\)  into the properties of a fluid element\n \\[\\frac{a_0}{L} \\rightarrow 0 \\qquad a_0 = \\text{discrete scale} \\qquad L = \\text{spatial scale of interest}\\]  Length scales smaller than the discrete scale will not be properly captured by the model, so scales like the particle radius will be meaningless in our fluid model.\nPlasma Definition #  A plasma is a quasi-neutral gas of charged and neutral particles which exhibit collective behavior. The particles (electrons, ions, neutrals) interact through EM fields and collisions.\nMathematically you would think that plasmas could be treated as individual particles. Doing so gives an N-body problem with classical interactions through the Lorentz force (Coulomb interaction removed by assumption of quasi-neutrality) and binary collisions. Each particle  \\( i \\)  has a well-defined mass  \\( m_i \\)  and charge  \\( q_i \\)  which do not change in time. The governing equations are\n \\[\\dv{\\vec v_i}{t} = \\frac{q_i}{m_i} (\\vec E \u0026#43; \\vec v_i \\cross \\vec B) \u0026#43; \\sum_{j \\neq i} \\left[ \\left. \\dv{\\vec v_{ij}}{t} \\right|_{coll} (\\vec r_i - \\vec r_j) \\right]\\]   \\[\\dv{\\vec r_i}{t} = \\vec v_i\\] \nThe fields E and B are described by the Maxwell equations\n \\[\\pdv{B}{t} = - \\curl E\\]   \\[\\frac{1}{c^2} \\pdv{E}{t} = \\curl B - \\mu_0 \\sum_i q_i v_i \\delta (r - r_i)\\]   \\[\\div B = 0\\]   \\[\\epsilon_0 \\div E = \\sum_i q_i \\delta (r - r_i)\\] \nKlimontovich Equation #  Re-writing the force relations as a statement of conservation in phase space, we get Klimontovich equation for species  \\( \\alpha \\) \n \\[\\dv{N_\\alpha}{t} = 0 = \\pdv{N_\\alpha}{t} \u0026#43; \\pdv{}{q} \\cdot (\\dot{q} N_\\alpha) \u0026#43; \\pdv{}{p} \\cdot (\\dot{p}N_\\alpha)\\]  The particle phase space is defined by  \\[N_\\alpha(p, q) = \\sum_i \\delta(p - p_i) \\delta(q - q_i)\\]  where  \\( p \\)  and  \\( q \\)  are generalized momentum and position coordinates. The resulting  \\( N(p) \\)  looks very spiky, with nonzero values only at the exact values inhabited by particles. Unfortunately, that means that only the tools of discrete mathematics are applicable to the distribution, and we\u0026rsquo;re forbidden from our favorite tool (calculus). To make the analysis possible, we can smooth over the discreteness by performing an ensemble average of the Klimontovich equation. This gives us a statistical description using smooth distribution functions:\n \\[f(x, v, t) \\qquad \\frac{(\\text{no. of particles})}{(\\text{unit distance})^3(\\text{unit velocity})^3}\\]  That is,  \\( f(x, v, t) \\)  is the number of particles at position  \\( x \\)  with velocity  \\( v \\)  at time  \\( t \\) . We also work with normalized distributions which give the probability of finding a particle.\nBy our definition, we can integrate to get the total number of particles at time  \\( t \\) .\n \\[\\int \\int \\dd x \\dd v f(x, v, t) = N(t)\\]  The ensemble averaging process works well if the the number of particles is very large\n \\[N \\gg 1\\]  See a kinetic theory text (e.g. Krall and Trivelpiece) for a full description of the ensemble averaging process.\nNow we can write our Klimontovich equation in terms of continuous quantities\n \\[\\pdv{f}{t} \u0026#43; \\left[ \\pdv{}{q} \\cdot (\\dot{q} f) \u0026#43; \\pdv{}{p} \\cdot (\\dot{p}f) \\right] = \\text{(cross terms)}\\]  The collision terms now have an infinite number of cross terms. We call this the BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy.\nExpressing as the Boltzmann equation (generally called the Boltzmann-Maxwell equation, since the solution requires solving for the electromagnetic fields of the Maxwell equations)\n \\[\\pdv{f_\\alpha}{t} \u0026#43; \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \u0026#43; \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} = \\left. \\pdv{f_\\alpha}{t} \\right|_{coll}\\]  We leave the collision term in. A lot of the work of kinetic theory is coming up with an applicable form of the collision operator which is appropriate but still simple enough to solve. We often write the collision operator as the product of binary collisions\n \\[ \\left. \\pdv{f_\\alpha}{t} \\right|_{coll} = \\sum_\\beta C_{\\alpha \\beta}\\]  Notice that the terms  \\( \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \\)  and  \\( \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\)  are advection equations, advecting in  \\( \\vec x \\)  and  \\( \\vec v \\)  respectively. If we ignore the collision term (set the RHS to zero) we have the Vlasov equation.\nNote that the fields E and B at any location are generated from the charges and currents of the entire plasma volume, including externally applied fields. As a result, there\u0026rsquo;s an inherent integrating process taking into account the sources across the whole volume that leads to long-range smoothly varying forces. This is in contrast to the collisional effects, which by their nature lead to very short range abrupt forces. It makes sense to make a distinction between the long-range electromagnetic forces and the short range collisional forces.\nBecause the Boltzmann-Maxwell model is inherently 6-dimensional, it is a very challenging model to implement. The B-M model provides a complete description, but it is often too detailed to solve.\nIf we solve the Vlasov equation for two parallel opposite beams, for example, we see that the\nAs it turns out, the integral, centroid, and variance are all that are required to fully describe a Maxwellian distribution\n \\[f_{M, \\alpha} (\\vec v) = n_0 \\left( \\frac{ m_\\alpha }{2 \\pi T_\\alpha} \\right)^{3/2} \\text{exp}\\left[ - \\frac{ \\frac{1}{2} m_\\alpha (v - v_\\alpha)^2}{T_\\alpha} \\right]\\]  In other words, we can write the Maxwellian distribution as  \\( f_M(n_0, \\vec v_\\alpha, T_\\alpha ; \\vec v) \\) . We care about Maxwellian distributions so much in plasma physics because it is the solution to the Boltzmann equation for  \\( \\left. \\pdv{f_\\alpha}{t} \\right|_{coll} = 0 \\)  (Vlasov equation). That\u0026rsquo;s not saying that there are no collisions, it is saying that there are so many collisions that the effect is isotropic and the overall force is zero.\nAnother feature of the Maxwellian distribution is  \\( \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} = 0 \\) . This is a famous result called the Boltzmann H-theorem, and says that any initial distribution will relax (and very quickly) to a Maxwellian distribution.\nBy replacing our velocity distribution with the associated Maxwell distribution, we arrive at a Plasma Fluid Model\n"});index.add({'id':2,'href':'/r/notes/UWAA558/03-plasma-fluid-model/','title':"Plasma Fluid Model",'section':"MHD Theory",'content':"Plasma Fluid Model #  We take velocity moments of each of the pieces of the kinetic model:\n distribution function,   document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\( f_\\alpha \\)   \\( \\rightarrow \\)  fluid variables Boltzmann Equation  \\( \\rightarrow \\)  governing equations describing the evolution of the fluid variables.  Starting with the zeroth moment (integral) of the distribution function:\n \\[\\int f_\\alpha (\\vec x, \\vec v, t) \\dd \\vec v = n_\\alpha(\\vec x, t)\\]  1st Moment (momentum):\n \\[m_\\alpha \\int \\vec v \\cdot f_\\alpha(\\vec x, \\vec v, t) \\dd \\vec v = \\vec p_\\alpha (\\vec x, t) = m_\\alpha n_\\alpha \\vec v_\\alpha\\]  which is to say that the velocity is the 1st moment divided by the zeroth moment\n \\[v_\\alpha = \\frac{\\int \\vec v f_\\alpha}{\\int f_\\alpha}\\]  2nd Moment:\n \\[\\int \\vec v \\vec v f_\\alpha (\\vec x, \\vec v, t) \\dd \\vec v = \\vec E_\\alpha(\\vec x, t) (\\text{energy tensor})\\]  We can simplify the 2nd moment by taking a reduced 2nd moment. This means that we\u0026rsquo;re going to insert a dot product\n \\[\\int \\vec v \\cdot \\vec v f_\\alpha (\\vec x, \\vec v, t) \\dd \\vec v\\]  Before moving forward, we want to define a \u0026ldquo;random\u0026rdquo; velocity. Note that\n \\[\\int \\vec v f_\\alpha - \\vec v_\\alpha \\int f_\\alpha = 0 \\rightarrow \\int (\\vec v - \\vec v_\\alpha) f_\\alpha \\dd \\vec v = 0\\]  We can define a random velocity  \\( \\vec w = \\vec v - \\vec v_\\alpha \\) . It is random in the sense that it is a fluctuation about the mean velocity, and when we integrate it we get zero. We can use this to define the energy tensor using the mean velocity to get a meaningful result. The pressure tensor is the second moment, using the random velocity\n \\[\\vec P_\\alpha = m_\\alpha \\int \\vec w \\vec w f_\\alpha \\dd \\vec v = P_\\alpha \\overline{I} \u0026#43; \\overline{\\Pi}_\\alpha\\]  where we\u0026rsquo;ve decomposed the pressure into an isotropic value  \\( P_\\alpha \\)  and what\u0026rsquo;s called the Braginskii stress tensor  \\( \\overline{\\Pi}_\\alpha \\) . The average isotropic pressure is given by the reduced 2nd moment:\n \\[P_\\alpha = n_\\alpha T_\\alpha = \\frac{1}{3} \\int m_\\alpha \\vec w \\cdot \\vec w f_\\alpha\\]  where the factor of  \\( 1/3 \\)  comes from the number of degrees of freedom in our system. It is related to the thermodynamic factor  \\( \\gamma \\)  where  \\( \\gamma = \\frac{DOF \u0026#43; 2}{DOF} \\) . Now we can define the temperature  \\( T_\\alpha \\)  as\n \\[T_\\alpha (\\vec x, t) = \\frac{1}{DOF} \\frac{\\int m_\\alpha \\vec w \\cdot \\vec w f_\\alpha \\dd \\vec v}{\\int f_\\alpha \\dd \\vec v}\\]  Now that we\u0026rsquo;ve got  \\( n_\\alpha \\) ,  \\( v_\\alpha \\) , and  \\( T_\\alpha \\)  we have what we need to define a Maxwellian distribution. Higher moments would be required to describe non-Maxwellian distribution functions. For example, the 3rd moment is called the skewness of the distribution. The 4th moment is the kurtosis. So on and so forth. These give a measure of degree of departure from a Maxwellian distribution, in which case it is often more useful to talk about the excess kurtosis, where the excess kurtosis of a Maxwellian is defined to be zero. You can continue to calculate the moment expansion, and in general it requires an infinite number of moments to describe an arbitrary distribution function. Because of the Boltzmann H-theorem and the tendency of plasmas to quickly relax to Maxwellian, we can usually get away with using just the first three moments.\nNow, what are the governing equations? We get these by taking moments of the Boltzmann equation. Let\u0026rsquo;s proceed carefully in sections, so we\u0026rsquo;ll integrate each piece of the BE in terms.\n0th Moment of Boltzmann Equation (Conservation) #   \\[\\int \\pdv{f_\\alpha}{t} \\dd \\vec v \u0026#43; \\int \\vec v \\cdot \\pdv{f_\\alpha}{x} \\dd v \u0026#43; \\int \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\int \\left. \\pdv{f_\\alpha}{t} \\right|_{coll} \\dd \\vec v\\]  First, we take  \\( \\int \\pdv{f_\\alpha}{t} \\dd \\vec v \\) . Because t and v are both independent variables, with an argument of sufficient smoothness we can reverse the order of integration and differentiation\n \\[\\int \\pdv{f_\\alpha}{t} \\dd \\vec v = \\pdv{}{t}\\int f_\\alpha \\dd v = \\pdv{n_\\alpha}{t}\\]  For  \\( \\int \\vec v \\cdot \\pdv{f_\\alpha}{x} \\dd v \\)  we can perform an integration by parts\n \\[ \\int \\vec v \\cdot \\pdv{f_\\alpha}{x} \\dd v = \\int \\pdv{}{\\vec x} \\cdot ( \\vec v f_\\alpha) \\dd \\vec v - \\int f_\\alpha \\pdv{}{\\vec x} \\cdot \\vec v \\dd \\vec v \\\\ = \\int \\pdv{}{\\vec x} \\cdot ( \\vec v f_\\alpha) \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int \\vec v f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot (n_\\alpha \\vec v_\\alpha) = \\div (n_\\alpha \\vec v_\\alpha)\\]  Once again, we\u0026rsquo;ve switched the order of integration of  \\( x \\)  and  \\( v \\) , which we can only do because we have specified that  \\( f \\)  is a distribution function, and as such meets the criterion of sufficient smoothness.\nFor the last part, we can write it as a surface integral\n \\[\\int \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\oint f_\\alpha \\frac{q_\\alpha}{m_\\alpha} ( \\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\dd S_v - \\int \\frac{q_\\alpha}{m_\\alpha} f_\\alpha \\pdv{}{\\vec v} \\cdot (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\dd \\vec v\\]  For  \\( f_\\alpha \\)  to be well-defined, we require  \\( \\lim_{v \\rightarrow \\infty} v^3 f_\\alpha = 0 \\)  so the surface term vanishes and we\u0026rsquo;re left with\n \\[\\int \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = - \\int \\frac{q_\\alpha}{m_\\alpha} f_\\alpha \\pdv{}{\\vec v} \\cdot (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\dd \\vec v \\]  Distributing the divergence through,\n \\[\\pdv{}{\\vec v} \\cdot \\vec E = 0\\]  because  \\( \\vec E \\)  is independent of  \\( \\vec v \\) .\n \\[\\pdv{}{\\vec v} \\cdot ( \\vec v \\cross \\vec B) = 0\\]  because  \\( \\vec v \\cross \\vec B \\)  is always orthogonal to  \\( \\vec v \\) .\nWe\u0026rsquo;ve been working through this in normal vector notation for familiarity, but there is another notation known as Einstein Tensor Notation using the lovely Levi-Civita symbol  \\( \\epsilon_{ijk} \\) .\n \\[\\epsilon_{ijk} = 1 \\qquad \\text{even permutations of i j k} \\\\ = -1 \\qquad \\text{odd permutations of i j k} \\\\ = 0 \\qquad \\text{any repeated indexes}\\]  We can write out vector products as products of indices and operators, and any repeated indices are implicitly summed:\n \\[\\vec v \\cross \\vec B = \\epsilon_{ijk} v_j B_k = \\sum_{jk} \\epsilon_{ijk} v_j B_k\\]  where  \\( \\epsilon_{ijk} \\)  is defined to be 1 for even permutations of ijk, -1 for negative permutations if ijk, and 0 for any repeated indices.\nWe can now write the derivative with respect to  \\( \\vec v \\)  as\n \\[\\pdv{}{\\vec v} \\cdot \\vec v \\cross \\vec B = \\partial _{v_i} \\epsilon_{ijk} v_j B_k \\\\ = \\epsilon_{ijk} (\\partial_{v_i} v_j) B_k\\]  We see that we\u0026rsquo;re taking the derivative of the j-th component of velocity with respect to the i-th component of velocity, and that\u0026rsquo;s how we can most easily point out that the quantity is zero without relying on properties of 3-vector products.\nWe\u0026rsquo;ll also want to use the divergence in Einstein tensor notation\n \\[\\div \\vec A = \\partial_i A_j \\delta_{ij} = \\partial_i A_i\\]  Finally, we come back to the collision term in the zeroth moment of the B-M equation\n \\[\\sum_\\beta \\int C_{\\alpha \\beta} \\dd \\vec v = 0\\]  We can say the collision term is zero by making a physical argument, rather than a mathematical one. We assert that collisions cannot create or destroy particles. This assumption is now baked into our equations going forward, but is not always true! Ionization, recombination, fusion reactions all create/destroy species.\nFinally, we\u0026rsquo;ve got\n \\[\\pdv{n_\\alpha}{t} \u0026#43; \\div ( n_\\alpha \\vec v_\\alpha) = 0 \\qquad \\text{continuity equation}\\]  So by taking the 0th moment of the Boltzmann Equation, we\u0026rsquo;ve arrived at the continuity equation for  \\( n_\\alpha \\)  by introducing  \\( \\vec v_\\alpha \\) .\n1st Moment of Boltzmann Equation #   \\[\\int \\vec v \\pdv{f_\\alpha}{t} \\dd \\vec v \u0026#43; \\int \\vec v \\vec v \\pdv{f_\\alpha}{\\vec x} \\dd \\vec v \u0026#43; \\int \\vec v \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\int \\left. \\vec v \\pdv{f_\\alpha}{t} \\right|_{coll}\\]  Term 1:\n \\[\\int \\vec v \\pdv{f_\\alpha}{t} \\dd \\vec v = \\pdv{}{t} \\int \\vec v f_\\alpha \\dd \\vec v - \\int \\pdv{\\vec v}{t} f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{t} \\left( n_\\alpha \\vec v_\\alpha \\right)\\]  Before we get to the second term, let\u0026rsquo;s have an aside about dyad math. An outer product  \\( \\vec A \\vec B \\)  gives a second-rank tensor. In Cartesian coordinates, it looks like\n \\[\\vec A \\vec B = \\begin{bmatrix} A_x B_x \u0026amp; A_x B_y \u0026amp; A_x B_z \\\\ A_y B_x \u0026amp; A_y B_y \u0026amp; A_y B_z \\\\ A_z B_x \u0026amp; A_z B_y \u0026amp; A_z B_z \\\\ \\end{bmatrix}\\]  A useful property to do with dot products:\n \\[\\vec A \\vec B \\cdot \\vec C = \\vec A ( \\vec B \\cdot \\vec C ) = (\\vec A \\vec B) \\cdot \\vec C\\]   \\[( \\vec V \\cdot \\grad) \\vec B = \\vec V \\cdot ( \\grad \\vec B)\\]  Back to term 2:\n \\[\\int \\vec v \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \\dd \\vec v = \\int \\vec v \\cdot \\pdv{}{\\vec x} ( \\vec v f_\\alpha) \\dd \\vec v - \\int \\vec v f_\\alpha \\pdv{}{\\vec x} \\cdot \\vec v \\dd \\vec v \\\\ = \\int \\pdv{}{\\vec x} \\cdot ( \\vec v \\vec v f_\\alpha) \\dd \\vec v - \\int \\vec v f_\\alpha \\cdot \\pdv{\\vec v}{\\vec x} \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int \\vec v \\vec v f_\\alpha \\dd \\vec v\\]  Re-expanding in terms of random velocity  \\( \\vec v = \\vec v_\\alpha \u0026#43; \\vec w \\) \n \\[= \\pdv{}{\\vec x} \\cdot \\int (\\vec v_\\alpha \u0026#43; \\vec w) (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int (\\vec v_\\alpha \\vec v_\\alpha \u0026#43; \\vec v_\\alpha \\vec w \u0026#43; \\vec w \\vec v_\\alpha \u0026#43; \\vec w \\vec w) f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\left[ \\vec v_\\alpha \\vec v_\\alpha \\int f_\\alpha \\dd \\vec v \u0026#43; \\vec v_\\alpha \\int \\vec w f_\\alpha \\dd \\vec v \u0026#43; \\left( \\int \\vec w f_\\alpha \\dd \\vec v \\right) \\vec v_\\alpha \u0026#43; \\int \\vec w \\vec w f_\\alpha f_\\alpha \\dd \\vec v \\right]\\]  The middle two terms are zero by the very definition of  \\( \\vec v_\\alpha \\) . The random velocity  \\( \\vec w \\)  is defined such that the integral of the random velocity over all phase space is zero.\n \\[= \\pdv{}{\\vec x} \\cdot \\left( \\vec v_\\alpha \\vec v_\\alpha n_\\alpha \\right) \u0026#43; \\pdv{}{\\vec x} \\cdot \\frac{ \\vec P_\\alpha}{m_\\alpha} \\\\ = \\div (n_\\alpha \\vec v_\\alpha \\vec v_\\alpha) \u0026#43; \\frac{1}{m_\\alpha} \\div \\vec P_\\alpha\\]  Substituting in the pressure tensor,\n \\[= \\div ( n_\\alpha \\vec v_\\alpha \\vec v_\\alpha ) \u0026#43; \\frac{1}{m_\\alpha} \\left( \\grad P_\\alpha \u0026#43; \\div \\vec \\Pi _\\alpha \\right)\\]  Moving on to term 3:\n \\[\\int \\vec v \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\int \\frac{q_\\alpha}{m_\\alpha} \\pdv{}{\\vec v} \\cdot \\left[( \\vec E \u0026#43; \\vec v \\cross \\vec B) f_\\alpha \\right] \\vec v \\dd \\vec v - \\int \\frac{q_\\alpha}{m_\\alpha} f_\\alpha \\pdv{}{\\vec v} \\cdot \\left[ \\vec E \u0026#43; \\vec v \\cross \\vec B \\right] \\vec v \\dd \\vec v \\\\ = \\int \\frac{q_\\alpha}{m_\\alpha} \\pdv{}{\\vec v} \\cdot \\left[ f_\\alpha (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\vec v \\right] \\dd \\vec v - \\int \\frac{q\\alpha}{m\\alpha} f ( \\vec E \u0026#43; \\vec v\\alpha \\cross \\vec B) \\cdot \\pdv{\\vec v}{\\vec v} \\dd \\vec v \\\\ = \\frac{q\\alpha}{m\\alpha} \\oint f\\alpha ( \\vec E \u0026#43; \\vec v\\alpha \\cross \\vec B) \\vec v \\cdot \\dd \\vec S_v - \\frac{q\\alpha}{m\\alpha} \\left[ \\vec E \\int f \\dd \\vec v\\alpha - \\vec B \\cross \\int \\vec v f\\alpha \\dd \\vec v \\right] \\\\ = - \\frac{q\\alpha}{m\\alpha} n\\alpha (\\vec E \u0026#43; v\\alpha \\cross \\vec B)\\]  Finally, the 4th term gives\n \\[ \\int \\left. \\vec v \\pdv{f_\\alpha}{t} \\right|_{coll} = \\sum_\\beta \\int \\vec v C_{\\alpha \\beta} \\dd \\vec v \\\\ = \\int \\vec v C_{\\alpha \\alpha} \\dd \\vec v \u0026#43; \\sum_{\\beta \\neq \\alpha} \\int \\vec v C_{\\alpha \\beta} \\dd \\vec v\\]  Collisions of like particles  \\( \\alpha \\)  do not result in a net change of momentum of species  \\( \\alpha \\) , so all we have left is the change in momentum due to collisions of unlike particles\n \\[= \\sum_{\\beta \\neq \\alpha} \\left[ \\vec v_{\\alpha} \\int C_{\\alpha \\beta} \\dd \\vec v \u0026#43; \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v \\right]\\]  For the same reason as before, we assert that collisions between particles  \\( \\alpha \\)  and  \\( \\beta \\)  do not lead to the creation or destruction of any species, so the 0th moment  \\( \\int C_{\\alpha \\beta} \\dd \\vec v = 0\\) . This leads to the conclusion that only random motion contributes to momentum transfer, not  \\( \\vec v_\\alpha \\) . Viscosity and friction are good examples of similar physical processes where bulk velocity does not transfer momentum, but random motion does.\nThe momentum transfer from  \\( \\alpha \\)  to  \\( \\beta \\)  must equal transfer from  \\( \\beta \\)  to  \\( \\alpha \\) . Momentum is globally conserved.\n \\[\\int \\vec w C_{\\alpha \\beta} \\dd \\vec v = - \\int \\vec w C_{\\beta \\alpha} \\dd \\vec v\\]  Now we can finally write out the full momentum equation\n \\[\\pdv{}{t} (n_\\alpha \\vec v_\\alpha ) \u0026#43; \\div (n_\\alpha \\vec v_\\alpha \\vec v_\\alpha) \u0026#43; \\frac{1}{m_\\alpha} \\div \\vec P_\\alpha - \\frac{q_\\alpha}{m_\\alpha} n_\\alpha ( \\vec E \u0026#43; \\vec v _\\alpha \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v\\]  The collision term is often represented as the momentum transfer vector  \\( \\vec R_{\\alpha \\beta} \\)  to  \\( \\alpha \\)  from  \\( \\beta \\) .\n \\[\\vec R_{\\alpha \\beta} \\equiv m_\\alpha \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v\\]   \\[\\pdv{}{t} (m_\\alpha n_\\alpha v_\\alpha) \u0026#43; \\div (m_\\alpha n_\\alpha \\vec v_\\alpha \\vec v_\\alpha) \u0026#43; \\grad \\vec P_{\\alpha} \u0026#43; \\div \\vec \\Pi_{\\alpha} - q_\\alpha n_\\alpha (\\vec E \u0026#43; \\vec v _\\alpha \\cross \\vec B) = \\sum_{\\beta} R_{\\alpha \\beta}\\] \nOnce again, we\u0026rsquo;ve written the above in a conservation law form. The terms that aren\u0026rsquo;t strictly conservation terms are the source term  \\( q_\\alpha n_\\alpha (\\vec E \u0026#43; \\vec v _\\alpha \\cross \\vec B) \\)  and sink term  \\( \\sum_{\\beta} R_{\\alpha \\beta} \\) .\nIntroducing the mass density\n \\[\\rho_\\alpha = m_\\alpha n_\\alpha \\qquad \\text{mass density}\\]   \\[\\pdv{}{t} (\\rho_\\alpha \\vec v _\\alpha) \u0026#43; \\div (\\rho_\\alpha \\vec v_\\alpha v_\\alpha \u0026#43; \\ldots \\\\ = \\rho_\\alpha \\pdv{\\vec v_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\left( \\pdv{\\rho_\\alpha}{t} \u0026#43; \\div \\rho_\\alpha \\vec v_\\alpha \\right) \u0026#43; \\rho_\\alpha \\vec v_\\alpha \\cdot \\grad \\vec v_\\alpha \u0026#43; \\ldots\\]  The term in parentheses is just the continuity equation, which is zero, and what\u0026rsquo;s left is a more usual form of the momentum equation\n \\[\\text{Momentum Equation:}\\\\ \\rho_\\alpha \\left(\\pdv{\\vec v_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad \\vec v_\\alpha \\right) \u0026#43; \\grad \\vec P_\\alpha \u0026#43; \\div \\vec \\Pi_\\alpha - q_\\alpha n_\\alpha (\\vec E \u0026#43; \\vec v_\\alpha \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\vec R_{\\alpha \\beta}\\]  So by taking the 1st moment of the BE, we arrive at the momentum equation for  \\( \\vec v_\\alpha \\)  and we have introduced  \\( \\vec P_\\alpha \\) .\nMomentum transfer (collisions) #  Looking back at  \\( \\vec R_{\\alpha \\beta} \\) , it\u0026rsquo;s worth noting that the actual situation is complicated by the potential for multi-body collisions. While it is true that the probability of, e.g. three-body collisions is small, but there are many three-body collisional processes which are extremely important in plasma physics. Three-body recombination is the primary loss term in the ionization balance, for example. For now the simplest model is to only include binary collisions that result in small angel deflections. If we suppose we only have two species with distributions that look like the following:\nthen we expect collisions between species to tend to drag the distributions towards each other. We should expect the collision-based momentum transfer to be proportional to the distance between the distributions, i.e. the total current density  \\( \\vec j \\) .\nFor an ion-electron plasma, the collision momentum transfer vector (to ions from electrons) is\n \\[\\vec R_{ie} = - n_e e \\eta \\vec j\\]  where\n \\[\\eta = \\frac{m_e \\nu_{ei}}{n e^2} \\qquad \\text{resistivity}\\]   \\[\\vec j = n e (\\vec v_i - \\vec v_e) \\qquad \\text {current density for Z = 1}\\]  2nd Moment of Boltzmann Equation #  Finally we\u0026rsquo;re on to our last expansion term. In general, we would say that the second moment would be\n \\[\\int \\vec v \\vec v \\pdv{f_\\alpha}{t} \\dd \\vec v = \\pdv{}{t} \\int \\vec v \\vec v f_\\alpha \\dd \\vec v = \\pdv{}{t} \\vec E_\\alpha / m_\\alpha \\rightarrow \\pdv{}{t} \\vec P_\\alpha\\]  Usually we think of the second moment as giving us energy, a scalar, rather than a tensor equation that we have here. Since the pressure tensor is symmetric, the energy equation will contain 6 independent quantities. Together with the continuity and velocity equations, we have 10 quantities in total, so we call this expansion a 10-moment expansion.\nInstead of the full tensor equation, we\u0026rsquo;re actually going to compute the reduced 2nd moment by contracting  \\( \\vec v \\cdot \\vec v \\) \n \\[\\int \\vec v \\cdot \\vec v \\pdv{f_\\alpha}{t} \\dd v\\]  So, let\u0026rsquo;s get after it:\n \\[\\int v^2 \\pdv{f_\\alpha}{t} \\dd \\vec v \u0026#43; \\int v^2 \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \\dd \\vec v \u0026#43; \\int v^2 \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v = \\left. \\int v^2 \\pdv{f_\\alpha}{t} \\right|_{coll} \\dd \\vec v\\]  Simplifying\u0026hellip; Term 1:\n \\[\\int v^2 \\pdv{f_\\alpha}{t} \\dd \\vec v = \\int \\pdv{}{t} (v^2 f_\\alpha) \\dd \\vec v - \\int \\pdv{v^2}{t} f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{t} \\int (\\vec v_\\alpha \u0026#43; \\vec w) \\cdot (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{t} \\left[ \\int v_\\alpha ^2 f_\\alpha \\dd \\vec v \u0026#43; 2 \\int \\vec v_\\alpha \\cdot \\vec w f_\\alpha \\dd \\vec v \u0026#43; \\int w^2 f_\\alpha \\dd \\vec v \\right] \\\\ = \\pdv{}{t} \\left[ v_\\alpha ^2 \\int f_\\alpha \\dd \\vec v \u0026#43; 2 \\vec v_\\alpha \\cdot \\int \\vec w f_\\alpha \\dd \\vec v \u0026#43; 3 \\frac{n_\\alpha T_\\alpha}{m_\\alpha} \\right] \\\\ = \\pdv{}{t} \\left[ n_\\alpha v_\\alpha ^2 \u0026#43; 3 \\frac{n_\\alpha T_\\alpha}{m_\\alpha} \\right]\\]  Term 2:\n \\[\\int v^2 \\vec v \\cdot \\pdv{f_\\alpha}{\\vec x} \\dd \\vec v = \\int \\pdv{}{\\vec x} \\cdot (v^2 \\vec v f_\\alpha) \\dd \\vec v - \\int f_\\alpha \\pdv{}{\\vec x} \\cdot (v^2 \\vec v) \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int ( \\vec v_\\alpha \u0026#43; \\vec w) \\cdot (\\vec v_\\alpha \u0026#43; \\vec w) \\vec w f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\int (v_\\alpha ^2 \u0026#43; 2 \\vec v_\\alpha \\cdot \\vec w \u0026#43; w^2) \\vec v f_\\alpha \\dd \\vec v \\\\ = \\pdv{}{\\vec x} \\cdot \\left[ \\int v_\\alpha ^2 (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \u0026#43; \\int 2 \\vec v_\\alpha \\cdot \\vec w (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \u0026#43; \\int w^2 (\\vec v_\\alpha \u0026#43; \\vec w) f_\\alpha \\dd \\vec v \\right] \\\\ = \\pdv{}{\\vec x} \\cdot \\left[ v_\\alpha ^2 \\vec v_\\alpha \\int f_\\alpha \\dd \\vec v \u0026#43; v_\\alpha ^2 \\int \\vec w f_\\alpha \\dd \\vec v \u0026#43; 2 \\vec v_\\alpha \\cdot \\left(\\int \\vec w f_\\alpha \\dd \\vec v \\right) \\vec v_\\alpha \\\\ \\qquad \u0026#43; 2 \\vec v_\\alpha \\cdot \\int \\vec w \\vec w f_\\alpha \\dd \\vec v \u0026#43; \\vec v_\\alpha \\int w^2 f_\\alpha \\dd \\vec v \u0026#43; \\int \\vec w w^2 f_\\alpha \\dd \\vec v \\right] \\\\ = \\pdv{}{\\vec x} \\cdot (n_\\alpha v_\\alpha ^2 \\vec v_\\alpha) \u0026#43; \\pdv{}{\\vec x} \\cdot 2 \\vec v_\\alpha \\cdot \\frac{\\vec P_\\alpha}{m_\\alpha} \u0026#43; \\pdv{}{\\vec x} \\cdot \\left(\\frac{3 n_\\alpha T_\\alpha \\vec v_\\alpha}{m_\\alpha} \\right) \u0026#43; \\pdv{}{\\vec x} \\cdot \\left( \\frac{2 \\vec h_\\alpha}{m_\\alpha} \\right)\\]  In keeping with the progression so far, the very last term would be something like a full 3rd moment  \\( \\int \\vec w \\vec w \\vec w f_\\alpha \\dd \\vec v \\) . Instead, we have a \u0026ldquo;contracted 3rd moment\u0026rdquo; of the distribution  \\( \\vec h_\\alpha \\)  which is actually the heat flux, or the random energy flux of random energy.\nTerm 3:\n \\[\\int v^2 \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} \\dd \\vec v \\\\ = \\oint f_\\alpha \\frac{q_\\alpha}{m_\\alpha} v^2 ( \\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\dd \\vec S_v - \\int f_\\alpha \\frac{q_\\alpha}{m_\\alpha} \\pdv{}{\\vec v} \\cdot \\left[ v^2 ( \\vec E \u0026#43; \\vec v \\cross \\vec B) \\right] \\dd \\vec v \\\\ = - \\frac{q_\\alpha}{m_\\alpha} \\int f_\\alpha \\left[ v^2 \\left(\\pdv{}{\\vec v} \\cdot \\vec E \\right) \u0026#43; \\left(\\vec E \\cdot \\pdv{}{\\vec v} v^2 \\right) \\right] \\dd \\vec v \\\\ \\qquad - \\frac{q_\\alpha}{m_\\alpha}\\int f_\\alpha \\left[v^2 \\pdv{}{\\vec v} \\cdot (\\vec v \\cross \\vec B) \u0026#43; \\vec v \\cross \\vec B \\cdot \\pdv{}{\\vec v} v^2 \\right] \\dd \\vec v \\\\ = - \\frac{q_\\alpha}{m_\\alpha} \\int f_\\alpha (2 \\vec E \\cdot \\vec v) \\dd \\vec v \\\\ = - \\frac{2 q_\\alpha}{m_\\alpha} n_\\alpha \\vec v_\\alpha \\cdot \\vec E\\]  We\u0026rsquo;ve ended up with the work done on a fluid by the electric field. As expected, there is no work term associated with the bulk magnetic field.\nTerm 4:\n \\[\\left. \\int v^2 \\pdv{f_\\alpha}{t} \\right|_{coll} \\dd \\vec v = \\int v^2 C_{\\alpha \\alpha} \\dd \\vec v \u0026#43; \\sum _{\\beta \\neq \\alpha} \\int v^2 C_{\\alpha \\beta} \\dd \\vec v \\]  We can make an energy conservation argument to get rid of the first term. The first term goes to 0 since collisions of like particles results in no net energy exchange.\n \\[= \\sum_{\\beta \\neq \\alpha} \\int ( \\vec v_\\alpha \u0026#43; \\vec w) \\cdot(\\vec v_\\alpha \u0026#43; \\vec w) C_{\\alpha \\beta} \\dd \\vec v \\\\ = \\sum_{\\beta \\neq \\alpha}\\left( v_\\alpha ^2 \\int C_{\\alpha \\beta} \\dd \\vec v \u0026#43; 2 \\vec v_\\alpha \\cdot \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v \u0026#43; \\int w^2 C_{\\alpha \\beta} \\dd \\vec v \\right)\\]  Since collisions (we assume) don\u0026rsquo;t create or destroy particles, the first term  \\( v_\\alpha ^2 \\int C_{\\alpha \\beta} \\dd \\vec v \\)  goes away. The second term  \\( \\vec v_\\alpha \\cdot \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v \\)  we interpret as a frictional force due to the relative motion between  \\( \\alpha \\)  and  \\( \\beta \\) \n \\[= \\sum_{\\beta \\neq \\alpha} 2 \\frac{\\vec v_\\alpha \\cdot \\vec R_{\\alpha \\beta}}{m_\\alpha} \u0026#43; 2 \\frac{Q_{\\alpha \\beta}}{m_\\alpha}\\]  where  \\( Q_{\\alpha \\beta} \\)  is the heat exchange term (heat generation term in Braginskii). It describes the heat exchange by random collisions of unlike particles, analogous to viscous heating.\n \\[Q_{\\alpha \\beta} = \\frac{1}{2} \\int m_\\alpha w^2 C_{\\alpha \\beta} \\dd \\vec v\\]  If we were to draw distribution functions for species  \\( \\alpha \\)  and  \\( \\beta \\)  (as shown), even though they have the same centroid they will interact by viscous heating. Species  \\( \\beta \\)  will heat up and  \\( \\alpha \\)  will cool\nIf we multiply by  \\( \\frac{1}{2} m_\\alpha \\)  then the energy equation becomes\n \\[\\text{Energy Equation: } \\\\ \\qquad \\pdv{}{t} \\left( \\frac{3}{2} n_\\alpha T_\\alpha \u0026#43; \\frac{1}{2} \\rho_\\alpha v_\\alpha ^2 \\right) \u0026#43; \\\\ \\div \\left(\\frac{3}{2} n_\\alpha T_\\alpha \u0026#43; \\frac{1}{2} \\rho_\\alpha v_\\alpha ^2 \\right) \\vec v_\\alpha \u0026#43; \\div (\\vec v_\\alpha \\cdot \\vec P_\\alpha) \u0026#43; \\div \\vec h_\\alpha - q_\\alpha n_\\alpha \\vec v_\\alpha \\cdot \\vec E \\\\ = \\sum_{\\beta \\neq \\alpha} (v_\\alpha \\cdot \\vec R_{\\alpha \\beta} \u0026#43; Q_{\\alpha \\beta})\\]  Notice that the total energy appears\n \\[E_\\alpha = \\frac{3}{2} n_\\alpha T_\\alpha \u0026#43; \\frac{1}{2} \\rho_\\alpha v_\\alpha ^2\\]  The factor of  \\( \\frac{3}{2} \\)  as usual comes from  \\( \\gamma \\) \n \\[\\frac{3}{2} = \\frac{1}{\\gamma - 1} = \\frac{1}{5/3 - 1}\\]  where\n \\[\\gamma = \\frac{DOF \u0026#43; 2}{DOF} = \\frac{5}{3}\\]  We can remove the kinetic portion of the energy equation by subtracting the product of the momentum equation with  \\( \\vec v_\\alpha \\) .\n \\[\\rho_\\alpha \\vec v_\\alpha \\cdot \\pdv{v_\\alpha}{t} = \\frac{1}{2} \\pdv{}{t} (\\rho_\\alpha v_\\alpha ^2)\\]   \\[\\pdv{}{t} \\left( \\frac{3}{2} n_\\alpha T_\\alpha \\right) \u0026#43; \\div \\left(\\frac{3}{2} n_\\alpha T_\\alpha \\vec v_\\alpha \\right) \u0026#43; \\div \\left( \\vec v_\\alpha \\cdot \\vec P_\\alpha \\right) - \\vec v_\\alpha \\cdot (\\div \\vec P_\\alpha) \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\]   \\[\\frac{3}{2} n_\\alpha \\left( \\pdv{T_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\div T_\\alpha \\right) \u0026#43; \\frac{3}{2} T_\\alpha \\left[ \\pdv{n_\\alpha}{t} \u0026#43; \\div (n_\\alpha \\vec v_\\alpha) \\right] \u0026#43; \\vec P_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\]   \\[\\frac{3}{2} n_\\alpha \\left( \\pdv{T_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\div T_\\alpha \\right) \u0026#43; \\vec P_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\] \nWhere  \\( \\vec P_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \\)  is a tensor contraction in two indices. It is a generalization of the dot product, i.e.  \\[\\vec P_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha = \\delta_{ik} \\delta_{jl} P_{ij} \\partial_k v_l \\\\ = P_{xx} \\partial_x v_x \u0026#43; P_{xy} \\partial_y v_x \u0026#43; \\ldots\\] \n \\[\\frac{3}{2} n_\\alpha \\left( \\pdv{T_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad T_\\alpha \\right) \u0026#43; P_\\alpha \\div \\vec v_\\alpha \u0026#43; \\vec \\Pi_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta}\\]  Now the continuity, momentum, and energy equations describe the evolution of each fluid  \\( \\alpha \\) . The only velocity information we have retained in taking moments is the centroid  \\( \\vec v_\\alpha \\)  and the width  \\( T_\\alpha \\) . This is the only information we will include in the 5-moment model (5N-moment plasma fluid model). The higher moments (describing skewness, kurtosis) are not evolved with the 5-moment model. There are fluid models that do evolve those moments, but we won\u0026rsquo;t touch on those here.\nClosure Problem #  We still need to \u0026ldquo;close\u0026rdquo; the fluid model (i.e. solving the Closure Problem). As we have been finding out, each additional moment of the Boltzmann Equation introduces the next-higher moment. In calculating the 0th moment, we introduce the 1st moment, etc. We need to address the closure problem by relating higher moment variables to lower moment variables. Applying a closure scheme (relating variables that are not evolved directly) is usually equivalent to making a statement on heat flow.\nIn our model, we are not evolving the heat flux  \\( \\vec h_\\alpha \\) , so we need to relate it to one of the other variables that we do evolve directly. We usually do that by writing down a conductivity relation\n \\[\\vec h_\\alpha = - \\kappa \\grad T_\\alpha\\]  for some conductivity  \\( \\kappa \\) .\nWe\u0026rsquo;re also only evolving the scalar pressure, so we need to relate the stress tensor  \\( \\vec \\Pi_\\alpha \\)  to the other variables. We can do that with a viscosity relation\n \\[\\vec \\Pi_\\alpha = \\nu \\grad \\vec v_\\alpha\\]  for some viscosity  \\( \\nu \\) . We call the introduced quantities  \\( \\kappa \\)  and  \\( \\nu \\)  the transport coefficients. They are usually derived by performing an expansion of the equations we\u0026rsquo;ve already discussed. They are what lead to \u0026ldquo;nondimensional\u0026rdquo; numbers that characterize the flow in our system.\nWe can also achieve closure by constraining portions of the system. In an isothermal system\n \\[\\text{isothermal: } \\quad T = \\text{const.}, p \\propto n\\]   \\[\\text{adiabatic: } \\quad p \\propto n^\\gamma\\]   \\[\\text{force-free, cold: } \\quad p = 0\\] \nThe expressions that lead us to a closure scheme are the equations of state. They are not time-dependent, they simply relate fluid parameters to other properties of the system.\nMaxwell\u0026rsquo;s equations to couple the electromagnetic terms to the fluid variables.  \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   \\[\\epsilon_0 \\pdv{\\vec E}{t} = \\frac{1}{\\mu_0} \\curl \\vec B - \\sum_{alpha} q_\\alpha n_\\alpha \\vec v_\\alpha \\]   \\[\\epsilon_0 \\div \\vec E = \\sum_\\alpha q_\\alpha n_\\alpha \\]   \\[\\div \\vec B = 0\\] \nThe fluid variables provide the source terms for Ampere\u0026rsquo;s law and Gauss' law, and we\u0026rsquo;ve already seen how the electromagnetic fields appear in the momentum equations as source terms for the fluid forces.\nStopping after the 2nd moment (5N-Moment Plasma Fluid Model) #  What assumptions have we made by stopping here? It\u0026rsquo;s important to know what they are and to recognize what they mean for the fluid model.\n Each species  \\( \\alpha \\)  is well-represented by a Maxwellian with a small perturbation. The pressure tensor is not strictly diagonal and the heat flux is not zero, so the small perturbations are what lead to the transport coefficients. This is due to the process by which we obtain the transport coefficients, called the Chapman-Enskog expansion. Kinetic effects (stream instabilities, counter-flow instabilities, etc.) are not captured.  The variables of the 5-moment model that we do evolve are  \\[n, v_x, v_y, v_z, T\\] \nWe can take additional moments. The 10N-moment equation evolves, in addition to the quantities in the 5N-moment model, we evolve all of the independent terms of the pressure tensor  \\[n, v_x, v_y, v_z, P_{xx}, P_{xy}, P_{xz} , P_{yy} , P_{yz} , P_{zz} \\] \nThe 13-N moment model has everything from the 10N-moment model plus the heat flux tensor  \\( h_x, h_y, h_z \\) \n"});index.add({'id':3,'href':'/r/notes/UWAA558/04-two-fluid-plasma-model/','title':"Two-Fluid Plasma Model",'section':"MHD Theory",'content':"Two-Fluid Plasma Model (ions-electrons) #  Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let\u0026rsquo;s start with a cold plasma approximation (   document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\( p = 0 \\) ) and neglect collisions. The momentum equation reduces to\n \\[m_\\alpha \\left( \\pdv{\\vec v_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad \\vec v_\\alpha \\right) - q_\\alpha (\\vec E \u0026#43; \\vec v_\\alpha \\cross \\vec B) = 0\\]  From here on out we can avoid some clutter (and wrist strain) by dropping the  \\( \\alpha \\)  subscripts and acknowledging that we have sets of equations for ions and electrons. Apply a perturbation to an equilibrium  \\( g = g_0 \u0026#43; g_1 \\) \n \\[m \\left(\\pdv{\\vec v_0}{t} \u0026#43; \\vec v_0 \\cdot \\grad \\vec v_0 \u0026#43; \\pdv{\\vec v_0}{t} \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_0 \u0026#43; \\vec v_0 \\cdot \\grad \\vec v_1 \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_1 \\right) \\\\ \\qquad - q (E_0 \u0026#43; E_1) - q (\\vec v_0 \\cross \\vec B_0 \u0026#43; \\vec v_1 \\cross \\vec B_0 \u0026#43; \\vec v_0 \\cross \\vec B_1 \u0026#43; \\vec v_1 \\cross \\vec B_1) = 0\\]  We can drop some terms because equilibrium has to satisfy the original equation. We can balance all of the subscript-0 terms and sum them to get zero.\n \\[m \\left( \\pdv{\\vec v_1}{t} \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_0 \u0026#43; \\vec v_0 \\cdot \\grad \\vec v_1 \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_1 \\right) \u0026#43; q (- E_1 - \\vec v_1 \\cross \\vec B_0 - \\vec v_0 \\cross \\vec B_1 - \\vec v_1 \\cross \\vec B_1)\\]  Let\u0026rsquo;s now make the assumption that the perturbation is small, that is  \\( g_1 \\ll g_0 \\) . That means that nonlinear products of perturbation terms are negligible (linearization process).\n \\[m \\left( \\pdv{\\vec v_1}{t} \u0026#43; \\vec v_0 \\cdot \\grad \\vec v_1 \u0026#43; \\vec v_1 \\cdot \\grad \\vec v_0 \\right) - q \\vec E_1 - q (\\vec v_1 \\cross \\vec B_0 \u0026#43; \\vec v_0 \\cross \\vec B_1) = 0\\]  Now, assume that the equilibrium is a static equilibrium, that is  \\( \\vec v_0 = 0 \\) . If we decompose into components that are parallel and perpendicular to the equilibrium magnetic field  \\( \\vec B_0 \\) , then\n \\[\\pdv{v_{1, \\parallel}}{t} - \\frac{q}{m} E_{1, \\parallel} = 0\\]   \\[\\pdv{\\vec v_{1, \\perp}}{t} - \\frac{q}{m} \\left(\\vec E_{1, \\perp} \u0026#43; B_0 \\vec v_{1, \\perp} \\cross \\vu z \\right) = 0\\] \nThe parallel component  \\( E_{1, \\parallel} \\)  will lead us to the ordinary wave (O-wave). Consideration of the more general case with perpendicular components will lead to the X-wave.\nThe plasma velocity is related to the fields through the current density (Maxwell equations). Faraday\u0026rsquo;s law gives\n \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   \\[\\rightarrow \\curl \\pdv{\\vec B}{t} = - \\curl \\curl \\vec E = - \\grad (\\div \\vec E) \u0026#43; \\nabla ^2 \\vec E \\] \nAmpere\u0026rsquo;s law gives\n \\[\\epsilon_0 \\pdv{\\vec E}{t} = \\frac{1}{\\mu_0} \\curl \\vec B - \\sum_{\\alpha} q_\\alpha n_\\alpha \\vec v_\\alpha\\]   \\[\\epsilon_0 \\pdv{^2 \\vec E }{t ^2} = \\frac{1}{\\mu_0} \\curl \\pdv{\\vec B}{t} \\\\ = \\frac{1}{\\mu_0} \\left[ \\nabla ^2 \\vec E - \\grad (\\div \\vec E) \\right] - \\sum_\\alpha q_\\alpha \\pdv{}{t} (n_\\alpha \\vec v_\\alpha) \\] \nSince this is a linear system, assume that the perturbed quantities have a wave-like structure. That is, the perturbed quantities  \\( g_1 \\)  are proportional to  \\( e^{i(\\omega t \u0026#43; \\vec k \\cdot \\vec r)} \\) . This lets us transform the spatial and temporal derivatives into factors of  \\( \\omega \\)  and  \\( \\vec k \\) \n \\[- \\epsilon_0 \\omega ^2 \\vec E_1 = - \\frac{1}{\\mu_0} \\left[k^2 \\vec E_1 - \\vec k (\\vec k \\cdot \\vec E_1) \\right] \u0026#43; i \\omega e n_0 \\vec v_1\\]  Let\u0026rsquo;s now consider only high frequency oscillations, assuming that only the electrons respond and the ions remain stationary. There\u0026rsquo;s nothing particularly complicated about including the ion response, this just lets us drop the  \\( \\alpha \\)  subscripts and focus on a single set of equations.\n \\[- i \\omega \\frac{e n_0}{\\epsilon_0} \\vec v_1 = (\\omega ^2 - c^2 k^2) \\vec E_1 \u0026#43; c^2 \\vec k (\\vec k \\cdot \\vec E_1)\\]  Now let\u0026rsquo;s apply the perturbed form to the linearized momentum equation\n \\[\\pdv{v_{1, \\parallel}}{t} - \\frac{q}{m} E_{1, \\parallel} = 0 \\\\ \\rightarrow i \\omega v_{1, \\parallel} = - \\frac{e}{m} E_{1, \\parallel} \\\\ \\rightarrow i \\omega \\frac{e n_0}{\\epsilon_0} v_{1, \\parallel} = \\frac{e^2 n_0}{\\epsilon_0 m} E_{1, \\parallel}\\]  Combine the momentum equation and the Maxwell equations to eliminate  \\( \\vec E_1 \\)  and  \\( \\vec v_1 \\) \n \\[\\frac{e^2 n_0}{\\epsilon_0 m} E_{1, \\parallel} = ( \\omega ^2 - c^2 k^2 ) E_{1, \\parallel} \u0026#43; c^2 k_{\\parallel} \\vec (k \\cdot \\vec E_1)\\]  Consider different possibilities for the  \\( \\vec k \\)  vector. If it is along the magnetic field  \\( \\vec k = k_{\\parallel} \\vu{e}_\\parallel \\)  (longitudinal wave) then\n \\[\\frac{e^2 n_0}{\\epsilon_0 m} = \\omega ^2 - c^2 k^2 \u0026#43; c^2 k^2 = \\omega_{pe}^2\\]  For  \\( \\vec k = k_{\\perp} \\vu e_\\perp \\)  (transverse wave) then we get the dispersion relation for the O-wave\n \\[\\text{dispersion relation for O-waves:} \\qquad \\omega^2 - c^2 k^2 = \\omega_p ^2\\]  The electric field is in the same direction as the magnetic field  \\( (\\vec E_1 = \\vec E_{1, \\parallel}) \\) , which means the O-wave is linearly polarized. At large  \\( k \\)  we just have regular light waves, but as we turn the frequency downwards we see a cut-off at the plasma frequency:\nIt turns out that the dispersion relation for the X-wave has the same cut-off, but also has another branch with a resonance\nThe two-fluid plasma model is highly reduced from the full kinetic model, but it is still too complete to be useful when studying gross plasma behavior. Further reductions of the model are possible by making asymptotic assumptions:\nLow-frequency Asymptotic Assumption #   Eliminate high frequency, short wavelength phenomena by using pre-Maxwell field equations. Formally, this is  \\( \\epsilon_0 \\rightarrow 0 \\) .  The direct consequences of the low-frequency approximation are\n \\[c^2 = \\frac{1}{\\epsilon_0 \\mu_0} \\qquad c \\rightarrow \\infty\\]   \\[\\omega_p ^2 = \\frac{n e^2}{\\epsilon_0 m} \\qquad \\omega_p \\rightarrow \\infty\\]   \\[\\lambda_D = \\frac{v_T}{\\omega_p} \\rightarrow 0\\] \nThis means that all phenomena will have  \\( \\omega \\ll \\omega_p \\) , limiting the frequencies we can resolve to the ion plasma frequency. The characteristic speeds will be limited by the speed of light\n \\[\\frac{\\omega}{k} \\ll c\\]  and all characteristic lengths will be much greater than the Debye length\n \\[x_0 \\gg \\lambda_D\\]  Looking at Gauss' law,  \\[\\epsilon_0 \\div \\vec E = \\sum_\\alpha q_\\alpha n_\\alpha \\rightarrow \\sum_\\alpha q_\\alpha n_\\alpha = 0\\] \nso we now have charge neutrality everywhere in the domain. For H plasma, locally we have  \\( n_e = n_i \\)  everywhere.\nLooking at Ampere\u0026rsquo;s law,\n \\[\\epsilon_0 \\pdv{\\vec E}{t} = \\frac{1}{\\mu_0} \\curl \\vec B - \\sum_\\alpha q_\\alpha n_\\alpha \\vec v_\\alpha = 0 \\\\ \\rightarrow \\vec j = \\frac{1}{\\mu_0} \\curl \\vec B\\]  Things we do not get from this approximation are  \\( \\vec E = 0 \\)  or  \\( \\pdv{\\vec E}{t} = 0 \\) . It does mean that plasma dynamics occur on a sufficiently large spatial scale that charge separation is small, and they occur on a sufficiently long temporal scale that electrons respond quickly.\nTiny electron asymptotic assumption #  2nd approximation: neglect electron inertia in the momentum equation. Formally, we let the electron mass  \\( m_e \\rightarrow 0 \\) \n \\[\\omega_{pe}^2 = \\frac{n e^2}{\\epsilon_0 m_e} \\rightarrow \\infty\\]   \\[\\omega_{c, e} = \\frac{e B}{m_e} \\rightarrow \\infty\\]   \\[v_{T, e} \\rightarrow \\infty\\]  The Larmor radius goes to zero  \\[r_{l, e} = \\frac{v_{T, e}}{\\omega_{c, e}} \\rightarrow 0\\]  Importantly, as the gyroradius  \\( r_{l, e} \\)  goes to 0 (because the thermal velocity goes as  \\( \\sqrt{m_e} \\)  and the cyclotron frequency goes as  \\( m_e \\) ), this means that the electrons are tied to the magnetic field. The skin depth is also small.  \\[\\delta_e = \\frac{c}{\\omega_{p, e}} \\rightarrow 0\\] \nSo all phenomena that we capture must have  \\( \\omega \\ll \\omega_{p, e} \\) ,  \\( \\omega \\ll \\omega_{c, e} \\) , and  \\( x_0 \\gg r_{L, e} \\) ,  \\( x_0 \\gg \\delta_e \\)  .\nThe electron momentum equation becomes\n \\[\\grad P_e \u0026#43; \\div \\vec \\Pi _e \u0026#43; e n_e (\\vec E \u0026#43; \\vec v \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\vec R_{\\alpha \\beta}\\]  The momentum equation is now a state equation, not an evolution equation. It now simply relates the dynamical variables to each other at any point in time.\nNow, note that along magnetic field lines electrons can travel long distances at very fast (finite) speeds which can produce low frequency, long wavelength phenomena. Neglecting electron inertia implies that electrons respond instantaneously, meaning we cannot capture these modes. An example of such a phenomena is drift waves.\nThe characteristic speeds  \\( c \\)  and  \\( v_{T, e} \\)  have disappeared from the model. Remaining is  \\( v_{T, i} \\) . This means that the ion dynamics dictate the plasma evolution.\n"});index.add({'id':4,'href':'/r/notes/UWAA558/05-mhd-model/','title':"Magnetohydrodynamic (MHD) Model",'section':"MHD Theory",'content':"Magnetohydrodynamic (MHD) Model #  Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD).\nFirst, define the MHD variables:\n  document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\[\\text{mass density:} \\qquad \\rho = n_i m_i \u0026#43; n_i m_e\\]   \\[\\text{fluid velocity:} \\qquad \\vec v = \\frac{n_i m_i \\vec v_i \u0026#43; n_e m_e \\vec v_e}{n_i m_i \u0026#43; n_e m_e}\\]   \\[\\text{current density:} \\qquad \\vec j = q_i n_i \\vec v_i - e n_e \\vec v_e \\rightarrow \\vec v_e = \\frac{q_i n_i \\vec v_i}{e n_e} - \\vec j / e n_e\\]   \\[\\text{total pressure:} \\qquad p = p_i \u0026#43; p_e = n_e T_e \u0026#43; n_i T_i\\]   \\[\\text{total temperature:} \\qquad T = \\frac{n_i T_i \u0026#43; n_e T_e}{(n_i \u0026#43; n_e)/2}\\]  Now let\u0026rsquo;s begin applying asymptotic approximations. For the mass density, applying the first approx (charge neutrality) we have\n \\[\\rho \\approx n (m_i \u0026#43; m_e)\\]  Using approx 2 (vanishing electron mass)\n \\[\\rho \\approx n m_i\\]  where  \\[n = n_i \\qquad n_e = Z n\\] \nThe center-of-mass velocity (charge neutrality) gives\n \\[\\vec v \\approx \\frac{m_i \\vec v_i \u0026#43; m_e Z \\vec v_e}{m_i \u0026#43; Z m_e}\\]  with small electron mass approximation:\n \\[\\vec v \\approx \\vec v_i\\]  The current density is (charge neutrality approx)\n \\[\\vec j \\approx Z e n (\\vec v_i - \\vec v_e) \\qquad \\vec v_e = \\vec v_i - \\vec j / Z e n\\]  The pressure and total temperature are (with charge neutrality)\n \\[P \\approx n (T_i \u0026#43; Z T_e)\\]   \\[T \\approx \\frac{T_i \u0026#43; Z T_e}{(1 \u0026#43; Z)/2}\\]  MHD Momentum Equation #  Now we combine the two-fluid equations with these asymptotic approximations to obtain the governing equations for the MHD variables. Multiplying the ion continuity equation by the ion mass gives\n \\[\\pdv{}{t} (m_i n_i) \u0026#43; \\div (m_i n_i \\vec v_i) = 0 \\\\ \\rightarrow \\pdv{\\rho}{t} \u0026#43; \\div (\\rho \\vec v) = 0 \\quad \\text{(MHD continuity eq.)}\\]  If we multiply the two-fluid continuity equations by the charge and sum them, we get\n \\[\\pdv{}{t}(q_i n_i - e n_e) \u0026#43; \\div (q_i n_i \\vec v_i - e n_e \\vec v_e) = 0 \\\\ \\rightarrow \\div \\vec j = 0 \\quad \\text{(no accumulation of charge)}\\]  If we add the electron and ion momentum equations and apply the small electron mass approximation, and recognizing that  \\( \\vec R_{ei} = - \\vec R_{ie} = n e \\eta \\vec j \\) \n \\[\\rho_i \\left(\\pdv{\\vec i}{t} \u0026#43; \\vec v_i \\cdot \\grad \\vec v_i \\right) \u0026#43; \\grad (P_i \u0026#43; P_e) \u0026#43; \\div (\\vec \\Pi _i \u0026#43; \\vec \\Pi_e) \\\\ \\qquad - (q_i n_i - e n_e) \\vec E - (q_i n_i \\vec v_i - e n_e \\vec v_e) \\cross \\vec B = 0\\]   \\[\\text{MHD Momentum Equation} \\\\ \\rightarrow \\rho \\left(\\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = - \\div (\\vec \\Pi _i \u0026#43; \\vec \\Pi _e)\\] \n!!! info \u0026ldquo;MHD Momentum Equation\u0026rdquo;\n \\[ \\rho \\left(\\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = - \\div (\\vec \\Pi _i \u0026#43; \\vec \\Pi _e) \\]   If we take the electron momentum equation, apply the small electron mass asymptotic approximation and introduce the current density, then we have\nMHD Ohm\u0026rsquo;s Law #   \\[e n_e (\\vec E \u0026#43; \\vec v_i \\cross \\vec B - \\vec j \\cross \\vec B / Z e n) = - \\grad P_e - \\div \\vec \\Pi_e \u0026#43; \\vec R_{ei}\\]  Substituting  \\( \\vec R_{ei} = e n_e \\eta \\vec j \\)   \\[\\text{MHD Generalized Ohms Law}\\\\ \\vec E \u0026#43; \\vec v \\cross \\vec B = \\eta \\vec j \u0026#43; \\frac{1}{Z e n} \\left(\\vec j \\cross \\vec B - \\grad p_e - \\div \\vec \\Pi_e \\right)\\] \n!!! info \u0026ldquo;MHD Generalized Ohms Law\u0026rdquo;\n \\[ \\vec E \u0026#43; \\vec v \\cross \\vec B = \\eta \\vec j \u0026#43; \\frac{1}{Z e n} \\left(\\vec j \\cross \\vec B - \\grad p_e - \\div \\vec \\Pi_e \\right) \\]   MHD Energy Equation #  The energy equation is found by adding the ion and electron energies, but first we need to manipulate them into a common form. The ion energy equation is\n \\[\\frac{1}{\\gamma - 1} \\left[ \\pdv{}{t} (n_i T_i) \u0026#43; \\div (n_i T_i \\vec v_i) \\right] \u0026#43; n_i T_i \\div \\vec v_i = RHS_i\\]  where  \\( RHS_i \\equiv - \\vec \\Pi_i \\cdot \\cdot \\grad \\vec v_i - \\div \\vec h_i \u0026#43; Q_{ie} \\) \n \\[\\frac{1}{\\gamma - 1} \\left( \\pdv{p_i}{t} \u0026#43; \\vec v_i \\cdot \\grad p_i \\right) \u0026#43; \\frac{\\gamma}{\\gamma - 1} p_i (\\div \\vec v_i) = RHS_i\\]  Multiply by  \\( \\frac{\\gamma - 1}{n_i ^\\gamma} \\)  and define the total derivative with respect to the ion velocity as\n \\[\\dv{}{t} \\equiv \\pdv{}{t} \u0026#43; \\vec v_i \\cdot \\grad\\]   \\[\\rightarrow \\dv{}{t} \\left( \\frac{p_i}{n_i ^\\gamma} \\right) - p_i \\dv{}{t} \\left( \\frac{1}{n_i ^\\gamma} \\right) \u0026#43; \\frac{\\gamma p_i}{n_i ^\\gamma} \\div \\vec v_i = \\frac{\\gamma - 1}{n_i ^\\gamma} RHS_i\\]  Simplify further by recognizing that carrying out the total derivative gives\n \\[\\dv{}{t} \\left( \\frac{p_i}{n_i ^\\gamma} \\right) - \\frac{\\gamma p_i}{n_i ^{\\gamma \u0026#43; 1}} \\dv{}{t} n_i \u0026#43; \\frac{\\gamma p_i}{n_i ^\\gamma} \\div \\vec v_i = \\frac{\\gamma - 1}{n_i ^\\gamma} RHS_i\\]  The continuity equation says  \\[\\dv{}{t} n_i = - n_i \\div \\vec v_i\\]  Multiply by  \\( m_i ^{- \\gamma} \\)  and use  \\( \\rho = n m_i \\)  and the resulting ion energy equation is\n \\[\\dv{}{t} \\left( \\frac{p_i}{\\rho ^\\gamma} \\right) = \\frac{\\gamma - 1}{\\rho ^\\gamma} RHS_i\\]  We want to get the electron energy equation in the same form. The steps are very similar:\n \\[\\dv{}{t} \\left( \\frac{p_e}{n_e ^\\gamma} \\right) \u0026#43; \\vec v_e \\cdot \\grad \\left( \\frac{p_e}{n_e ^\\gamma} \\right) = \\frac{\\gamma - 1}{n_e ^\\gamma} RHS_e\\]  Since we\u0026rsquo;ve defined our center-of-mass reference frame to be that of the ions, we can not use the same total derivative\n \\[\\pdv{}{t} \u0026#43; \\vec v_e \\cdot \\grad \\neq \\dv{}{t}\\]  We now apply the charge neutrality approximation\n \\[n_e = Z n \\qquad \\vec v_e = \\vec v_i - \\frac{1}{Z e n} \\vec j\\]   \\[\\rightarrow \\dv{}{t} \\left( \\frac{p_e}{Z^\\gamma n^\\gamma} \\right) \u0026#43; \\vec v_i \\cdot \\grad \\left( \\frac{p_e}{Z^\\gamma n^\\gamma} \\right) = \\frac{1}{Z e n} \\vec j \\cdot \\grad \\frac{p_e}{Z^\\gamma n^\\gamma} \u0026#43; \\frac{\\gamma - 1}{Z^\\gamma n^\\gamma} RHS_e\\]  Multiply by  \\( (Z / m_i)^\\gamma \\)  and the result is\n \\[\\dv{}{t} \\frac{p_e}{\\rho ^\\gamma} = \\frac{1}{Z e n} \\vec j \\cdot \\grad \\frac{p_e}{\\rho^\\gamma} \u0026#43; \\frac{\\gamma - 1}{\\rho^\\gamma} RHS_e\\]  Finally we can add the ion energy equation to the electron energy equation to get\n \\[\\dv{}{t} \\left( \\frac{p}{\\rho^\\gamma} \\right) = \\\\ \\frac{\\gamma - 1}{\\rho^\\gamma} \\left[ Q_{ie} \u0026#43; Q_{ei} - \\div (\\vec h_i \u0026#43; \\vec h_e) - \\vec \\pi_i \\cdot \\cdot \\grad \\vec v_i - \\vec \\Pi_e \\cdot \\cdot \\grad \\vec v_e \\right] \u0026#43; \\frac{\\vec j}{Z e n} \\cdot \\grad \\frac{p_e}{\\rho^\\gamma}\\]  !!! info \u0026ldquo;MHD Energy Equation\u0026rdquo;\n \\[ \\dv{}{t} \\left( \\frac{p}{\\rho^\\gamma} \\right) = \\frac{\\gamma - 1}{\\rho^\\gamma} \\left[ Q_{ie} \u0026#43; Q_{ei} - \\div (\\vec h_i \u0026#43; \\vec h_e) - \\vec \\pi_i \\cdot \\cdot \\grad \\vec v_i - \\vec \\Pi_e \\cdot \\cdot \\grad \\vec v_e \\right] \u0026#43; \\frac{\\vec j}{Z e n} \\cdot \\grad \\frac{p_e}{\\rho^\\gamma} \\]   Obviously, we\u0026rsquo;ve retained a number of terms that are specific to the behavior of the electrons. It is possible to incorporate the electron behavior by using a single-fluid MHD model with two temperatures  \\( T_i \\neq T_e \\) . One can imagine a hierarchy of models, in which the most simplified is the single-fluid MHD model in which you evolve  \\( \\rho \\) ,  \\( \\vec v \\) , and  \\( T \\) . Moving up a level, you have a MHD model with two temperatures in which you evolve  \\( \\rho \\) ,  \\( \\vec v \\) ,  \\( T_i \\) , and  \\( T_e \\) . Upwards from there you move back into the realm of multi-fluid models.\nNow to relate the fields back to source terms. The low-frequency Maxwell\u0026rsquo;s equations are\n \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   \\[\\vec j = \\frac{1}{\\mu_0} \\curl \\vec B\\]   \\[\\div \\vec B = \\div \\vec E = 0\\] \nLike in the multi-fluid plasma model, we still need to close the system by expressing some of our variables using equations of state ( \\( \\vec h \\) ,  \\( \\vec \\Pi \\) ).\nTo simplify further, we can make some assumptions about heat flow\n \\[\\text{isothermal} \\qquad \\rightarrow p \\propto n \\qquad T = \\text{const.} \\qquad \\gamma = 1\\]   \\[\\text{adiabatic} \\qquad \\rightarrow p \\propto n^\\gamma\\]   \\[\\text{cols plasma / force-free} \\qquad \\rightarrow p = \\text{const.} \\qquad \\gamma = 0\\] \nIdeal MHD Model #  The extended MHD equations are simpler than the two-fluid model, but they can still be quite complicated. We can often still get useful analysis from further reductions. The ideal MHD model is such a reduction that we can get by dropping (with justification) several terms from the extended model. We justify the simplifications by comparing the magnitude of the neglected terms to the terms that are retained.\nRecall the characteristic speed is  \\( v_{T, i} \\) . If we say that the characteristic length plasma length is  \\( L \\) , then we can define characteristic time  \\( \\tau = L / v_{T, i} \\) .\nThe derivation of the two-fluid plasma model assumed a Maxwellian velocity distribution. We need the velocity distribution to thermalize, reach local thermodynamic equilibrium, and become Maxwellian. This means that we need many collisions, in fact so many collisions occurring frequently enough that we can ignore collisional effects. There must then be many collisions during the characteristic time  \\( \\tau \\) .\nFor ions to be thermalized,\n \\[\\frac{\\tau_{ii}}{\\tau} \\ll 1\\]  And similarly for electrons\n \\[\\frac{\\tau_{e}}{\\tau} \\ll 1\\]  The continuity equation remains unchanged from the extended MHD model\n!!! info \u0026ldquo;Ideal MHD Continuity Equation\u0026rdquo;\n \\[ \\pdv{\\rho}{t} \u0026#43; \\div (\\rho \\vec v) = 0 \\]   The momentum equation is\n \\[\\rho \\left( \\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = - \\div (\\vec \\Pi_e \u0026#43; \\vec \\Pi_e)\\]  Drop the anisotropic pressure\n!!! info \u0026ldquo;Ideal MHD Momentum Equation\u0026rdquo;\n \\[ \\rho \\left( \\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = 0 \\]   The generalized Ohm\u0026rsquo;s law is\n \\[\\vec E \u0026#43; \\vec v \\cross \\vec B = \\eta \\vec j \u0026#43; \\frac{1}{Z e n} \\left( \\vec j \\cross B - \\grad p_e - \\div \\vec \\Pi_e \\right)\\]  We\u0026rsquo;re going to drop the entire right hand side\n!!! info \u0026ldquo;Ideal MHD Generalized Ohm\u0026rsquo;s Law\u0026rdquo;\n \\[ \\vec E \u0026#43; \\vec v \\cross \\vec B = 0 \\]   For the energy equation we have\n \\[\\dv{}{t} \\left( \\frac{p}{\\rho^\\gamma} \\right) = \\frac{\\gamma - 1}{\\rho^\\gamma} [Q_{ie} \u0026#43; Q_{ei} - \\div (\\vec h_i \u0026#43; \\vec h_e) - \\vec \\Pi_i \\cdot \\cdot \\grad \\vec v_i - \\vec \\Pi_e \\cdot \\cdot \\grad \\vec v_e] \u0026#43; \\frac{\\vec j}{Z e n} \\cdot \\grad \\frac{p_e}{\\rho^\\gamma}\\]  We neglect the entire right-hand side\n!!! info \u0026ldquo;Ideal MHD Energy Equation\u0026rdquo;\n \\[ \\pdv{\\rho}{t} \u0026#43; \\div (p \\vec v) = (1 - \\gamma) p \\div \\vec v \\]   Collision/Pressure terms #  If we assume that the ions and electrons are in thermal equilibrium \\( T_i = T_e \\) , we can relate the collision times\n \\[\\tau_{ee} : \\tau_{ii} : \\tau_{ei} = 1 : \\left( \\frac{m_i}{m_e} \\right) ^{1/2} : \\frac{m_i}{m_e}\\]  The collision times are specifically collisional relaxation times of the Boltzmann equation\n \\[\\left. \\pdv{f}{t} \\right|_{coll} = \\frac{f - f_{\\text{Maxwellian}}}{\\tau_{\\alpha \\beta}}\\]  For electrons, the thermalization condition is much stricter for the ions\n \\[\\frac{\\tau_{ee}}{\\tau} = \\left( \\frac{m_e}{m_i} \\right)^{1/2} \\frac{\\tau_{ii}}{\\tau} \\ll 1\\]  Neglect the anisotropic pressure tensor in the momentum and generalized Ohm\u0026rsquo;s law,  \\( \\div \\vec \\Pi \\) .  \\( \\vec \\Pi \\)  is primarily the shear stress tensor. The ion thermal speed gives us a characteristic velocity for the plasma, so we use it to characterize the shear stress\n \\[\\vec \\Pi_{i, max} \\sim 2 \\mu \\left( \\pdv{u}{x} - \\frac{1}{3} \\div \\vec v \\right) \\sim \\mu \\frac{v_{T, i}}{L}\\]  Standard treatments of the viscosity (Braginskii, etc.) show that viscosity scales with the number density, temperature, and collision time\n \\[\\mu \\sim n T_i \\tau_{ii} \\sim p_i \\tau_{ii}\\]   \\[\\Pi_{i, max} \\sim p_i \\frac{\\tau_{ii} v_{T, i}}{L}\\]  The specific term we want to get rid of is  \\( \\div \\vec \\Pi \\) , so let\u0026rsquo;s compare it to a term we want to keep  \\( \\grad P \\) \n \\[\\frac{\\div \\vec \\Pi}{\\grad p} \\sim \\frac{p_i \\tau_{ii} v_{T, i} / L^2}{p_i / L} \\sim \\frac{\\tau_{ii}}{\\tau}\\]  So, to neglect the anisotropic pressure term in the momentum equation, once again we require\n \\[\\frac{\\tau_{ii}}{\\tau} \\ll 1\\]  In other words, as long as the plasma is collision-dominated, we can drop the ion anisotropic pressure term. What about associated the electron term? If you can assume  \\( T_i \\approx T_e \\) . Then  \\( p_i \\approx p_e \\)  for a neutral plasma, and\n \\[\\frac{\\div \\vec \\Pi_e}{\\grad p} \\sim \\frac{p_i \\tau_{ee} v_{T, i} / L^2}{p_i / L} \\sim \\frac{\\tau_{ee}}{\\tau} \\ll 1\\]  Magnetic terms #  In the generalized Ohm\u0026rsquo;s law, the diamagnetic drift term is\n \\[\\frac{\\grad p_e}{Z e n} \\sim \\frac{n T_e / L}{Z en} \\sim \\frac{T_i / L}{Ze} \\sim \\frac{m_i v_{T, i}^2}{L Z e}\\]  Compare  \\( \\grad p_e / Z e n \\)  to a term that we\u0026rsquo;re going to keep, which is the dynamo term  \\( \\vec v \\cross \\vec B \\) \n \\[\\frac{ \\grad p_e / Z en}{|\\vec v \\cross \\vec B|} \\sim \\frac{m_i v_{T, i}^2}{LZe}{v_{T, i} B} \\sim \\frac{m_i}{ZeB}\\frac{v_{T, i}}{L} = \\frac{v_{T, i}}{\\omega_{c, i}} \\frac{1}{L} \\sim \\frac{r_{L, i}}{L} \\ll 1\\]  So to neglect the diamagnetic drift term, we need the plasma to be well-magnetized. This means the Larmor radius must be much less than the plasma characteristic length  \\( r_{L, i} \\ll L \\) \nNow what can we do with the Hall term  \\( \\frac{\\vec j \\cross \\vec B}{Zen} \\) . For a static plasma (or one with subsonic flows):\n \\[\\rho (\\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v) - \\grad p - \\vec j \\cross \\vec B = 0\\]  so by \u0026ldquo;subsonic\u0026rdquo; we mean that the static pressure is much larger than the dynamic pressure and we can discard the  \\( \\vec v \\)  terms.\n \\[\\vec j \\cross \\vec B \\approx \\grad p\\]  Comparing the Hall term to the dynamo term  \\( \\vec v \\cross \\vec B \\)  also gives the same requirement\n \\[\\frac{r_{L, i}}{L} \\ll 1\\]  How well do some real plasmas hold up to the requirements of ideal MHD? Consider field nulls in a Z-pinch or an FRC, and weakly magnetized plasmas (such as those in a Hall thruster). Clearly, the magnetization requirement does not hold up at all points in space, and ideal MHD does not necessarily apply across the whole domain.\nResistivity #  We neglect resistivity and the resistive electric field  \\( \\eta \\vec j \\)  in the generalized Ohm\u0026rsquo;s law\n \\[\\eta \\vec j \\sim \\eta \\frac{\\grad p}{B} \\sim \\frac{ \\eta n T}{LB}\\]   \\[\\eta = \\frac{m_e \\nu_{ei}}{n e^2} = \\frac{m_e}{n e^2 \\tau_{ei}} = \\frac{m_e ^2/ m_i}{n e^2 \\tau_{ee}}\\]   \\[\\rightarrow \\eta \\vec j \\sim \\frac{m_e ^2}{e^2 L B} \\frac{v_{T, i} ^2}{\\tau_{ee}}\\]  How does it compare to  \\( \\vec v \\cross \\vec B \\) \n \\[\\frac{\\eta \\vec j}{|\\vec v \\cross \\vec B|} \\sim \\left(\\frac{m_e}{m_i} \\right)^2 \\frac{m_i}{e ^2 B^2} \\frac{v_{T, i} ^2}{L^2} \\frac{L}{v_{T, i} \\tau_{ee}} \\sim \\left( \\frac{m_e}{m_i}\\right)^2 \\frac{v_{T, i} ^2}{\\omega_{c, i} ^2} \\frac{1}{L^2} \\sim \\left( \\frac{m_e}{m_i} \\right) ^{3/2} \\left( \\frac{r_{L, i}}{L} \\right) ^{2} \\frac{\\tau}{\\tau_{ii}} \\ll 1\\]  This now places a lower limit on  \\( \\tau_{ii} \\) ; to neglect resistivity  \\( \\tau_{ii} \\)  cannot be too low\n \\[\\left( \\frac{m_e}{m_i} \\right) ^{3/2} \\left( \\frac{r_{L, i}}{L} \\right) ^{2} \\ll \\frac{\\tau_{ii}}{\\tau} \\ll 1\\]  Heating sources #  Neglect the collisional heating sources  \\( Q_{ei} \\) ,  \\( Q_{ie} \\)  in the energy equation. We do that by assuming that they are equal and opposite, which only happens when the temperatures are equal. In other words, we are again assuming local thermodynamic equilibrium between electrons and ions.\n \\[1 \\gg \\frac{\\tau_{ei}}{\\tau} = \\left( \\frac{m_i}{m_e} \\right)^{1/2} \\frac{\\tau_{ii}}{\\tau} \\rightarrow \\frac{\\tau_{ii}}{\\tau} \\ll \\left( \\frac{m_e}{m_i} \\right) ^{1/2}\\]  This is a much more restrictive condition than ion collisionality. Alternatively, we could track two temperature independently.\nWe also neglect the heat flux terms  \\( \\div (\\vec h_i \u0026#43; \\vec h_e) \\)  in the energy equation. Consider parallel (to the magnetic field) heat conduction which dominates since  \\( \\kappa_\\perp \\ll \\kappa_\\parallel \\) , so\n \\[\\div \\vec h \\approx \\grad_\\parallel \\cdot (\\kappa_\\parallel \\grad_\\parallel T)\\]   \\[\\grad_\\parallel \\cdot \\left[ ( \\kappa_{\\parallel, i} \u0026#43; \\kappa_{\\parallel, e}) \\grad_\\parallel T \\right]\\]   \\[\\kappa_{\\parallel, e} \\sim \\frac{n T_e}{m_e} \\tau_{ee} \\qquad \\text{ and } \\qquad \\kappa_{\\parallel, i} \\sim \\frac{n T_i}{m_i} \\tau_{ii}\\]   \\[\\frac{\\kappa_{\\parallel, e}}{\\kappa_{\\parallel, i}} \\sim \\frac{\\tau_{ee}}{\\tau_{ii}} \\frac{m_i}{m_e} \\sim \\left( \\frac{m_i}{m_e} \\right) ^{1/2}\\]  Compare the thermal conductivity to the rate of pressure change  \\( \\pdv{p}{t} \\) \n \\[\\frac{\\grad_\\parallel \\cdot (\\kappa_{\\parallel, e} \\grad_\\parallel T)}{\\pdv{p}{t}} \\sim \\frac{n T^2 \\tau_{ee}/m_e L^2}{nT/\\tau} \\sim \\left( \\frac{m_i}{m_e} \\right)^{1/2} \\frac{\\tau_{ii}}{\\tau} \\ll 1\\]  We\u0026rsquo;re back to the same requirement that we have ion-electron local thermodynamic equilibrium.\nAnisotropic pressure terms in energy equation #  Neglect the anisotropic pressure terms in the energy equation. That is,  \\( \\vec \\Pi_i \\cdot \\cdot \\grad \\vec v_i \\)  and  \\( \\vec \\Pi_e \\cdot \\cdot \\grad \\vec v_e \\) . Skipping ahead, the result is\n \\[\\frac{\\tau_{ii}}{\\tau} \\ll 1 \\]   \\[\\left( \\frac{m_e}{m_i} \\right)^{1/2} \\frac{\\tau_{ii}}{\\tau} \\frac{r_{L, i}}{L} \\ll 1\\] \nElectron convection term #  Lastly, we neglect the electron convection term in the energy equation  \\( \\frac{\\vec j}{Zen} \\cdot \\grad \\frac{P_e}{\\rho^\\gamma} \\) . The result is\n \\[\\frac{r_{L, i}}{L} \\ll 1\\]  Conservation Law Form of MHD #   \\[\\pdv{}{t} q \u0026#43; \\div \\vec f = 0\\]  We can express momentum in conservation law form:\n \\[\\pdv{}{t} (\\rho \\vec v) \u0026#43; \\div \\left[ \\rho \\vec v \\vec v - \\frac{ \\vec B \\vec B}{\\mu_0} \u0026#43; (p \u0026#43; \\frac{B^2}{2 \\mu_0})\\vec 1 \\right] = 0\\]  The conservation law form for the magnetic field looks like\n \\[\\pdv{ \\vec B}{t} \u0026#43; \\div \\left[ \\vec v \\vec B - \\vec B \\vec v \\right ] = 0\\]  And of course the energy equation for\n \\[E = \\frac{1}{\\gamma - 1} p \u0026#43; \\frac{1}{2} \\rho v^2 \u0026#43; \\frac{ B^2}{2 \\mu_0}\\]   \\[\\pdv{E}{t} \u0026#43; \\div \\left[ \\left( E \u0026#43; p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) - \\left( \\frac{ \\vec B \\cdot \\vec v}{\\mu_0} \\right) \\vec B \\right] = 0\\]  Conservation law forms are particularly useful when considering equilibrium steady-state force balance. This means that in steady-state equilibrium we have\n \\[\\div \\left[ \\rho \\vec v \\vec v - \\frac{ \\vec B \\vec B}{\\mu_0} \u0026#43; \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\vec 1 \\right] = 0\\]  We can use this relationship and integrate over various volumes to determine the relationship between the various force balance terms\nExamples of equilibrium plasma confinement #  For a plasma in static equilibrium  \\( \\vec v = 0 \\rightarrow \\vec j \\cross \\vec B = \\grad p \\) \n \\[ \\div \\vec T = 0 = \\div \\left( \\right) \\]   \\[\\vec j \\cross \\vec B = \\frac{1}{\\mu_0} (\\curl \\vec B) \\cross \\vec B = \\frac{1}{\\mu_0} (\\vec B \\cdot \\grad \\vec B - \\frac{1}{2} \\grad B^2) \\\\ = \\frac{1}{\\mu_0} ( B^2 \\vu e_B \\cdot \\grad \\vu e_B \u0026#43; \\frac{1}{2} \\vu e_B \\vu e_B \\cdot \\grad B^2 - \\frac{\\grad B^2}{2} )\\]  where  \\( \\vu e_B = \\vec B / B \\) \n \\( \\vu e_B \\cdot \\grad \\vu e_B \\)  is the curvature of  \\( \\vec B \\) . Write it like a curvature\n \\[\\vec K = - \\vu r / R_c\\]   \\[\\vu e_B \\vu e_B \\cdot \\grad B^2\\]  is gradient of  \\( B^2 \\)  that is parallel to B. Multiply by  \\( e_B \\)  gives the component of gradient along  \\( e_B \\) . The difference between that and  \\( \\grad B^2 / 2 \\)  gives you the perpendicular gradient\n \\[\\vec j \\cross \\vec B = \\frac{1}{\\mu_0} (B^2 \\vec \\kappa - \\frac{1}{2} \\grad _\\perp B^2) = \\grad p = \\grad_\\perp p\\]  identify  \\( B^2 \\vec \\kappa \\)  is magnetic tension resulting from having a bent magnetic field line.  \\( \\frac{1}{2} \\grad_\\perp B^2 \\)  is magnetic pressure. They have to balance the plasma pressure at equilibrium.\nFor example, consider a cylindrical plasma that\u0026rsquo;s in equilibrium with a helical magnetic field  \\[\\vec B = B_\\theta (r) \\vu \\theta \u0026#43; B_z (r) \\vu z\\] \nHow is plasma pressure profile determined by the different components of the magnetic field? If we want to maximize the amount of pressure we confine, what should be maximized/minimized?\n \\[\\frac{B^2}{\\mu_0} \\vu \\kappa = \\grad _\\perp ( p \u0026#43; \\frac{B^2}{2 \\mu_0} ) \\]   \\( B_z \\)  is straight and has no curvature, so the only magnetic tension comes from  \\( B_\\theta \\) , so the magnetic tension from  \\( B_\\theta \\)  must balance the total pressure.\nThe role of  \\( B_z \\)  is displacing plasma pressure. The utility in defining  \\( \\beta \\)  as\n \\[\\beta = \\frac{\\text{plasma pressure}}{\\text{magnetic pressure}}\\]  Conditions of Ideal MHD Validity #  The conditions for ideal MHD to be valid are\n High Ion Collisionality:  \\( \\frac{\\tau_{ii}}{\\tau} \\ll 1 \\)  Small ion Larmor radius:  \\( \\frac{r_{L, i}}{L} \\ll 1 \\)  Low resistivity:  \\( \\left(\\frac{m_e}{m_i} \\right)^{3/2} \\left( \\frac{r_{L, i}}{L} \\right)^2 \\ll 1 \\)   For a given plasma in force balance, we can relate the plasma pressure to the magnetic pressure\n \\[\\beta = \\frac{n T}{B^2 / 2 \\mu_0} = 4 \\times 10^{-16} \\frac{n_{cm^{-3}} T_{keV}}{B^2 _{T}}\\]  Ion collision time is (Spitzer collisionality)\n \\[\\tau_{ii} = 2.09 \\times 10^{7} \\frac{T_{eV} ^{3/2} \\mu ^{1/2}}{\\ln \\Lambda n_{cm ^{-3}}} \\left[\\text{s}\\right]\\]  where  \\( \\mu \\equiv m_i / m_p \\) .\nPutting the conditions for ideal MHD in terms of  \\( \\beta \\)  and  \\( \\tau_{ii} \\) ,\n High collisionality:  \\[\\frac{\\tau_{ii}}{\\tau} = 2.14 \\times 10^{12} \\frac{T^2}{n L} \\ll 1 \\]   \\[\\rightarrow T_{eV} \\ll 6.8 \\times 10^{-7} L_{cm} ^{1/2} n_{cm^{-3}} ^{1/2}\\]  Small gyroradius:  \\[\\frac{r_{L, i}}{L} = 2.3 \\times 10^7 \\frac{1}{Z L} \\sqrt{ \\frac{\\mu B}{n}} \\ll 1\\]   \\[\\rightarrow n_{cm^{-3}} \\gg 5.3 \\times 10^{14} \\frac{\\mu B_{T}}{Z^2 L_{cm} ^2}\\]  Low resistivity:  \\[\\left( \\frac{m_e}{m_i} \\right) ^{3/2} \\left( \\frac{r_{L, i}}{L} \\right) ^2 \\frac{\\tau}{\\tau_{ii}} = 5.65 \\frac{\\mu B}{Z^2 L T^2}\\]   \\[\\rightarrow T_{eV} \\gg 2.4 \\frac{\\mu ^{3/2} \\beta^{1/2}}{Z L _{cm} ^{1.2}}\\]   Perpendicular MHD #  For most configurations for magnetic fusion confinement, we are able to satisfy 2. and 3. but often have densities much too low to meet the high collisionality constraint. However, in practice ideal MHD does accurately model macroscopic behavior of many plasmas. At the same time, magnetized / fusion plasmas are often largely collisionless. We can understand why by re-writing the collisionality requirement as\n \\[1 \\gg \\frac{\\tau_{ii}}{\\tau} = \\frac{\\tau_{ii} v_{T, i}}{\\tau v_{T, i}} \\sim \\frac{\\lambda}{L}\\]  where  \\( \\lambda \\)  is the mean free path. The ratio  \\( \\lambda / L \\)  is also known as the Knudsen number. In magnetized plasmas, the mean free path is often very long, but the path between collisions can cover a great distance only by following magnetic field lines. Motion  \\( \\perp \\)  to the magnetic field is constrained by  \\( r_{L, i} \\) . This suggests an approach wherein we divide the plasma model into a 1-D kinetic model to be solved along the magnetic field and a 2-D MHD model  \\( \\perp \\)  to the magnetic field.\nWe consider diffusivity (terms like viscosity, conductivity) parallel and perpendicular to  \\( \\vec B \\)   \\[k_\\parallel \\sim \\frac{\\lambda ^2}{\\tau_{ii}}\\]   \\[k_\\perp \\sim \\frac{r_{L, i} ^2}{\\tau_{ii}}\\]   \\[\\frac{k_\\parallel}{k_\\perp} \\sim \\left( \\frac{\\lambda}{r_{L, i}} \\right)^2 \\sim (\\omega_{c, i} \\tau_{ii} )^2\\] \nThis changes one of our conditions for validity. Specifically, we can write\n \\[\\frac{r_{L, i}}{L} \\frac{1}{ \\omega_{c, i} \\tau_{i i}} = 254 \\frac{\\mu \\beta}{Z^2 L_{cm} T_{eV} ^2}\\]  or\n \\[T_{eV} \\gg 16 \\frac{\\mu ^{1/2} \\beta^{1/2}}{Z L^{1/2}}\\]  This is now only slightly more restrictive than the low-resistivity condition. In fact, most of the plasmas that we looked at before (large tokamaks \u0026amp; toruses, propulsion systems) which had temperatures above the validity range now fall comfortably within the region of validity. The perpendicular MHD model is equivalent to a collisionless model, giving a much wider applicability than the collisional MHD model.\nWhat about the parallel component? Our collisionality condition still isn\u0026rsquo;t valid parallel to the field. Basically, we ignore the parallel component, which is the same as assuming that  \\( \\rho \\) ,  \\( T \\) ,  \\( p \\)  are constant along magnetic field lines, with  \\( \\div \\vec v = 0 \\)  . Let\u0026rsquo;s write out the expressions for collision-less MHD and for ideal MHD and compare:\n    Collisionless MHD Ideal MHD     Continuity  \\( \\pdv{\\rho}{t} = 0 \\)   \\( \\pdv{\\rho}{t} = - \\rho \\div \\vec v \\)    Momentum  \\( \\rho \\dv{\\vec v_\\perp}{t} = \\vec j \\cross \\vec B - \\grad_\\perp p \\)   \\( \\rho \\dv{\\vec v}{t} = \\vec j \\cross \\vec B - \\grad p \\)    Parallel constraint  \\( \\vec B \\cdot \\grad \\frac{v_\\parallel}{B} = - \\div \\vec v _\\perp \\)     Energy  \\( \\dv{p}{t} = 0 \\)   \\( \\dv{p}{t} = - \\gamma p \\div \\vec v \\)     Collisionless MHD reproduces many of the effects of ideal MHD but has a wider region of validity. Corollary: ideal MHD is accurate beyond its region of validity, unless results lead to parallel gradients. For example, we know that MHD is not valid when representing confinement of a plasma confined in a magnetic mirror, which is an inherently kinetic phenomenon. But ideal MHD can generate parallel gradients within its region of validity, and we need to be careful. Ideal MHD does not require different models  \\( \\parallel \\)  and  \\( \\perp \\)  to the magnetic field, and is therefore preferred. We will continue to use ideal MHD outside of its region of validity.\n"});index.add({'id':5,'href':'/r/notes/UWAA558/06-boundary-conditions/','title':"Boundary Conditions",'section':"MHD Theory",'content':"Boundary Conditions #  Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region.\nPerfectly Conducting Wall #  For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:\n  document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\[\\left. \\vec E_t \\right|_{wall} = 0 \\quad \\rightarrow \\quad \\left. \\vu n \\cross \\vec E \\right| _{wall} = 0\\]  Applying Faraday\u0026rsquo;s law at the wall,\n \\[\\left. \\vu n \\cdot \\pdv{\\vec B}{t} \\right|_{wall} = \\left. - \\vu n \\cdot \\curl \\vec E \\right|_{wall} = \\left. \\div (\\vu n \\cross \\vec E) \\right| _{wall} = 0\\]   \\[\\pdv{}{t} \\vu n \\cdot \\vec B |_{wall} = 0\\]  If initially there is no normal magnetic field, then  \\[\\vu n \\cdot \\vec B|_{wall} = 0 \\quad \\text{if initially true}\\] \nAnd of course, for an impermeable wall,\n \\[\\vu n \\cdot \\vec v |_{wall} = 0\\]  Is this a sufficient set of boundary conditions? Think back to the governing equations in conservation form  \\[\\pdv{}{t} \\vec Q \u0026#43; \\div \\vec F = 0\\]  The boundary conditions come into play when defining  \\( \\vec F \\)  at the boundary. In particular, we need to know what  \\( \\dd \\vec S \\cdot \\vec F |_{wall} \\)  is. In our governing equations, this will involve  \\( \\vec E \\) ,  \\( \\vec B \\) , and  \\( \\vec v \\) .\nInsulating Boundary #  As a slight modification, an insulating boundary can have a tangential electric field. Consider a simple geometry of parallel electrodes with an insulating wall between them.\nFrom Ohm\u0026rsquo;s law  \\[\\vec E \u0026#43; \\vec v \\cross \\vec B = 0\\]  so the only way an electric field tangential to the wall can exist is if  \\( \\vu n \\cdot \\vec v \\neq 0 \\) .\nFor either a perfectly conducting or an insulating boundary, the other variables are arbitrary:  \\( \\rho \\)  ,  \\( p \\) ,  \\( \\vec v_t \\) ,  \\( \\vec B_t \\) .\nVacuum Region #  The plasma (radius  \\( R_p \\) ) is supported by a region of vacuum out to a perfectly conducting wall at some radius  \\( R_w \\) . We assume that there is no plasma in the vacuum region. The governing equations in vacuum are just Maxwell\u0026rsquo;s equations\n \\[\\curl \\vec B_{vac} = 0 \\qquad \\text{and} \\qquad \\div \\vec B_{vac} = 0\\]  At the wall,  \\[\\vu n \\cross \\vec E |_{wall} = 0\\]   \\[\\left. \\vu n \\cdot \\pdv{\\vec B}{t} \\right|_{wall} = 0\\] \nWhat happens at the plasma-vacuum interface? We need to specify jump conditions and continuity conditions. Let\u0026rsquo;s use square brackets to signify a jump:\n \\[\\left[ X \\right] = \\left. X \\right|_{R_p \u0026#43; dr} - \\left. X \\right|_{R_p - dr}\\]  The normal magnetic field has to be continuous.\n \\[[\\vu n \\cdot \\vec B]_{R_p} = 0\\]  The tangential magnetic field jump is given by the surface current density at the jump\n \\[\\left[ \\vu n \\cross \\vec B \\right] _{R_p} = \\mu_0 \\vec K \\]  Integrating  \\( \\grad_\\perp (p \u0026#43; \\frac{B^2}{2 \\mu_0}) = \\frac{B^2}{\\mu_0} \\vec \\kappa \\)  over a differential volume across the surface gives\n \\[\\left[ p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right] _{R_p} = 0\\]  The plasma shape is determined self-consistently by the wall shape and surface current. This is a free-boundary problem. Another option is to specify the plasma shape, and then determine the required wall shape. This is a fixed-boundary problem.\nThe most realistic case includes externally applied magnetic fields coming from source coils, perhaps computed by Biot-Savart law. The vacuum magnetic field is then  \\( \\vec B_{vac} = \\vec B_{ext} \u0026#43; \\vec B_{plasma} \\) . The crazy coil shapes in the stellarator design come from the 3D geometry computations solving this problem.\nConservation of Magnetic Flux (\u0026ldquo;Frozen-In\u0026rdquo; Flux) #  Locally,  \\( \\vec E \u0026#43; \\vec v \\cross \\vec B = 0 \\)  with Faraday\u0026rsquo;s law  \\[\\pdv{B}{t} = - \\curl \\vec E = - B \\div \\vec v \u0026#43; \\vec B \\cdot \\grad \\vec v - \\vec v \\cdot \\grad B\\]  From the continuity equation,  \\[\\pdv{\\rho}{t} \u0026#43; \\vec v \\cdot \\grad \\rho = - \\rho \\div \\vec v\\]  Combining we find that  \\[\\dv{\\vec B}{t} = \\frac{\\vec B}{\\rho} \\dv{\\rho}{t} \u0026#43; \\vec B \\cdot \\grad \\vec v\\]   \\[\\rightarrow \\dv{}{t} \\left( \\frac{\\vec B}{\\rho} \\right) = \\frac{\\vec B}{\\rho} \\cdot \\grad \\vec v\\] \nThis says that the field and plasma density move together. Locally, if the magnetic field increases then mass density increases, such that the ratio  \\( \\vec B / \\rho \\)  remains constant. In the direction parallel to the magnetic field we have a term that involves field line twisting, which is a bit more complicated, but in the perpendicular direction  \\[\\dv{}{t} \\left( \\frac{\\vec B}{\\rho} \\right) _\\perp = 0\\] \nIf we consider globally the magnetic flux through a moving surface S at velocity  \\( \\vec u \\) . The magnetic flux penetrating the surface is  \\[\\Psi = \\int \\vec B \\cdot \\dd \\vec S\\]  or  \\[\\dv{\\Psi}{t} = \\int \\dv{\\vec B}{t} \\cdot \\vu n \\dd S\\]   \\[= \\int \\pdv{\\vec B}{t}\\cdot \\vu n \\dd S \u0026#43; \\oint \\vec B \\cross \\vec u \\dd \\vec l\\]  Using Faraday\u0026rsquo;s law\n \\[\\dv{\\Psi}{t} = \\int - \\curl \\vec E \\cdot \\vu n \\dd S \u0026#43; \\oint \\vec B \\cross \\vec u \\cdot \\dd \\vec l\\]   \\[= \\oint (- \\vec E \u0026#43; \\vec B \\cross \\vec u) \\cdot \\dd \\vec l\\]  Using the electric field from Ohm\u0026rsquo;s law  \\[\\dv{\\Psi}{t} = \\oint(\\vec v - \\vec u) \\cross \\vec B \\cdot \\dd \\vec l\\]  This tells us that if the surface moves with the plasma  \\( \\vec u = \\vec v \\)  then  \\[\\dv{\\Psi}{t} = 0\\]  the flux through the surface is constant, and the flux is a constant of the topology. This is a direct consequence of ideal MHD. If we add even a small amount of resistivity, we dramatically alter the results in a process called \u0026ldquo;tearing\u0026rdquo; where the magnetic field \u0026ldquo;tears\u0026rdquo; and reconnects with itself.\n"});index.add({'id':6,'href':'/r/notes/UWAA558/07-equilibrium-for-fusion/','title':"Equilibrium for Fusion",'section':"MHD Theory",'content':"Equilibrium for Fusion (   document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\( \\beta \\) ) #  For a fusion device we would like to determine a magnetic configuration that confines plasma while it fuses. At fusion temperatures, the power required to maintain the equilibrium will be substantial. For a device to be useful, the power required to sustain the equilibrium must be less than the power released from fusion. Important loss terms for a confined plasma are transport (thermal conduction primarily) and radiation terms. The scaling factors are  \\( P_{Brem} \\sim n^2 T^{1/2} \\)  and  \\( P_{cycl} \\sim n^2 T^2 \\)  for radiation, and  \\( P_L \\sim \\frac{3nT}{\\tau_E} \\)  for thermal losses.\nWe know that the fusion source term will primarily come from the DT fusion reaction\n \\[\\text{D} \u0026#43; \\text{T} \\rightarrow \\text{He}^4 (3.5\\, MeV) \u0026#43; \\text{n} (14.1\\, MeV)\\]  The primary fusion reaction releases an  \\( \\alpha \\) -particle and a high-energy neutron. The concept of ignition is that the neutron leaves the plasma, and the  \\( \\alpha \\)  (with energy  \\( E_\\alpha = 3.5 MeV) \\)  remains to heat the plasma.\n \\[P_\\alpha = \\frac{1}{4} n^2 \\langle \\sigma v \\rangle E_\\alpha \\qquad \\text{(assuming} \\quad n_D = n_T = n/2 \\text{)}\\]   \\[P_\\alpha \u0026gt; P_L \\quad \\rightarrow \\quad n \\tau_E \u0026gt; \\frac{12 T}{E_\\alpha \\langle \\sigma v \\rangle}\\]  To sustain fusion, we set the fusion heating term above the thermal loss term. The reaction cross-section  \\( \\sigma \\)  can be maximized to give the Lawson criterion\n \\[n \\tau_E \u0026gt; 10^{14} s / cm^3\\]  The Lawson criterion only applies at fusion temperatures, but it is a useful parameter even outside of ignition since it gives a ratio of fusion power to lost power\n    \\( T_i (keV) \\)   \\( \\langle \\sigma v \\rangle (cm^3 / s) \\)  Required  \\( n \\tau_E (s / cm^3) \\)      1  \\( 7 \\cdot 10^{-21} \\)   \\( 5 \\cdot 10^{17} \\)    5  \\( 1.4 \\cdot 10^{-17} \\)   \\( 1.2 \\cdot 10^{15} \\)    20  \\( 4.3 \\cdot 10^{-16} \\)   \\( 1.6 \\cdot 10^{14} \\)    60  \\( 8.7 \\cdot 10^{-16} \\)   \\( 2.4 \\cdot 10^{14} \\)     We can see that the required  \\( n \\tau_E \\)  actually has a minimum around  \\( 20 keV \\)  (at least, as far as the data in the table goes). Even though the maximum cross-section is at a much higher temperature, what we\u0026rsquo;re really concerned with is the ratio of the fusion source term to the thermal loss term, which is linear in temperature.\nMHD equilibrium does not place a limit on the density  \\( n \\) . Instead, it places a limit on  \\( \\beta \\)  in order to achieve equilibrium force-balance  \\( (\\beta = 1) \\) \n \\[\\beta = \\frac{n (T_e \u0026#43; T_i)}{B^2 / 2 \\mu_0} \\rightarrow n = \\frac{ \\beta B^2}{4 \\mu_0 T}\\]  In this form, we can more clearly see what our options are to achieve MHD equilibrium. Some devices (large-scale tokamaks) are able to achieve the requisite confinement time at a low  \\( \\beta \\)  by making use of very strong magnetic fields. Other devices are able to make use of more modest magnetic fields by working at a higher  \\( \\beta \\) .\nTherefore,\n \\[\\tau_E \u0026gt; \\frac{1}{\\beta B^2} \\frac{48 \\mu_0}{E_\\alpha} \\frac{T^2}{\\langle \\sigma v \\rangle}\\]  The term  \\( \\frac{T^2}{\\langle \\sigma v \\rangle} \\)  has a minimum at  \\( 10-20 keV \\) . At 15 keV and a magnetic field of  \\( 5T \\)  (many actual components cannot reasonably exceed such magnetic fields) then\n \\[\\tau_E \u0026gt; \\frac{0.1}{\\beta} \\text{s}\\]  For a large-scale toroidal device with  \\( \\beta = 1\\% \\) , the confinement time  \\( \\tau_E \u0026gt; 10s \\) . If we consider a common diffusivity (how fast energy will leave due to thermal conductivity)  \\( D_E \\approx 1 m^2 / s\\) , so for a characteristic radius  \\( a \\) \n \\[\\tau_E \\approx \\frac{a^2}{4 D_E} \\rightarrow a \u0026gt; 6.3 \\text{m}\\]  This gives you a sense of why low- \\( \\beta \\)  devices need to be so large. Instead, if we consider  \\( \\beta \\sim 50\\% \\) ,  \\( \\tau_E \u0026gt; 0.2 \\text{s} \\)  and\n \\[\\beta \\sim 50\\% \\rightarrow a \u0026gt; 0.9 \\text{m}\\]  When you consider that the cost of a device (to first order) scales with the volume of the device, achieving a high  \\( \\beta \\)  is very important for fusion equilibrium. However, when we consider MHD stability we are generally forced into lower  \\( \\beta \\)  to avoid destructive instabilities. Configuration optimization is the process of balancing this trade-off.\nVirial Theorem #  Application of the virial theorem to energy balance for the stress tensor  \\( \\vec T \\)  tells us that MHD equilibria must be supported by externally supplied currents. Many times you\u0026rsquo;ll hear of theoretical designs for compact toroid devices which can maintain stability under their own currents, but they are the MHD stability equivalent of a perpetual motion machine. A compact toroid cannot exist unsupported.\nWriting static equilibrium:\n \\[\\div \\left[ - \\frac{ \\vec B \\vec B}{\\mu_0} \u0026#43; \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0}\\right) \\vec I \\right] = \\div \\vec T = 0\\]  If we define the direction of the magnetic field to be  \\( \\vu e _B = \\vu z \\)  then\n \\[\\vec T = p_\\perp ( \\vec I - \\vu e_B \\vu e_B ) \u0026#43; p_\\parallel \\vu e_B \\vu e_B \\\\ = \\begin{bmatrix} p_\\perp \u0026amp; 0 \u0026amp; 0 \\\\ 0 \u0026amp; p_\\perp \u0026amp; 0 \\\\ 0 \u0026amp; 0 \u0026amp; p_\\parallel \\end{bmatrix}\\]  where\n \\[p_\\perp = p \u0026#43; \\frac{B^2}{2 \\mu_0}\\]  and  \\[p_\\parallel = p - \\frac{B^2}{2 \\mu_0}\\] \nA gradient vector identity gives\n \\[\\div (\\vec r \\cdot \\vec T) = \\vec r \\cdot ( \\div \\vec T) \u0026#43; \\vec T \\cdot \\cdot \\grad \\vec r\\]  Integrating this expression over a volume and assuming that the volume contains a confined MHD equilibrium that is self-contained and self-supported:\n \\[\\int_V \\div ( \\vec r \\cdot \\vec T) \\dd V = \\int_V (\\vec r \\cdot \\overbrace{\\cancel{(\\div \\vec T)}}^{\\text{MHD equil.}} \u0026#43; \\vec T \\cdot \\cdot \\grad \\vec r) \\dd V\\]   \\[\\grad \\vec r = \\vec I\\]  so  \\[\\vec T \\cdot \\cdot \\grad \\vec r = p_\\perp \u0026#43; p_\\perp \u0026#43; p_\\parallel \\\\ = 3p \u0026#43; \\frac{B^2}{2 \\mu_0}\\] \n \\[\\int_V (3p \u0026#43; \\frac{B^2}{2 \\mu_0} ) \\dd V = \\int _V \\div ( \\vec r \\cdot \\vec T) \\dd V = \\oint _S (\\vec r \\cdot \\vec T) \\cdot \\vu n \\dd S \\\\ = \\oint _S \\left[ \\vec r \\cdot \\vec I p_\\perp \u0026#43; \\vec r \\cdot \\vu e_B \\vu e_B (p_\\parallel - p_\\perp) \\right]\\cdot \\vu n \\dd S \\\\ =\\oint \\left[ \\left( \\cancel{p} \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\vu r \\cdot \\vu n - \\frac{B^2}{\\mu_0} (\\vec r \\cdot \\vu e_B)(\\vu e_B \\cdot \\vu n) \\right] \\dd S\\]  Beyond where the plasma is contained, the pressure does not contribute  \\( p = 0 \\) . If all current sources are contained in the configuration, the magnetic field  \\( \\sim 1/r^3 \\)  for a dipole,  \\( \\sim 1/r^4 \\)  for a quadrupole, etc. Therefore the right-hand side will fall off like\n \\[RHS \\propto \\oint_S B^2 r \\dd S \\propto \\left( \\frac{1}{r^3} \\right) ^2 r r^2 \\propto \\frac{1}{r^3} \\text{(dipole)}\\]  so  \\( RHS \\rightarrow 0 \\)  as  \\( r \\rightarrow \\infty \\) . But what about the left-hand side? Both of the terms in the volume integral are positive definite, so the LHS must be positive finite and the equality can\u0026rsquo;t possibly hold. The assumption that the plasma is self-contained must be invalid. This tells us that we must have external currents.\nMagnetic Flux Surfaces #  The vast difference in thermal conductivity parallel and perpendicular to the magnetic field in a plasma confinement configuration leads to an avoidance of any open field lines. Magnetic equilibria are generally toroidal to eliminate end losses from open configurations. In general fusion confinement devices, magnetic field lines lie on a set of closed nested toroidal surfaces. This means that we can no longer describe any equilibria in a solely 1D geometry. The minor radius is no longer the only important scale length.\nFrom  \\( \\vec j \\cross \\vec B = \\grad p \\) , we know that the pressure gradient is perpendicular to  \\( \\vec j \\)  and  \\( \\vec B \\) , and therefore both  \\( \\vec B \\)  and  \\( \\vec j \\)  lie on surfaces of uniform pressure. We call these toroidal surfaces either magnetic surfaces or flux surfaces. We can use these surfaces to build a 1-dimensional description.\nAs a brief aside, some geometrical vocabulary will be useful when describing toroidal geometry. A toroid is any surface of revolution with a hole in the middle. A torus is the particular case of a toroid in which the revolved figure is a circle.\nWe will define our global toroidal coordinate system to consist of the major axis  \\( (z) \\) , the distance from the major axis  \\( (R) \\) , and the azimuthal angle around the major axis  \\( (\\phi) \\) .\nWe will also make use of a poloidal coordinate system measured by minor radius (distance from the minor axis)  \\( r \\)  and the poloidal angle from the minor axis  \\( \\theta \\) . We will generally refer to a point on the torus relative to the major axis  \\( (R, z, \\phi) \\) , or relative to the minor axis  \\( (r, \\theta, \\phi) \\) , or in spherical coordinates  \\( (R, \\theta, \\phi) \\) . The major radius  \\( R_0 \\)  is the distance from the major axis to the minor axis. The minor radius  \\( a \\)  is the characteristic distance from the minor axis to the exterior of the revolved figure. Usually we will find symmetry under  \\( \\phi \\) .\nThe aspect ratio of a torus is the ratio of the major radius to the minor radius.\n \\[A = \\frac{R_0}{a}\\]  When we talk about a \u0026ldquo;toroidal surface,\u0026rdquo; we mean a cross-section of the toroidal rotation. When we talk about a \u0026ldquo;poloidal surface\u0026rdquo; we mean a surface which is coplanar with the minor axis:\nThe poloidal flux is determined by the size of the poloidal surface and the poloidal magnetic field:\n \\[\\Psi _p = \\int_{S_p} \\vec B \\cdot \\dd \\vec S\\]  and the toroidal flux is determined by the size of a toroidal surface and the toroidal magnetic field:\n \\[\\Psi_t = \\int_{S_t} \\vec B \\cdot \\dd \\vec S\\]  Considering the poloidal flux, we can see that if we expand the size of the surface towards the minor radius, the flux will increase until eventually we come to a point where the flux begins to decrease. The position of this maximum is called the magnetic axis, which does not necessarily correspond to the minor axis. In fact, it is generally displaced from the minor axis.\nTo refer back to something more familiar, we\u0026rsquo;ll define the same terms for a cylindrical geometry  \\((r, \\theta, z) \\) . An axial surface corresponds with a toroidal surface, and an azimuthal surface corresponds with a poloidal surface:\nIf we consider the trajectory of a single field line, what sorts of surfaces will it trace out? What surface will contain the field line? As it turns out, there are three options:\n Rational surface - the field line closes on itself, and it does so after a finite number of revolutions. One way to quickly visualize such a surface is to draw a Poincaré puncture plot. Choose a toroidal plane and plot a point wherever the field line punctures the surface. A Poincaré puncture plot of a rational surface contains a finite number of points and no continuous curves. Ergodic surface - the field line completely covers an entire surface, which is to say the field line punctures any toroidal surface an infinite number of times. In other words, it never closes on itself and defines an irrational curve. Stochastic region - In this case, there is no definite surface and the field line fills a volume.  Generally rational surfaces and ergodic surfaces are largely equivalent, but by introducing a small amount of resistivity a rational surface can lead to magnetic islands. One can imagine the addition of resistivity equivalent to allowing a small degree of motion of the magnetic field lines. In an ergodic surface, a flux surface is defined by a single (irrational) field line. If it moves toward itself in one location it will necessarily move away from itself in another location. But in a rational surface, different field lines can lie on the same constant pressure surface and will tend to move towards each other. By concentrating into magnetic islands, the flux surfaces are now more closely spaced, and the pressure gradient increases (a bad thing!)\nSurface quantities: Since pressure, current (not current density!), and flux (not field!) are constant along a flux surface, it is convenient to use flux  \\( \\Psi_p \\)  as a coordinate. A particular poloidal flux itself uniquely determines a poloidal surface with constant pressure and current. The flux surface quantities are  \\( p, \\, \\Psi_p, \\, \\Psi_t, \\, I_p = \\int_{S_p} \\vec j \\cdot \\dd \\vec S, \\, I_t = \\int_{S_t} \\vec j \\cdot \\dd \\vec S\\) .\nSurface quantities are not independent. The poloidal current  \\( I_p \\)  affects the toroidal field  \\( B_t \\)  and toroidal flux  \\( \\Psi_t \\) . The toroidal current affects the poloidal field  \\( B_p \\)  and poloidal flux  \\( \\Psi_p \\) .\n \\[B_t = \\vec B \\cdot \\vu \\phi\\]   \\[\\vec B_p = B_\\theta \\vu \\theta \u0026#43; B_z \\vu z \\\\ = B_r \\vu R \u0026#43; B_z \\vu z\\]  Toroidal Force Balance #  Toroidal equilibria solves the end losses of linear equilibrium, but generates a new force which must be balanced. This is a result of the virial theorem.\nPoloidal Fields (Tire Tube Pressure Force) #  If you think of a flexible bike tire being inflated, as the pressure within the inner tube increases, the major radius will increase! Why is this? The pressure within the tire is isotropic. As we pump up the tire, the pressure increase causes a force imbalance with the atmospheric pressure that causes the tube to expand. To simplify, we can consider a square tube with inner \u0026ldquo;radius\u0026rdquo;  \\( a \\) . The radial force scales with the pressure over the outer and inner surfaces\n \\[F_r = p S_{outer} - p S_{inner} \\\\ \\sim p (R_0 \u0026#43; a) a - p (R_0 - a) a \\sim 2 a^2 p\\]  !!! error \u0026ldquo;Missing content here on how to balance radial forces for purely poloidal fields (axisymmetric) with a conducting wall\u0026rdquo;\nAlternatively, we can use vertical external field coils to increase the field strength near the outer wall. They will also tend to center the plasma between the coils, so they have the advantage of preventing the plasma from drifting upwards or downwards\nToroidal Fields #  Now let\u0026rsquo;s consider driving a toroidal field in the plasma using a central coil on the major axis. Driving a toroidal field  \\( B_\\phi \\)  will also create a poloidal current  \\( j_\\theta \\) . We know that the driven field will decay as  \\( 1/R \\)  through the plasma. In actuality, the relationship is not perfectly  \\( 1/R \\)  because of the generated poloidal current, which will modify the field within the plasma. Depending on the orientation of  \\( j_\\theta \\) , it could either increase or decrease the poloidal field within the plasma. These are two different operations by which the plasma interacts with the magnetic field. When the induced plasma current tends to increase the field, we have a paramagnetic effect. In the opposite case (decrease) there is a diamagnetic effect. Generally, plasmas are diamagnetic, but there are certain situations where they become paramagnetic.\nHere again, we have a larger field on the in-bore side than on the out-bore side, so there will be a force imbalance tending to push the plasma towards larger radii. We might consider surrounding the plasma with a conducting wall, as we did previously, but we run into a difficulty determining a current distribution in the wall which would balance the radial effect. Any current distribution must be circular; because of the geometry of a torus such a current distribution would have the same force contribution on both the inside and outside edges. There is no way to stabilize the distribution with purely toroidal fields.\n"});index.add({'id':7,'href':'/r/notes/UWAA558/08-1d-equilibria/','title':"1-D Equilibria",'section':"MHD Theory",'content':"1-Dimensional Equilibria #  The   document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\( \\theta \\) -pinch #  In a  \\( \\theta \\)  pinch, we have an applied axial field generated by a driven azimuthal current distribution. The way these usually work is that you begin with a plasma generated by some pre-ionization process and zero field. Then you crank up the current to drive an azimuthal current in the plasma (in the opposite direction as the external current).\n \\[\\vec j \\cross \\vec B = \\grad p \\quad \\rightarrow \\quad j_\\theta B_z = \\dv{p}{r}\\]   \\[j_\\theta = - \\frac{1}{\\mu_0} \\dv{B_z}{r}\\]   \\[\\dv{p}{r} = - \\frac{1}{\\mu_0} B_z \\dv{B_z}{r} = - \\dv{}{r} \\left( \\frac{B_z ^2}{2 \\mu_0} \\right)\\]   \\[p \u0026#43; \\frac{B_z ^2}{2 \\mu_0} = \\text{constant} = \\frac{B_0 ^2}{2 \\mu_0}\\] \nAt equilibrium, the magnetic pressure balances the plasma pressure. If we say that the pressure is\n \\[p = p_0 e^{- r^2 / a^2}\\]  with  \\( p_0 \\)  the pressure on-axis, then we can solve for the axial field\n \\[B_z = B_0 (1 - B_0 e^{-r^2/a^2})^{1/2}\\]  We can define the peak  \\( \\beta \\)  to be the ratio of the on-axis pressure to the maximum magnetic field\n \\[\\beta_0 = \\frac{p_0}{B_0 ^2 / 2 \\mu_0}\\]  By definition, the peak  \\( \\beta \\)  will always be  \\( \\leq 1 \\) . We can define  \\( \\langle \\beta \\rangle \\) \n \\[\\langle \\beta \\rangle = \\frac{ \\langle p \\rangle }{B_a ^2 / 2 \\mu_0}\\]  where  \\( B_a \\)  is a characteristic field value, typically taken to be at the plasma edge.\n \\[\\langle \\beta \\rangle = \\frac{2 \\mu_0}{B_0 ^2 \\pi a^2} \\int _0 ^a 2 \\pi r p \\, \\dd r\\]   \\[= \\frac{2}{a^2} \\int_0 ^a \\frac{ rp}{B_0 ^2/2 \\mu_0} \\dd r = \\frac{2}{a^2} \\int_0 ^a \\left( 1 - \\frac{B_z ^2}{B_0 ^2} \\right) r \\dd r\\] \nIn this form, we can see that because  \\( B_z \\)  will everywhere be less than  \\( B_0 \\) , we can increase  \\( \\langle \\beta \\rangle \\)  by driving  \\( B_z \\)  as low as possible. In this particular example,  \\( \\langle \\beta \\rangle / \\beta_0 = 63\\% \\) .\nZ-pinch #  In the case of a Z-pinch, we only have an applied axial current.\n \\[\\vec j = j_z (r) \\vu z\\]  For force balance\n \\[\\vec j \\cross \\vec B = \\grad p \\quad \\rightarrow \\quad - j_z B_\\theta = \\dv{p}{r}\\]   \\[j_z = \\frac{1}{\\mu_0} \\frac{1}{r} \\dv{}{r} ( r B_\\theta) \\] \n \\[- \\dv{p}{r} = \\frac{B_\\theta}{\\mu_0 r} \\dv{}{r} ( r B_\\theta)\\]  If we find it convenient we can separate this into a magnetic pressure term\n \\[- \\dv{}{r} \\left( p \u0026#43; \\frac{B_\\theta ^2}{2 \\mu_0} \\right) = \\frac{B_\\theta ^2}{\\mu_0 r}\\]  Bennett Profile #  An example of an achievable distribution is the Bennett profile, which has a diffuse form\n \\[B_\\theta = \\frac{\\mu_0 I}{2 \\pi} \\frac{r}{r^2 \u0026#43; a^2}\\]   \\[j_z = \\frac{I}{\\pi} \\frac{a^2}{(r^2 \u0026#43; a^2) ^2}\\] \n \\[p = \\frac{\\mu_0 I^2}{8 \\pi ^2} \\frac{a^2}{(r^2 \u0026#43; a^2)^2}\\]  Interestingly,  \\( j \\propto p \\) . For a uniform temperature,  \\( j \\propto n \\) . Since current density is the product of  \\( \\vec v \\)  and  \\( n \\) , this says that we have a uniform drift velocity and all particles are drifting with the same velocity at all points along the profile. If we consider what the equilibrium profile looks like for a Bennett profile:\nSo for  \\( r \u0026lt; a \\)  we have magnetic tension and pressure which balance the plasma pressure. For  \\( r \\geq a \\)  we have magnetic tension which balances both plasma pressure and magnetic pressure.\nThe Z-pinch  \\( \\langle \\beta \\rangle \\) \n \\[\\langle \\beta \\rangle \\equiv \\frac{ \\langle p \\rangle}{B_a ^2 / 2 \\mu_0} \\\\ = \\frac{2 \\mu_0}{B_a ^2 \\pi a^2} \\int_0 ^a 2 \\pi r p \\dd r\\]  If we multiply the force balance by  \\( r^2 \\)  and integrate\n \\[\\int_0 ^a r^2 \\dv{p}{r} \\dd r \u0026#43; \\frac{1}{\\mu_0} \\int_0 ^a r B_\\theta \\dv{}{r} (r B_\\theta) \\dd r = 0 \\\\ 0 = \\left[ r^2 p \\right] _0 ^a - \\int_0 ^a p \\dd (r^2) \u0026#43; \\left[ \\frac{(r B_\\theta)^2}{2 \\mu_0} \\right] _0 ^a\\]  If we have a discrete pinch such that  \\( p(a) = 0 \\)  then the first term vanishes.\n \\[\\int_0 ^a 2 r p \\dd r = \\frac{(a B_a)^2}{2 \\mu_0}\\]  If we substitute our definition of  \\( \\langle \\beta \\rangle \\) , we find  \\( \\langle \\beta \\rangle = 1 \\) . For a diffuse pinch in which  \\( p(a) \\neq 0 \\)  we end up with  \\( \\langle \\beta \\rangle \\leq 1 \\)  and we have a wall-supported plasma. Ideal confinement ( \\( \\langle \\beta \\rangle = 1 \\) ) is a very nice property and is what makes the Z-pinch configuration so interesting.\nStability Considerations #  Instability results if there exists a plasma displacement that leads to a lower energy state. There are several ways to provide stability in the context of MHD. The two most common are magnetic shear and magnetic well.\nMagnetic Shear #  In ideal MHD, magnetic field lines can not break or tear. Let\u0026rsquo;s consider some flux surface containing field lines  \\( \\vec B_3 \\) . Behind it, we have another flux surface containing field lines  \\( B_2 \\)  which are not parallel to  \\( \\vec B_3 \\) , and the same for  \\( \\vec B_1 \\) .\nBecause the field lines are a different angles to each other, these flux surfaces can not interpenetrate. In other words, if the flux surface pressures are  \\( P_1 \u0026gt; P_2 \u0026gt; P_3 \\) , we can maintain the pressure gradient and prevent the flux surfaces from moving each other. What prevents the surfaces from achieving a lower energy state is the magnetic shear between flux surfaces.\nWithout shear, the surfaces can interpenetrate and exchange positions. In the case of a toroidal geometry, magnetic shear is defined by the rotational transform  \\( \\iota \\) , or by the safety factor\n \\[q = \\frac{2 \\pi}{\\iota} \\]  Generally speaking,  \\( q \\)  is generally referenced for tokamaks and  \\( \\iota \\)  is referenced for stellarators. Another way of picturing the safety factor in a toroidal geometry is\n \\[q \\equiv \\frac{\\text{no. of windings long way}}{\\text{no. of windings short way}} \\\\ = \\frac{ \\dv{\\psi_t}{V}}{\\dv{\\phi_p}{V}} = \\dv{\\phi_t}{\\phi_p} \\\\ = \\frac{n}{m} = \\frac{\\text{toroidal transits}}{\\text{poloidal transits}}\\]  In a cylindrical (1D) geometry it is just\n \\[q = \\frac{\\text{longitudinal transits}}{\\text{azimuthal transits}}\\]  Let\u0026rsquo;s calculate the safety factor for a toroidal geometry:\n \\[\\dv{\\phi_p}{V} = \\frac{ B_\\theta 2 \\pi R \\dd r}{2 \\pi R_0 2 \\pi r \\dd r} \\\\ = \\frac{ B_\\theta}{2 \\pi r} \\frac{R}{R_0}\\]   \\[\\dv{\\phi_t}{V} = \\frac{B_\\phi 2 \\pi r \\dd r}{2 \\pi R_0 2 \\pi r \\dd r} \\\\ = \\frac{B_\\phi}{2 \\pi R_0}\\]   \\[q = \\frac{r B_\\phi}{R B_\\theta}\\]  In a cylindrical geometry the analysis is even simpler\n \\[q = 2 \\pi \\frac{r B_z}{L B_\\theta}\\]  As a note, it would appear that  \\( q \\rightarrow 0 \\)  at the magnetic axis as  \\( r \\rightarrow 0 \\) , but in general  \\( B_\\theta \\rightarrow 0 \\)  as well, and the safety factor is generally bounded at  \\( r \\rightarrow 0 \\) \n \\( q \\)  is a flux surface quantity.\nWe care about magnetic shear. How does that relate to the safety factor? Magnetic shear is defined as\n \\[s \\equiv 2 \\frac{\\dd q / q}{\\dd V /V} = 2 \\dv{\\ln (q)}{\\ln(V)}\\]  Even a uniform  \\( B_z \\)  or  \\( B_\\theta \\)  produces a finite magnetic shear because of the way that  \\( r \\)  and  \\( B_\\theta \\)  change. The safety factor is often considered synonymous with magnetic shear, and often we don\u0026rsquo;t even compute  \\( s \\) .\nShear is generally a stabilizing effect. Interchange between flux surfaces can be prevented/inhibited by shear, or by making it energetically unfavorable. Shear tends to stabilize current-driven instabilities.\nMagnetic Well #  As before, we can consider two adjacent flux surfaces  \\( B_1, P_1 \\)  and  \\( B_2, P_2 \\) . If  \\( B_2 \u0026gt; B_1 \\)  and  \\( P_2 \u0026gt; P_1 \\) , the interchange is energetically favorable. But if  \\( B_2 \u0026gt; B_1 \\)  and  \\( P_2 \u0026lt; P_1 \\)  then the interchange may be unfavorable without any magnetic shear.\nConsider a plasma confined by an externally applied magnetic field generated by a coil  \\( I \\) \nOn the left side, the magnetic field gradient is in the same direction as the plasma pressure gradient, which is a destabilizing configuration. Flux surfaces are able to interchange easily, and the magnetic field is described as having bad curvature. On the right side, the gradients are in the same direction and the magnetic field has a good curvature.\nWe can define the \u0026ldquo;wellness\u0026rdquo;  \\( W \\)  as\n \\[W \\equiv \\frac{ \\text{total pressure change relative to mag. pressure}}{\\text{relative volume change}} \\\\ = \\frac{\\dd \\langle p \u0026#43; B^2/2 \\mu_0 \\rangle / \\langle B^2/2 \\mu_0 \\rangle}{\\dd V / V}\\]  where the angle brackets indicate a quantity integrated along a field line  \\[\\langle Q \\rangle \\equiv \\frac{\\int_0 ^L \\frac{ Q \\dd l}{|B|}}{\\int_0 ^l \\frac{\\dd l}{|B|}}\\] \nFor a stabilizing effect, the wellness must be greater than 0. This means that the magnetic pressure must increase faster than the pressure decreases to prevent pressure-driven instabilities.\nSince  \\( W \\)  is evaluated along a field line, it is also a surface quantity.\nApplication to 1D Equilibria #   \\( \\theta \\) -pinch: Since  \\( B_\\theta = 0 \\) ,  \\( q \\rightarrow \\infty \\) , which really just means  \\( q \\)  is not well defined for a  \\( \\theta \\) -pinch. If we consider some small  \\( \\delta B_\\theta \\) , we get a very large  \\( q \\) . From a magnetic shear perspective, a  \\( \\theta \\) -pinch has very large values of shear and very good stability properties.\nThe wellness is\n \\[W = \\frac{V}{\\langle B^2 \\rangle} \\dv{}{V} \\langle 2 \\mu_0 p \u0026#43; B^2 \\rangle \\\\ = \\frac{\\pi r^2 L}{B_z ^2} \\frac{1}{2 \\pi r L} \\dv{}{r} (2 \\mu_0 p \u0026#43; B_z ^2) \\\\ = \\frac{\\mu_0 r}{B_z ^2} \\dv{}{r} \\left( p \u0026#43; \\frac{B_z ^2}{2 \\mu_0} \\right) = 0 \\]  so a  \\( \\theta \\) -pinch has neutral magnetic well.\nVacuum case:\n \\[W = \\frac{\\mu_0 r}{ B_z ^2} \\dv{}{r} \\left( \\frac{ B_z ^2}{2 \\mu_0} \\right) = 0\\]  So vacuum magnetic fields also have neutral wellness. This leads to a general result sometimes referred to as \u0026ldquo;a plasma cannot dig its own well.\u0026rdquo; In other words, by introducing plasma to a magnetic configuration, it cannot make the configuration more stable than it was. Plasmas make stability more challenging, not less.\nZ-pinch:\nSince  \\( B_z = 0 \\) ,  \\( q = 0 \\)  and there is no magnetic shear. Even for a small value of  \\( \\delta B_z \\)  you still get a small  \\( q \\) . The magnetic well properties of a Z-pinch are\n \\[W = \\frac{V}{\\langle B^2 \\rangle} \\dv{}{V} \\langle 2 \\mu_0 p \u0026#43; B^2 \\rangle \\\\ = \\frac{\\mu_0 r}{B_\\theta ^2} \\dv{}{r} \\left( p \u0026#43; \\frac{B_\\theta ^2}{2 \\mu_0} \\right) \\\\ = \\frac{\\mu_0 r}{B_\\theta ^2} \\left( - \\frac{ B_\\theta ^2}{ \\mu_0 r} \\right) = -1\\]  Recall that  \\( W \u0026gt; 0 \\)  for stability, so the Z-pinch has negative magnetic well and provides no pressure stability.\nIn summary,\n Both  \\( \\theta \\) - and Z-pinch have high  \\( \\beta \\)   \\( \\theta \\) -pinch is stable Z-pinch is unstable End losses in a  \\( \\theta \\)  pinch enormous since  \\( k_\\parallel \\gg k_\\perp \\)   Screw Pinch #  A natural extension is to combine a moderate toroidal field and a moderate poloidal field to produce a screw pinch configuration.\n \\[\\curl \\vec B = \\mu_0 \\vec j\\]   \\[\\rightarrow j_\\theta = - \\frac{1}{\\mu_0 } \\dv{B_z}{r} \\]   \\[j_z = \\frac{1}{\\mu_0 r} \\dv{}{r} (r B_\\theta)\\] \nFor static MHD equilibrium\n \\[\\vec j \\cross \\vec B = \\grad p\\]   \\[\\dv{}{r} \\left( p \u0026#43; \\frac{ B_\\theta ^2 \u0026#43; B_z ^2}{2 \\mu_0} \\right) = - \\frac{B_\\theta ^2}{\\mu_0 r}\\] \nWe can define a toroidal  \\( \\beta \\)  where  \\( B_0 = B_z (a) \\)   \\[\\beta_t = \\frac{\\langle p \\rangle }{B_0 ^2 / 2 \\mu_0} \\\\ = \\frac{2 \\mu_0}{ B_0 ^2} \\frac{1}{\\pi a^2} \\int _0 ^a 2 \\pi r p \\dd r\\] \nand in the poloidal direction with  \\( B_{\\theta, a} = B_\\theta (a) = \\frac{\\mu_0 I}{2 \\pi a} \\) \n \\[\\beta_p = \\frac{\\langle p \\rangle}{B_{\\theta, a} ^2 / 2 \\mu_0} \\\\ = \\frac{8 \\pi ^2 a^2}{ \\mu_0 I_0 ^2} \\left( \\frac{1}{\\pi a^2} \\int _0 ^a 2 \\pi r p \\dd r \\right) \\\\ = \\frac{16 \\pi ^2}{\\mu_0 I_0 ^2} \\int_0 ^a r p \\dd r\\]  To proceed we can multiply the force balance by  \\( r^2 \\)  and integrate\n \\[\\underbrace{\\int_0 ^a r^2 \\pdv{p}{r} \\dd r}_{(1)} \u0026#43; \\underbrace{\\int _0 ^a \\pdv{}{r} \\left( \\frac{ B_\\theta ^2 \u0026#43; B_z ^2 }{2 \\mu_0} \\right) \\dd r}_{(2)} \u0026#43; \\underbrace{\\int_0 ^a r^2 \\frac{B_\\theta ^2}{\\mu_0 r} \\dd r}_{(3)} = 0\\]   \\[(1) = \\int_0 ^a r^2 \\dd p = \\left. r^2 p \\right|_0 ^a - \\int_0 ^a p \\dd (r^2) = - \\int_0 ^a 2 r p \\dd r\\]   \\[(2) = \\int_0 ^a r^2 \\dd \\left( \\frac{B_\\theta ^2 \u0026#43; B_z ^2}{2 \\mu_0 } \\right) \\\\ = \\left. r^2 \\left( \\frac{ B_\\theta ^2 \u0026#43; B_z ^2}{2 \\mu_0 } \\right) \\right|_0 ^a - \\int_0 ^a r \\left( \\frac{ B_\\theta ^2 \u0026#43; B_z ^2}{\\mu_0} \\right) \\dd r\\]   \\[(3) = \\int_0 ^a r \\frac{B_\\theta ^2}{\\mu_0} \\dd r\\]  Combining we have\n \\[- \\int_0 ^a 2 r p \\dd r \u0026#43; \\frac{a^2 B_{\\theta, a}^2}{2 \\mu_0} \u0026#43; \\overbrace{\\frac{a^2 B_0 ^2}{2 \\mu_0}}^{B_0 = B_z(r = a)} - \\int_0 ^a r \\frac{B_z ^2}{\\mu_0} \\dd r = 0 \\\\ - \\int_0 ^a 2 r p \\dd r \u0026#43; \\frac{ \\mu_0 I_0 ^2}{8 \\pi ^2} \u0026#43; \\int_0 ^a r \\left( \\frac{ B_\\theta ^2 - B_z ^2}{\\mu_0} \\right) \\dd r = 0\\]  Dividing  \\( \\int_0 ^a 2 r p \\dd r \\)  gives\n \\[\\left[ \\frac{16 \\pi^2}{\\mu_0 I^2} \\int_0 ^a r p \\dd r \\right] ^{-1} \u0026#43; \\left[\\frac{4 \\mu_0}{B_0 ^2 a^2} \\int_0 ^a r p \\dd r \\right] ^{-1} \\frac{2}{a^2} \\int_0 ^a \\left(1 - \\frac{B_z ^2}{B_0 ^2} \\right) r \\dd r = 1\\]  or\n \\[\\frac{1}{\\beta_p} \u0026#43; \\frac{\\alpha_t}{\\beta_t} = 1\\]  where  \\[\\alpha_t = \\frac{2}{a^2} \\int_0 ^a \\left( 1 - \\frac{B_z^2}{B_0 ^2} \\right) r \\dd r\\]  is the diamagnetism.\n \\[\\beta _p = \\left( 1 - \\frac{\\alpha_t}{\\beta_t} \\right) ^{-1}\\]  If we have a diamagnetic current, then  \\( \\alpha \u0026gt; 0 \\) . This maximizes confinement, since we have confinement in the azimuthal field, as well as the axial field. The limit where you have a skin current such that  \\( B_z = 0 \\)  inside the plasma results in the best confinement and  \\( \\alpha _t = 1 \\) .\nLooking at the safety factor,\n \\[q = \\frac{2 \\pi r B_z}{L B_\\theta}\\]  If we look at the edge  \\( r = a \\) ,  \\[q_a = \\frac{2 \\pi a B_0}{L B_{\\theta, a}}\\] \nAs it turns out, this value of the edge safety factor is critically important, and for stability we require that  \\( q_a \u0026gt; 1 \\) .\nThe magnetic shear of a screw pinch is\n \\[s = 2 \\frac{V}{q} \\dv{q}{V} \\\\ V = \\pi r^2 L \\\\ \\dd V = 2 \\pi L r \\dd r = \\frac{2 V}{r} \\dd r \\\\ s = \\frac{r}{q} \\dv{q}{r}\\]  The shear can be adjusted by changing the applied axial field.\nThe magnetic well is\n \\[W = \\frac{V}{B^2} \\dv{(2 \\mu_0 p \u0026#43; B^2)}{V} \\\\ = \\frac{\\mu_0 r}{B^2} \\dv{}{r} \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\\\ = \\frac{\\mu_0 r}{B^2} \\left( - \\frac{B_\\theta ^2}{\\mu_0 r} \\right) \\\\ = - \\frac{B_\\theta ^2}{B^2} \\\\ = - \\frac{B_\\theta ^2}{B_\\theta ^2 \u0026#43; B_z ^2} \\\\ = - \\left( 1 \u0026#43; \\frac{B_z ^2}{B_\\theta ^2} \\right) ^{-1}\\]  So the well is always less than zero, but adding  \\( B_z \\)  improves the well.\nBy combining the properties of  \\( \\theta \\) -pinch and Z-pinch, we are able to sacrifice some  \\( \\beta \\)  to achieve better stability properties. Of course, we have not addressed the end losses in any way; to do that, we need to connect the ends.\n"});index.add({'id':8,'href':'/r/notes/UWAA558/09-2d-equilibria/','title':"2D Equilibria",'section':"MHD Theory",'content':"2D Equilibria #  Let\u0026rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:\n  document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\[\\vec j \\cross \\vec B = j_\\theta B_z - j_z B_\\theta \\\\ = \\grad p = \\dv{p}{r}\\]  we move to an axisymmetric 2-dimensional torus, replacing our cylindrical coordinate system with a toroidal one\n \\[\\vec j \\cross \\vec B = \\vec j_\\theta \\cross \\vec B_\\phi \u0026#43; \\vec j_\\phi \\cross \\vec B_\\theta = \\grad p\\]  Eventually, the toroidal force balance will lead to the Grad-Shafranov Equation, which tells us how we can solve for a general equilibrium that solves  \\( \\vec j \\cross \\vec B = \\grad p \\) .\nLet\u0026rsquo;s consider how we might achieve such a configuration. A toroidal magnetic field can be achieved by driving current through a poloidal coil. A more complicated problem is how to drive toroidal current. In general this is done by means of a transformer, where the plasma itself is the secondary circuit. Driving a time-varying current through the primary induces a toroidal current through the plasma. This is called a transformer drive for current.\nGrad-Shafranov equation #  Computing  \\( j_\\theta \\)  and  \\( B_\\phi \\)  can be computationally difficult in a toroidal geometry, so let\u0026rsquo;s do some work towards simplifying our force balance expression. The toroidal magnetic vector potential is defined as\n \\[\\vec B_\\theta = \\curl \\vec A_\\phi\\]  If we integrate  \\( B_\\theta \\)  over a poloidal surface, Stokes' theorem gives\n \\[\\int _{S_p} \\curl \\vec A_\\phi \\cdot \\dd \\vec S = \\oint \\vec A_\\phi \\cdot \\dd \\vec l \\\\ = \\int _{S_p} B_\\theta \\cdot \\dd \\vec S = \\Psi _p \\]  If the equilibrium is axisymmetric,  \\( A_\\phi \\)  must be uniform along  \\( \\dd l \\) , so\n \\[A_\\phi \\vu \\phi \\cdot \\oint \\dd \\vec l = A_\\phi 2 \\pi R = \\Psi _p \\\\ \\rightarrow A_\\phi = \\frac{ \\Psi_p}{R} \\vu \\phi\\]  where we absorb the factor of  \\( 2 \\pi \\)  into the poloidal flux  \\( \\Psi _p \\) . After some manipulation, we can relate  \\( B_\\theta \\)  to the poloidal flux\n \\[\\vec B_\\theta = \\curl \\vec A_\\phi = - \\frac{ \\vu R}{R} \\pdv{\\Psi}{z} \u0026#43; \\frac{\\vu z}{R} \\pdv{\\Psi}{R}\\]   \\[\\mu_0 j_\\phi \\cross B_\\theta = (\\curl \\vec B_\\theta) \\cross \\vec B_\\theta \\\\ = \\curl \\vec B_\\theta \\cross \\left( \\grad \\Psi \\cross \\frac{\\vu \\phi}{R} \\right) \\\\ = - \\left[ \\pdv{}{R} \\left( \\frac{1}{R} \\pdv{\\Psi}{R} \\right) \u0026#43; \\frac{1}{R} \\pdv{\\Psi ^2}{z^2} \\right] \\cdot \\left[\\frac{ \\grad \\Psi}{R} ( \\vu \\phi \\cdot \\vu \\phi) - \\frac{ \\phi}{R} \\cancel{(\\grad \\Psi \\cdot \\vu \\phi)} \\right]\\]  That gives the first component of  \\( \\grad p \\) , now let\u0026rsquo;s do the other one\n \\[\\mu_0 \\vec j_\\theta \\cross \\vec B_\\phi = ( \\curl \\vec B_\\phi) \\cross \\vec B_\\phi \\\\ = \\left[ - \\vu R \\pdv{B_\\phi}{z} \u0026#43; \\vu z \\frac{1}{R} \\pdv{}{R} ( R B_\\phi) \\right] \\cross \\vec B_\\phi \\\\ = - \\frac{B_\\phi}{R} \\left[ \\vu R \\pdv{}{R} (R B_\\phi) \u0026#43; \\vu z \\pdv{}{z} (R B_\\phi) \\right] \\\\ = - \\frac{B_\\phi}{R} \\grad (R B_\\phi)\\]  Finally, since pressure is a flux surface quantity we can write\n \\[\\grad p = \\dv{p}{\\Psi} \\grad \\Psi = p\u0026#39; \\grad \\Psi\\]  The toroidal force balance now looks like\n \\[\\mu_0 p\u0026#39; \\grad \\Psi = - \\frac{1}{R} \\left( \\pdv{}{R} \\frac{1}{R} \\pdv{\\Psi}{R} \u0026#43; \\frac{1}{R} \\pdv{^2 \\Psi}{z^2} \\right) \\grad \\Psi - \\frac{B_\\phi}{R} \\grad(R B_\\phi)\\]  We notice that the only vector quantities here are  \\( \\grad \\Psi \\)  and  \\( \\grad (R B_\\phi) \\) , so  \\( \\grad (R B_\\phi) \\)  must be parallel to  \\( \\grad \\Psi \\)  and is a flux surface quantity. We can define our new flux surface quantity as\n \\[F(\\Phi) \\equiv R B_\\phi = \\frac{\\mu_0 I_\\theta}{2 \\pi} = \\frac{\\mu_0}{2 \\pi} \\int_{S_p} \\vec j_\\theta \\cdot \\dd \\vec S\\]   \\[\\grad F = \\dv{F}{\\Psi} \\grad \\Psi = F\u0026#39; \\grad \\Psi\\]  Now each term in the toroidal force balance has a factor of  \\( \\grad \\Psi \\)  attached. Let\u0026rsquo;s multiply through by  \\( R^2 \\)  and factor out the gradient to arrive at the Grad-Shafranov equation:\n \\[R^2 \\mu_0 p\u0026#39; = - \\Delta ^\\star \\Psi - F F\u0026#39;\\]  where  \\[\\Delta ^\\star \\equiv R \\pdv{}{R} \\frac{1}{R} \\pdv{}{R} \u0026#43; \\pdv{^2}{z^2}\\] \nTo solve the Grad-Shafranov equation, you solve for  \\( \\Psi(R, z) \\) , which determines  \\( p(\\Psi) \\)  and  \\( F(\\Psi) \\) , which directly gives you  \\( p(R, z) \\)  and  \\( F(R, z) \\)  and completely defines the equilibrium.\nYou can solve for the other terms as well. Since  \\( \\vec B_\\theta = \\frac{\\grad \\Psi}{R} \\cross \\vu \\phi \\) \n \\[\\vec j_\\phi = - \\frac{1}{\\mu_0 R} \\Delta ^\\star \\Psi \\vu \\phi\\]  and since  \\( \\vec B_\\phi = \\frac{F}{R} \\vu \\phi \\) \n \\[\\vec j_\\theta = - \\frac{1}{\\mu_0 R} \\grad (R B_\\phi) \\cross \\vu \\phi\\]  For the G-S equation to be solvable, you need to specify the equilibrium by specifying  \\( p(\\Phi) \\)  and  \\( F(\\Phi) \\) . In practice, this is usually done by making experimental measurements to determine  \\( p \\)  and  \\( F \\) . A common code that does this is called EFIT, which takes the boundary conditions of the magnetic field and measurements of temperature, density to perform a least-squares fit to solve the G-S equation.\nIn general, the Grad-Shafranov equation leads to a matrix equation\n \\[\\overline \\vec A \\vec \\Psi \u0026#43; \\vec f(\\Psi) = \\vec g\\]  Depending on the conditions we place on  \\( \\Psi \\) ,  \\( \\vec f(\\Psi) \\)  can be a nonlinear function.\nSolutions to the Grad-Shafranov equation #  In the limit that  \\( \\vec j \\parallel \\vec B \\) , then  \\( \\vec j \\cross \\vec B = 0 = \\grad p \\rightarrow p\u0026#39; = 0 \\) . These are called force-free states. In the G-S equation, the pressure term vanishes and we\u0026rsquo;re left with\n \\[\\Delta ^\\star \\Psi \u0026#43; F F\u0026#39; = 0\\]  Spheromaks and RFPs are examples of nearly force-free states in which the current is nearly parallel to the magnetic field. Notice that in completely force-free states,  \\( \\langle \\beta \\rangle = 0 \\) .\nAnother interesting limit is the case where  \\( F F\u0026#39; \\gg \\Delta ^\\star \\Psi \\) . Now we have  \\[\\grad p \\approx \\vec j_\\theta \\cross \\vec B_\\phi\\]  which looks like a  \\( \\theta \\) -pinch which has been connected at the ends. Remember from the previous section that we can not maintain radial force balance with purely toroidal fields, so the toroidal current is not zero (hence the  \\( \\approx \\) ) but is just high enough to maintain radial force balance. This sort of configuration is called a high- \\( \\beta \\)  tokamak.\nThe other limit is  \\( F F\u0026#39; \\ll \\Delta ^\\star \\Psi \\)   \\[\\grad p \\approx \\vec j_\\phi \\cross \\vec B_\\theta\\]  which looks like an end-connected z-pinch. This configuration is usually called an Ohmically heated Tokamak, and the majority of currently operating tokamaks operate this way. As we know, a purely poloidal field has very bad stability properties, so  \\( \\vec B_\\phi \\)  needs to be added to provide stability. The toroidal  \\( \\beta \\)  is very small  \\[\\beta _t \\ll 1 \\qquad \\beta _p \\approx 1\\] \nStability Considerations #  The same stability factors exist in 2D equilibria that we found for 1D equilibria:\nMagnetic shear - the safety factor  \\( q(\\Psi) = \\frac{\\Delta \\phi}{\\Delta \\theta} \\)  for  \\( \\Delta \\theta = 2 \\pi \\) . We can calculate  \\( q \\)  more easily by integrating along a flux surface in the poloidal plane:\n \\[q(\\Psi) = \\frac{1}{2 \\pi} \\int_0 ^{\\Delta \\phi} \\dd \\phi \\\\ = \\frac{1}{2 \\pi} \\int _0 ^{2 \\pi} \\dv{\\phi}{\\theta} \\dd \\theta \\\\ = \\frac{1}{2 \\pi} \\int_0 ^{2 \\pi} \\dd \\theta \\left. \\frac{r B_\\phi}{R B_\\theta} \\right|_{\\text{along flux surf.}} \\\\ = \\frac{F(\\Psi)}{2 \\pi} \\oint \\frac{ \\dd l_p}{R^2 B_\\theta} \\qquad \\dd l_p = r \\dd \\theta\\]  Magnetic well: similarly we can get the magnetic well factor by integrating around a flux surface in the poloidal plane\n \\[\\langle Q \\rangle = \\frac{\\oint \\frac{Q \\dd l_p}{B_\\theta}}{\\oint \\frac{ \\dd l_p}{B_\\theta}}\\]  Shafranov Shift #  Remember that when we had an equilibrium which had a toroidal current and a corresponding poloidal magnetic field, and a poloidal magnetic field, then radial force balance will tend to shift the configuration outwards away from the major axis and a conducting wall or external coil will be required to maintain the equilibrium. The radial force balance is really achieved by  \\( \\vec j_\\phi \\cross B_p \\) \nAs we move towards the magnetic axis,  \\( B_p \\rightarrow 0 \\)  by definition. With less poloidal field to balance the radial force imbalance, there is more radial expansion. This means that inner portion of the plasma (inner flux surfaces) must shift radially further to achieve radial force balance.\nThe shift increases with plasma pressure. This effect is further enhanced if we have low poloidal fields, for example in the high- \\( \\beta \\)  tokamak configurations.\nLow aspect ratios also enhance the effect. Recall that the radial force imbalance increases with smaller aspect ratio, leading to a larger shift.\n"});index.add({'id':9,'href':'/r/notes/UWAA558/10-equilibrium-of-3d-configurations/','title':"Equilibrium of 3D Configurations",'section':"MHD Theory",'content':"Equilibrium of 3D Configurations #  In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated.\nSome general features of 3D equilibria are:\n No net toroidal current. This means that they tend to be steady-state configurations. Radial confinement is accomplished by toroidal fields, as in the end-connected   document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\( \\theta \\) -pinch. As we saw, toroidal fields cannot provide radial confinement in a purely axisymmetric configuration, but radial variation with  \\( \\phi \\)  can provide confinement. Toroidal effect (radial force balance) is generated by helical magnetic fields. You can do an expansion of the magnetic field into a toroidal component, and a helical component that traces out a twisted shape as you move around the torus. These twisted shapes are what lead to radial confinement.  ELMO Bumpy Torus (EBT) #  In contrast to most other 3D configurations, even though the EBT is a 3D equilibrium, it has no helical windings.\nSince there are no helical windings, we have to provide radial stability in another way. In the EBT configuration, you also drive hot poloidal electron rings (driven by electron cyclotron resonance) to provide both stability and heating.\nStellarator #  The stellarator configuration is composed of a number of helical current lines (generated by helical coils with alternating currents), and a net toroidal field driven by poloidal coils. The direction of the currents alternate, for a total of  \\( 2l \\)  current lines.\nThe result is a net magnetic field with a ratio such that  \\( B_{\\text{helical}} \\gg B_\\phi \\) \nStellarators raise some very complicated engineering challenges both in the design and construction of the complicated geometry. It is also very difficult to maintain no net current within the plasma, especially during start-up. As you add plasma, you raise from zero  \\( \\beta \\)  to a finite  \\( \\beta \\) , introducing things like bootstrap currents that need to be balanced.\nTorsatron #  Similar to a stellarator, the torsatron does not have alternating currents. All of the helical current lines are in the same direction. There are also no toroidal field coils.\nThe engineering is slightly simpler, but it is slightly less efficient at generating the helical magnetic field.\nThe flux surfaces in stellarators and torsatrons have geometrical cross-sections depending on the number  \\( l \\)  of helical current lines. About the current lines, the flux surfaces are nearly circular. The flux surfaces within the plasma volume are determined by the separatrix of the helical coil fields.\n"});index.add({'id':10,'href':'/r/notes/UWAA558/11-mhd-stability/','title':"MHD Stability",'section':"MHD Theory",'content':"MHD Stability #  Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear. We can define the initial deviation from equilibrium to be a linear phenomenon. As usual, we perform this linearization by letting   document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\( Q(r, t) = Q_0 \u0026#43; Q_1(r, t) \\)  with  \\( Q_1 \\)  being a small first-order perturbation. Since the equilibrium is both time and space independent, the general form of the perturbation is\n \\[Q_1(r, t) = \\vu Q_1 e^{-i (\\omega t - \\vec k \\cdot \\vec r)}\\]   \\[\\grad p_0 = \\vec j_0 \\cross \\vec B_0\\]   \\[p = p_0 \u0026#43; p_1 \\qquad \\rho = \\rho_0 \u0026#43; \\rho_1\\]   \\[\\vec j = \\vec j_0 \u0026#43; \\vec j_1 \\qquad \\vec B = \\vec B_0 \u0026#43; \\vec B_1\\] \nand for a static equilibrium  \\[\\vec v = \\vec v_1\\] \nIn our momentum equations of the perturbed quantities, we assume that the static equilibrium holds, so most of the equilibrium terms drop out. We can define a velocity displacement  \\( \\vec \\xi = \\int_0 ^t \\vec v_1 \\dd t \\) . As we integrate the field and pressure in time,\n \\[\\pdv{B_1}{t} = \\curl (\\vec v_1 \\cross \\vec B_0)\\]   \\[\\int \\pdv{B_1}{t} = \\vec B_1 = \\curl \\int \\vec v_1 \\cross \\vec B_0 \\dd t \\\\ = \\curl (\\vec \\xi \\cross \\vec B_0)\\] \nIf we do the same for the pressure equation, we get\n \\[p_1 = - \\vec \\xi \\cdot \\grad p_0 - \\Gamma p_0 \\div \\vec \\xi\\]  where  \\( \\Gamma \\)  is the ratio of specific heats, to avoid confusion with typical perturbation growth rate  \\( \\gamma \\) .\nIf we combine all of these together, substituting into the momentum equation, we can express the perturbation entirely in terms of  \\( \\vec \\xi \\)  and the equilibrium properties:\n \\[\\rho_0 \\pdv{ ^2 \\vec \\xi }{t^2} = \\grad (\\vec \\xi \\cdot \\grad p_0 \u0026#43; \\Gamma p_0 \\div \\vec \\xi)\\\\ \u0026#43; \\frac{1}{\\mu_0} \\left[(\\curl \\vec B_0) \\cross \\curl (\\vec \\xi \\cross \\vec B_0) \\right] \\\\ \u0026#43; \\frac{1}{\\mu_0} \\left[ \\curl \\curl ( \\vec \\xi \\cross \\vec B_0) \\cross \\vec B_0 \\right]\\]  We define the right-hand-side as the linearized forcing function of our equilibrium  \\[\\rho_0 \\pdv{ ^2 \\vec \\xi }{t^2} = \\vec F(\\vec \\xi _i , p_0, \\vec B_0)\\] \nFor a linear force function, we can also write it in terms of a spring constant tensor\n \\[\\rho_0 \\pdv{ ^2 \\vec \\xi }{t^2} = \\vec F(\\vec \\xi) = - \\overline \\vec K \\cdot \\vec \\xi\\]  We can determine the stability behavior of a configuration by specifying an initial condition\n \\[\\vec \\xi (t = 0) = 0 \\qquad \\text{and} \\qquad \\left. \\pdv{\\xi}{t} \\right| _{t = 0} = f(\\vec r)\\]  and boundary conditions. A boundary condition may be a rigid wall  \\[\\vec \\xi \\cdot \\vu n |_{wall} = 0\\] \nOne way we can tell whether a given solution is unstable is to assume a variation of the form  \\[\\vec \\xi \\propto e^{-i \\omega t}\\]  If  \\( \\omega^2 \u0026gt; 0 \\) , the displacement will oscillate in time without growth, and if  \\( \\omega^2 \u0026lt; 0 \\)  then the displacement will grow. In other words, if  \\( \\omega \\)  is real, then the mode is stable, and if  \\( \\omega \\)  is imaginary then the mode is unstable. The eigenvalue equation to be solved is\n \\[- \\omega ^2 \\rho_0 \\vec \\xi = \\vec F(\\vec \\xi)\\]  which we can write as a matrix equation  \\[\\overline \\vec A\\, \\overline X = \\lambda \\overline X\\]   \\[\\frac{1}{\\rho_0} \\vec F (\\vec \\xi) = - \\omega ^2 \\vec \\xi\\] \nFor any arbitrary linear forcing function, we might get an infinite number of eigenvalues. How do we know which ones to look at? It turns out that the linearized force function  \\( \\vec F(\\vec \\xi) \\)  has the property of being self-adjoint, so  \\[\\int \\vec \\eta \\cdot \\vec F(\\vec \\xi) \\dd V = \\int \\xi \\cdot \\vec F( \\vec \\eta) \\dd V\\]  where  \\( \\vec \\eta \\)  and  \\( \\vec \\xi \\)  are arbitrary displacements that satisfy the same boundary conditions. If  \\( \\vec F \\)  is self-adjoint, then the system is Hermitian, which guarantees that we get real eigenvalues ( \\( \\omega^2 \\) ) , orthogonal eigenfunctions, and most importantly we are guaranteed to have an ordered spectrum of eigenvalues. That is to say  \\( \\omega_0 ^2 \u0026lt; \\omega _1 ^2 \u0026lt; \\omega _2 ^2 \u0026lt; \\ldots \\) . This means that the eigenvalue of the lowest mode is guaranteed to be the most negative, and therefore dictates the stability of the system. If the lowest eigenvalue is negative, then the system is necessarily unstable, and if the lowest eigenvalue is positive, then we are guaranteed that all modes are stable.\nBecause  \\( \\vec F \\)  is self-adjoint, we can make use of the energy principle to write the variation in the sum of the kinetic and potential energy as:\n \\[0 = \\dv{}{t} \\left[ \\frac{1}{2} \\int \\rho_0 \\left( \\pdv{\\vec \\xi}{t} \\right) ^2 \\dd V - \\frac{1}{2} \\int \\vec \\xi \\cdot \\vec F ( \\vec \\xi) \\dd V \\right]\\]  The kinetic energy term will always be positive, so we can formulate the stability based on the potential energy, often called a  \\( \\delta W \\)  approach  \\[\\delta W = - \\frac{1}{2} \\int \\vec \\xi \\cdot \\vec F ( \\vec \\xi) \\dd V\\]  is the change in potential energy due to a displacement  \\( \\xi \\) . If the potential energy decreases due to a displacement  \\( \\xi \\) , then the kinetic energy must necessarily increase, so  \\( \\delta W \u0026lt; 0 \\)  indicates instability.\nWe can write the change in kinetic energy for our normal mode decomposition as  \\[\\delta T = \\frac{1}{2} \\int \\rho _0 \\left( \\pdv{\\xi}{t} \\right) ^2 \\dd V = - \\frac{1}{2} \\omega ^2 \\int \\rho _0 \\vec \\xi ^ \\star \\cdot \\vec \\xi \\dd V \\\\ = - \\delta W = \\frac{1}{2} \\int \\vec \\xi ^\\star \\cdot \\vec F(\\vec \\xi) \\dd V\\]   \\[\\omega^2 = \\frac{- \\int \\xi ^\\star \\cdot \\vec F \\dd V}{\\int \\rho_0 \\xi ^\\star \\cdot \\xi \\dd V} = \\frac{\\delta W}{\\frac{1}{2} \\int \\rho_0 \\xi ^\\star \\cdot \\xi \\dd V}\\] \nThe denominator is strictly positive, so the sign of  \\( \\omega^2 \\)  is determined by the sign of  \\( \\delta W \\) \n \\[\\delta W \u0026lt; 0 \\rightarrow \\omega^2 \u0026lt; 0 \\rightarrow \\text{unstable} \\\\ \\delta W \u0026gt; 0 \\rightarrow \\omega^2 \u0026gt; 0 \\rightarrow \\text{stable}\\]  Analyzing the form of  \\( \\delta W \\)  (within the plasma volume)\n \\[\\delta W = \\frac{1}{2} \\int_{plasma} \\dd V \\Gamma p_0(\\div \\vec \\xi) ^2 \u0026#43; \\vec \\xi \\cdot \\grad p_0 (\\div \\vec \\xi) \\qquad \\qquad \\\\ \\qquad \\qquad \u0026#43; \\frac{1}{\\mu_0} \\left[ \\curl ( \\vec \\xi \\cross \\vec B_0) \\right]^2 \\\\ \\qquad \\qquad - \\frac{1}{\\mu_0} \\left[\\vec \\xi \\cross ( \\curl \\vec B_0) \\right] \\cdot \\left[ \\curl ( \\vec \\xi \\cross \\vec B_0) \\right]\\]  Generally speaking, the plasma volume does not extend to infinity, and we care very much about the boundary. The total  \\( \\delta W \\)  is the sum of that in the plasma volume  \\( \\delta W_F \\) , the surface  \\( \\delta W_S \\) , and the vacuum region  \\( \\delta W_V \\) . The vacuum term looks like\n \\[\\delta W_V = \\frac{1}{2} \\int _{vac} \\dd V \\frac{ (\\curl ( \\vec \\xi \\cross \\vec B_0))^2}{\\mu_0} = \\int_{vac} \\dd V \\frac{\\vec B_1 ^2}{\\mu_0} \u0026gt; 0\\]  so the vacuum term is always positive, and has a stabilizing influence. The surface contribution offsets this\n \\[\\delta W_S = \\frac{1}{2} \\oint \\dd S ( \\vu n \\cdot \\vec \\xi) ^2 \\left[ \\left[ \\grad \\left( p_0 \u0026#43; \\frac{B_0^2}{2 \\mu_0} \\right) \\right] \\right] \\cdot \\vu n\\]  Instabilities can be characterized as:\n Internal/fixed boundary  \\( \\delta W = \\delta W_F \\)  External/free boundary  \\( \\delta W = \\delta W_F \u0026#43; \\delta W_S \u0026#43; \\delta W_V \\)   The plasma portion can be re-written slightly as\n \\[\\delta W_F = \\frac{1}{2} \\int \\dd V \\frac{ |B_{1, \\perp}|^2}{\\mu_0} \\quad \\leftarrow \\text{Shear Alfven} \\\\ \u0026#43; \\mu_0 \\left| \\frac{B_{1, \\parallel}}{\\mu_0} - \\frac{B_0 \\xi \\cdot \\grad p_0}{B_0} ^2 \\right|^2 \\quad \\leftarrow \\text{Fast magnetosonic} \\\\ \u0026#43; \\Gamma p_0 |\\div \\xi|^2 \\quad \\leftarrow \\text{Acoustic}\\\\ \u0026#43; \\frac{\\vec j_0 \\cdot \\vec B_0}{B_0 ^2} (\\vec B_0 \\cross \\vec \\xi) \\cdot \\vec B_1 \\quad \\leftarrow \\text{Current-driven (kink)} \\\\ - 2 ( \\vec \\xi \\cdot \\grad p_0)(\\vec \\xi \\cdot \\vec \\kappa) \\quad \\leftarrow \\text{pressure-driven (interchange/balooning)}\\]  where  \\( \\vec \\kappa \\)  is the curvature vector  \\( \\vu e_B \\cdot \\grad \\vu e_B \\) . If we look at each of these terms, the first three terms are all going to be stabilizing effects, which means that all instability is going to come from the last two terms, the current-driven instability term and the pressure-driven instability term.\nGoing back to the screw pinch,  \\[\\dv{p}{r} = j_\\theta B_z - j_z B_\\theta\\]  we have current in the same direction as magnetic field ( \\( j_\\theta \\)  with  \\( B_\\theta \\)  and  \\( j_z \\)  with  \\( B_z \\) ), so kink instabilities are possible. We also have a pressure gradient, so interchange instabilities are also possible.\nAs a concrete example, look at the pressure driven instability term in a Z-pinch.  \\[\\kappa = - \\frac{ \\vu r}{r}\\]   \\[\\vec \\xi = \\xi _r \\vu r\\]   \\[\\delta W_{F, pressure} = \\int \\dd V \\xi _r \\dv{p_0}{r} \\frac{\\xi_r}{r} \\\\ = \\int \\dd V \\frac{2 \\xi _r ^2}{r} \\dv{p_0}{r}\\] \nIn a Z-pinch, it is always the case that  \\( \\dv{p_0}{r} \u0026lt; 0 \\) . As shown by Kadomtsev (1965) it turns out that these modes can be stabilized by adding  \\( B_z \\) , but this also introduces kink modes.\nGoing back to our stabilizing quantities of wellness and shear, current-driven instabilities are generally managed through shear, and pressure-driven instabilities are stabilized by well.\n"});index.add({'id':11,'href':'/r/notes/UWAA558/formulary/','title':"Formulary",'section':"MHD Theory",'content':"Formulary #  Example #  {{\u0026lt; katex [display] [class=\u0026#34;text-center\u0026#34;] \u0026gt;}} f(x) = \\int_{-\\infty}^\\infty\\hat f(\\xi)\\,e^{2 \\pi i \\xi x}\\,d\\xi {{\u0026lt; /katex \u0026gt;}}     document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: {  \u0026quot;\\\\lcm\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{lcm}}\u0026quot;, \u0026quot;\\\\sen\u0026quot;: '\\\\text{sen}\\\\,', \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\mathop{\\\\mathrm{d} #1}\u0026quot;, \u0026quot;\\\\abs\u0026quot;: \u0026quot;\\\\lvert #1 \\\\rvert\u0026quot;, \u0026quot;\\\\dd\u0026quot;: \u0026quot;\\\\text{d}\u0026quot;, \u0026quot;\\\\cross\u0026quot;: \u0026quot;\\\\times\u0026quot;, \u0026quot;\\\\pdv\u0026quot;: \u0026quot;\\\\frac{\\\\partial #1}{\\\\partial #2}\u0026quot;, \u0026quot;\\\\curl\u0026quot;: \u0026quot;\\\\nabla \\\\cross #1\u0026quot;, \u0026quot;\\\\vu\u0026quot;: \u0026quot;\\\\mathbf{\\\\hat{#1}}\u0026quot;, \u0026quot;\\\\vdot\u0026quot;: \u0026quot;\\\\dot\u0026quot;, \u0026quot;\\\\div\u0026quot;: \u0026quot;\\\\nabla \\\\cdot #1\u0026quot;, \u0026quot;\\\\grad\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, \u0026quot;\\\\dv\u0026quot;: \u0026quot;\\\\frac{d\\\\,#1}{d\\\\,#2}\u0026quot;, \u0026quot;\\\\gr\u0026quot;: \u0026quot;\\\\gamma\u0026quot;, \u0026quot;\\\\vec\u0026quot;: \u0026quot;\\\\mathbf{#1}\u0026quot;, \u0026quot;\\\\tagl\u0026quot;: \u0026quot;\\\\tag{#1}\u0026quot;, \u0026quot;\\\\eqref\u0026quot;: \u0026quot;(#1)\u0026quot;, \u0026quot;\\\\laplacian\u0026quot;: \u0026quot;\\\\mathbf{\\\\nabla ^2}\u0026quot;, \u0026quot;\\\\label\u0026quot;: \u0026quot;{}\u0026quot;, \u0026quot;\\\\negadelta\u0026quot;: \u0026quot;\\\\nabla\u0026quot;, } });  }); [f(x) = \\int_{-\\infty}^\\infty\\hat f(\\xi),e^{2 \\pi i \\xi x},d\\xi] \n  \u0026ldquo;Kinetic Description\u0026rdquo;\n \\[\\dv{\\vec v}{t} = \\frac{q_i}{m_i} (\\vec E \u0026#43; \\vec v_i \\cross \\vec B) \u0026#43; \\sum_{j \\neq i} \\left[ \\left. \\dv{\\vec v_{ij}}{t} \\right|_{coll} \\delta(\\vec r_i - \\vec r_j) \\right]\\]   \\[\\pdv{\\vec B}{t} = - \\curl \\vec E\\]   \\[\\frac{1}{c^2} \\pdv{\\vec E}{t} = \\curl \\vec B - \\mu_0 \\sum_i q_i \\vec v_i \\delta (\\vec r - \\vec r_i)\\]   \\[\\div \\vec B = 0\\]   \\[\\div \\vec E = \\frac{1}{\\epsilon_0} \\sum_i q_i \\delta (\\vec r - \\vec r_i)\\]  Klimontovich equation:  \\[\\dv{N}{t} = 0 = \\pdv{N}{t} \u0026#43; \\pdv{}{q_i} \\cdot (\\dot{q_i} N) \\\\ N \\equiv \\sum_i \\delta (p - p_i) \\delta(q - q_i)\\]   !!! info \u0026ldquo;Plasma Fluid Description\u0026rdquo;\nBoltzmann Equation  \\[ \\pdv{f_\\alpha}{t} \u0026#43; \\vec v \\cdot \\pdv{f_\\alpha}{t} \u0026#43; \\frac{q_\\alpha}{m_\\alpha} (\\vec E \u0026#43; \\vec v \\cross \\vec B) \\cdot \\pdv{f_\\alpha}{\\vec v} = \\left. \\pdv{f_\\alpha}{t} \\right|_{coll} = \\sum_{\\beta \\neq \\alpha} C_{\\alpha \\beta} \\]  Maxwellian distribution:  \\[ f_\\alpha (\\vec v) = n_\\alpha \\left( \\frac{m_\\alpha}{2 \\pi T} \\right)^{3/2} e^{- \\frac{m_\\alpha(\\vec v - \\vec v_\\alpha)^2}{2T}} \\]  Moments of fluid model (moments of distribution  \\( \\rightarrow \\)  moments of Boltzmann equation:  \\[ \\text{Continuity:} \\qquad n_\\alpha = \\int f_\\alpha \\dd \\vec v \\\\ \\quad \\rightarrow \\pdv{n_\\alpha}{t} \u0026#43; \\div (n_\\alpha \\vec v_\\alpha) = 0 \\]   \\[ \\text{Momentum:} \\qquad n_\\alpha \\vec v_\\alpha = \\int \\vec v f_\\alpha \\dd v \\\\ \\quad \\rightarrow \\quad \\pdv{}{t} (n_\\alpha \\vec v_\\alpha ) \u0026#43; \\div (n_\\alpha \\vec v_\\alpha \\vec v_\\alpha) \u0026#43; \\frac{1}{m_\\alpha} \\div \\vec P_\\alpha - \\frac{q_\\alpha}{m_\\alpha} n_\\alpha ( \\vec E \u0026#43; \\vec v _\\alpha \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\int \\vec w C_{\\alpha \\beta} \\dd \\vec v \\]   \\[ \\rightarrow \\rho_\\alpha \\left(\\pdv{\\vec v_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad \\vec v_\\alpha \\right) \u0026#43; \\grad \\vec P_\\alpha \u0026#43; \\div \\vec \\Pi_\\alpha - q_\\alpha n_\\alpha (\\vec E \u0026#43; \\vec v_\\alpha \\cross \\vec B) = \\sum_{\\beta \\neq \\alpha} \\vec R_{\\alpha \\beta} \\]   \\[ \\text{Energy:} \\qquad \\int \\vec v \\vec v \\pdv{f_\\alpha}{t} \\dd \\vec v = \\pdv{}{t} \\int \\vec v \\vec v f_\\alpha \\dd \\vec v = \\pdv{}{t} \\vec E_\\alpha / m_\\alpha \\rightarrow \\pdv{}{t} \\vec P_\\alpha \\]   \\[ \\rightarrow \\quad \\frac{3}{2} n_\\alpha \\left( \\pdv{T_\\alpha}{t} \u0026#43; \\vec v_\\alpha \\cdot \\grad T_\\alpha \\right) \u0026#43; P_\\alpha \\div \\vec v_\\alpha \u0026#43; \\vec \\Pi_\\alpha \\cdot \\cdot \\grad \\vec v_\\alpha \u0026#43; \\div \\vec h_\\alpha = \\sum_{\\beta \\neq \\alpha} Q_{\\alpha \\beta} \\]  Closure relations  \\[ \\vec h_\\alpha = - \\kappa \\grad T_\\alpha \\]   \\[ \\overline \\Pi_ \\alpha = \\nu \\grad \\vec v_\\alpha \\]   Ideal MHD #  !!! info \u0026ldquo;Ideal MHD\u0026rdquo;\nContinuity:  \\[ \\pdv{\\rho}{t} \u0026#43; \\div (\\rho \\vec v) = 0 \\]  Momentum:  \\[ \\rho \\left( \\pdv{\\vec v}{t} \u0026#43; \\vec v \\cdot \\grad \\vec v \\right) \u0026#43; \\grad p - \\vec j \\cross \\vec B = 0 \\]  Generalized Ohm's Law  \\[ \\vec E \u0026#43; \\vec v \\cross \\vec B = \\frac{1}{Zen}\\cancel{(\\vec j \\cross \\vec B - \\grad p_e)} = 0 \\]  Energy  \\[ \\dv{}{t} \\left( \\frac{p}{\\rho^\\gamma} \\right) = 0 \\]   !!! info \u0026ldquo;Lawson Criterion\u0026rdquo;\n \\[ n \\tau_E \u0026gt; 10^4 s/m^3 \\]   !!! info \u0026ldquo;Conservation Law Form of Ideal MHD\u0026rdquo;\nContinuity:  \\[ \\pdv{\\rho}{t} \u0026#43; \\div (\\rho \\vec v) = 0 \\]  Momentum:  \\[ \\pdv{(\\rho \\vec v)}{t} \u0026#43; \\div \\left[ \\rho \\vec v \\vec v - \\frac{\\vec B \\vec B}{\\mu_0} \u0026#43; \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\overline{I} \\right] = 0 \\]  Energy:  \\[ \\pdv{\\epsilon}{t} \u0026#43; \\div \\left[ \\left( \\epsilon \u0026#43; p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) \\vec v - (\\vec B \\cdot \\vec v) \\frac{\\vec B}{\\mu_0} \\right] = 0 \\]   \\[ \\pdv{\\vec B}{t} \u0026#43; \\div ( \\vec v \\vec B - \\vec B \\vec v) = 0 \\]  where  \\[ \\epsilon = \\frac{1}{\\gamma - 1} p \u0026#43; \\frac{1}{2} \\rho v^2 \u0026#43; \\frac{B^2}{2\\mu_0} \\]  Static Equilibrium:  \\[ \\vec j \\cross \\vec B = \\grad p \\]   \\[ \\frac{B^2}{\\mu_0} \\vec K = \\grad_\\perp (p \u0026#43; \\frac{B^2}{2 \\mu_0}) \\]   \\[ \\vec K \\equiv \\frac{\\vec B}{|B|} \\cdot \\grad \\frac{ \\vec B}{|B|} \\]  Conservation of flux:  \\[ \\vec E \u0026#43; \\vec v \\cross \\vec B = 0 \\]   \\[ \\pdv{\\vec B}{t} = - \\curl \\vec E \\]   \\[ \\rightarrow \\dv{}{t} \\left( \\frac{\\vec B}{\\rho} \\right) = \\frac{\\vec B}{\\rho} \\cdot \\grad \\vec v \\]   1D Equilibria #  !!! info \u0026ldquo; \\( \\theta \\) -pinch\u0026rdquo;\n \\[ B_\\theta = 0 \\]   \\[ j_\\theta B_z = \\dv{p}{r} \\]   \\[ j_\\theta = - \\frac{1}{\\mu_0} \\dv{B_z}{r} \\]   \\[ \\rightarrow p \u0026#43; \\frac{B_z ^2}{2 \\mu_0} = \\frac{B_0 ^2}{2 \\mu_0} \\]   \\[ \\langle \\beta \\rangle = \\frac{2}{a^2} \\int_0 ^a \\frac{r p}{B_0 ^2 / 2 \\mu_0} \\dd r \\]   \\[ q = \\infty \\]   \\[ W = \\frac{\\mu_0 r}{B_z ^2} \\dv{}{r} \\left( p \u0026#43; \\frac{B_z ^2}{2 \\mu_0} \\right) = 0 \\]   !!! info \u0026ldquo;Z-pinch\u0026rdquo;\n \\[ B_z =0 \\]   \\[ \\grad p = \\dv{p}{r} = - j_z B_\\theta \\]   \\[ - \\dv{}{r} \\left( p \u0026#43; \\frac{B_\\theta ^2}{2 \\mu_0} \\right) = \\frac{B_\\theta ^2}{\\mu_0 r} \\]   \\[ \\langle \\beta \\rangle = \\frac{2 \\mu_0}{B_0 ^2 \\pi a^2} \\int _0 ^a 2 \\pi r p \\dd r = 1 \\quad \\text{ if } \\quad p(a) = 0 \\]   \\[ q = S = 0 \\]   \\[ W = 1 \\]   !!! info \u0026ldquo;Screw pinch\u0026rdquo;\n \\[ \\dv{}{r} \\left( p \u0026#43; \\frac{B^2}{2 \\mu_0} \\right) = - \\frac{B_\\theta ^2}{\\mu_0 r} \\]   \\[ \\beta_t = \\frac{2 \\mu_0}{B_z (a) ^2} \\left( \\frac{1}{\\pi a^2} \\int_0 ^a 2 \\pi r p \\dd r \\right) \\]   \\[ \\beta_p = \\left( 1 - \\frac{\\alpha_t}{\\beta _t} \\right)^{-1} \\qquad \\alpha_t \\equiv \\frac{2}{a^2} \\int_0 ^a \\left(1 - \\frac{B_z ^2}{B_0 ^2} \\right) r \\dd r \\]   \\[ q = \\frac{2 \\pi r B_z}{L B_\\theta} \\]   \\[ q_a = \\frac{4 \\pi ^2 a^2 B_0}{\\mu_0 I_a} \\]   \\[ S = \\frac{r}{q} \\dv{q}{r} \\]   \\[ W = - \\frac{B_\\theta ^2}{B_\\theta ^2 \u0026#43; B_z ^2} \\]   !!! info \u0026ldquo;Stability\u0026rdquo;\nShear:  \\[ S = 2 \\frac{ dq / q}{dV / V} = 2 \\frac{d \\ln q}{d \\ln V} \\]   \\[ q = \\frac{\\text{\\# long windings}}{\\text{\\# short windings}} = \\dv{\\psi_t}{\\psi_p} \\]  Shear for toroid  \\[ q = \\frac{r B_\\phi}{R B_\\theta} \\]  Shear for cylinder  \\[ q = 2 \\pi \\frac{r B_z}{L B_\\theta} \\]  Well  \\[ W = \\frac{ d \\langle p \u0026#43; B^2 / 2 \\mu_0 \\rangle / \\langle B^2 / 2 \\mu_0 \\rangle}{dV / V} \\]  For stabilization,  \\( B^2 / 2 \\mu_0 \\)  should increase faster than  \\( p \\)  decreases  2D Equilibria #  !!! info \u0026ldquo;Grad-Shafranov Equation: Static toroidal equilibrium\u0026rdquo;\n \\[ \\grad p = \\vec j_\\theta \\cross \\vec B_\\phi \u0026#43; \\vec j_\\phi \\cross \\vec B_\\theta \\]   \\[ A_\\phi = \\frac{\\phi}{R} \\vu \\phi \\]   \\[ \\phi = \\frac{\\phi_p}{2 \\pi} \\]   \\[ \\vec B_\\theta = - \\frac{\\vu R}{R} \\pdv{\\psi}{z} \u0026#43; \\frac{ \\vu z}{R} \\pdv{\\psi}{R} = \\frac{ \\grad \\psi}{R} \\cross \\vu \\phi \\]   \\[ F \\equiv R B_\\phi \\]   \\[ \\Delta ^\\star \\equiv R \\pdv{}{R} \\frac{1}{R} \\pdv{}{R} \u0026#43; \\pdv{^2}{z^2} \\]   \\[ \\Delta ^\\star \\psi = \\pdv{^2 \\psi}{z^2} \u0026#43; \\pdv{^2 \\psi}{R^2} - \\frac{1}{R} \\pdv{\\psi}{R} \\]   \\[ \\vec j_\\phi = - \\frac{1}{\\mu_0 R} \\Delta ^\\star \\psi \\vu \\phi \\]   \\[ \\vec j_\\theta = \\frac{1}{\\mu_0 R} \\grad (F) \\cross \\vu \\phi \\]   \\[ R^2 \\mu_0 \\pdv{p}{\\psi} = - \\Delta ^\\star \\psi - F \\pdv{F}{\\psi} \\]   \\[ q(\\psi) = \\frac{F(\\psi)}{2 \\pi} \\oint_{p} \\frac{r \\dd \\theta}{R^2 B_\\theta} \\]  Limits:  \\[ \\text{Force-free:} \\qquad \\vec j \\parallel \\vec B \\]   \\[ \\rightarrow \\Delta ^\\star \\psi \u0026#43; F F\u0026#39; = 0 \\]   \\[ \\text{Connected $\\theta$ pinch:} \\qquad FF\u0026#39; \\gg \\Delta ^\\star \\psi \\]   \\[ \\rightarrow \\grad p \\approx \\vec j_\\theta \\cross \\vec B_\\phi \\]   \\[ \\text{Connected Z-pinch:} \\qquad FF\u0026#39; \\ll \\Delta ^\\star \\phi \\]   \\[ \\rightarrow \\grad p \\approx j_\\phi \\cross B_\\theta \\]   MHD Stability #  !!! info \u0026ldquo;Linear stability\u0026rdquo;\n \\[ \\pdv{\\rho_1}{t} = - \\vec v_1 \\grad \\rho_0 - \\rho_0 \\div \\vec v_1 \\]   \\[ \\pdv{\\vec B_1}{t} = \\curl ( \\vec v_1 \\cross \\vec B_0) \\]   \\[ \\rho_0 \\pdv{\\vec v_1}{t} = - \\grad p_1 \u0026#43; \\vec j_0 \\cross \\vec B_1 - \\vec j_1 \\cross \\vec B_0 \\]   \\[ \\pdv{p_1}{t} = - \\vec v_1 \\cdot \\grad p_0 - \\gamma p_0 \\div \\vec v_1 \\]  For linear perturbation  \\( \\vec \\xi = \\int_0 ^t \\vec v_1 \\dd t \\)  the momentum equation becomes  \\[ \\rho_0 \\pdv{^2 \\xi}{t^2} = \\vec F(\\xi) \\]  where  \\[ F(\\xi) = \\grad (\\xi \\cdot \\grad p_0 \u0026#43; \\gamma p_0 \\div \\xi) \u0026#43; \\frac{1}{\\mu_0} \\left[ ( \\curl \\vec B_0) \\cross \\curl (\\xi \\cross \\vec B_0) \u0026#43; \\curl \\curl (\\xi \\cross \\vec B_0) \\cross \\vec B_0 \\right] \\]  Eigenvalues of  \\( \\frac{1}{\\rho_0} \\vec F (\\xi) = \\omega^2 \\xi \\)  are real and ordered. Only need to check  \\( n=0 \\)  to determine stability/instability of configuration.  !!! info \u0026ldquo; \\( \\delta W \\)  Approach\u0026rdquo;\n \\( \\delta W = \\)  change in potential energy due to a displacement  \\( \\xi \\)   \\[ \\delta W \u0026lt; 0 \\rightarrow \\text{instability} \\]   \\[ \\delta W = - \\frac{1}{2} \\int \\xi \\cdot F(\\xi) \\dd V = \\delta W_F \u0026#43; \\delta W_S \u0026#43; \\delta W_V \\]  Surface term:  \\[ \\delta W_s = \\frac{1}{2} \\oint \\dd S (\\vu n \\cdot \\xi) ^2 \\left( \\vu n \\cdot \\grad p_0 \u0026#43; \\left[ \\vu n \\cdot \\grad \\frac{B_0 ^2}{2 \\mu_0} \\right]_{jump} \\right) \\]  Vacuum term:  \\[ \\delta W_V = \\int_{vac} \\dd V \\frac{B_1 ^2}{\\mu_0} \\]  Plasma (free) term:  \\[ \\delta W_F = \\frac{1}{2} \\int \\dd V \\frac{ |B_{1, \\perp}|^2}{\\mu_0} \\quad \\leftarrow \\text{Shear Alfven} \\\\ \u0026#43; \\mu_0 \\left| \\frac{B_{1, \\parallel}}{\\mu_0} - \\frac{B_0 \\xi \\cdot \\grad p_0}{B_0} ^2 \\right|^2 \\quad \\leftarrow \\text{Fast magnetosonic} \\\\ \u0026#43; \\Gamma p_0 |\\div \\xi|^2 \\quad \\leftarrow \\text{Acoustic}\\\\ \u0026#43; \\frac{\\vec j_0 \\cdot \\vec B_0}{B_0 ^2} (\\vec B_0 \\cross \\vec \\xi) \\cdot \\vec B_1 \\quad \\leftarrow \\text{Current-driven (kink)} \\\\ - 2 ( \\vec \\xi \\cdot \\grad p_0)(\\vec \\xi \\cdot \\vec \\kappa) \\quad \\leftarrow \\text{pressure-driven (interchange/balooning)} \\]  Shear Alfven, fast magnetosonic, and acoustic modes are stabilizing. Current-driven and pressure-driven modes can lead to instability.  "});index.add({'id':12,'href':'/r/notes/working/crews2018/','title':"Crews (2018)",'section':"Working",'content':"Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities #  Shielded potential of a test electron:\n  document.addEventListener(\"DOMContentLoaded\", function () { renderMathInElement(document.body, { delimiters: [ { left: \"$$\", right: \"$$\", display: true }, { left: \"\\\\[\", right: \"\\\\]\", display: true }, { left: \"$\", right: \"$\", display: false }, { left: \"\\\\(\", right: \"\\\\)\", display: false }], macros: { \"\\\\lcm\": \"\\\\mathop{\\\\mathrm{lcm}}\", \"\\\\sen\": '\\\\text{sen}\\\\,', \"\\\\dd\": \"\\\\mathop{\\\\mathrm{d} #1}\", \"\\\\abs\": \"\\\\lvert #1 \\\\rvert\", \"\\\\dd\": \"\\\\text{d}\", \"\\\\cross\": \"\\\\times\", \"\\\\pdv\": \"\\\\frac{\\\\partial #1}{\\\\partial #2}\", \"\\\\curl\": \"\\\\nabla \\\\cross #1\", \"\\\\vu\": \"\\\\mathbf{\\\\hat{#1}}\", \"\\\\vdot\": \"\\\\dot\", \"\\\\div\": \"\\\\nabla \\\\cdot #1\", \"\\\\grad\": \"\\\\nabla\", \"\\\\dv\": \"\\\\frac{d\\\\,#1}{d\\\\,#2}\", \"\\\\gr\": \"\\\\gamma\", \"\\\\vec\": \"\\\\mathbf{#1}\", \"\\\\tagl\": \"\\\\tag{#1}\", \"\\\\eqref\": \"(#1)\", \"\\\\laplacian\": \"\\\\mathbf{\\\\nabla ^2}\", \"\\\\label\": \"{}\", \"\\\\negadelta\": \"\\\\nabla\", } }); }); \\[\\phi(r) = \\frac{-e}{4 \\pi \\epsilon_0 r} e ^{- r / \\lambda_D}\\]  where the Debye length is  \\( \\lambda_D = \\sqrt{\\frac{\\epsilon_0 T_e}{ n_e e}} \\) . The mean free path between large-angle collisions is estimated as\n \\[\\lambda_{mfp} \\sim \\frac{\\epsilon_0 T_e ^2}{\\phi_e n_e \\log ( \\Lambda)}\\]  where  \\(\\phi_e = e^2 / 4 \\pi \\epsilon_0\\)  are the constants from the Coulomb force law.\nSmooth out the discreteness of particles via spatial average over small volumes:\n \\[\\rho \\rightarrow \\langle \\rho_c \\rangle \\qquad \\vec E \\rightarrow \\langle \\vec E \\rangle \u0026#43; \\delta \\vec E\\]  The mean field  \\(\\langle \\vec E \\rangle\\)  is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales  \\(\\lambda_{mfp} / \\lambda_D\\) \n \\[\\frac{\\lambda_{mfp}}{\\lambda_D} \\sim \\frac{T_e ^{3/2}}{n_e ^{1/2}}\\]  Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified.\nPhase space mechanics #  To arrive at a kinetic equation governing the collisionless mechanics, consider the one-dimensional motion of a single particle\n \\[\\dot{x} = v \\qquad \\dot v = F(x)\\]  and define the phase space coordinates as  \\(\\vec{\\dot r} = \\vec F \\equiv [ v, F(x) ]\\) . The flux vector is similar to the velocity field of a fluid flow. The streamlines of  \\(\\vec F\\)  are the streamlines which a particle will follow if the flux is constant in time. Phase flow is always analogous to that of an incompressible fluid because the flux divergence is zero:\n \\[\\div [v, F(x)] = \\pdv{v}{x} \u0026#43; \\pdv{F(x)}{v}\\]  If the phase fluid density is given by a function  \\(f(x, v, t)\\)  where  \\(t\\)  is the time parameter, because any instantiation of a particle can not leave the phase plane, the probability density will be conserved. We can write a conservation law:  \\[\\pdv{f(x, v, t)}{t} = - \\div ( f (x, v, t) \\vec F)\\]  and due to the flow\u0026rsquo;s incompressibility  \\[\\div (f \\vec F) = f ( \\div \\vec F) \u0026#43; \\vec F \\cdot \\grad f = \\vec F \\cdot \\grad f \\\\ \\rightarrow \\pdv{f}{t} = - \\left[ v, F(x) \\right] \\cdot \\left[ \\pdv{f}{x}, \\pdv{f}{v} \\right] \\\\ \\rightarrow \\pdv{f}{t} \u0026#43; v \\pdv{f}{x} \u0026#43; F(x) \\pdv{f}{v} = 0\\] \n"});})();
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+      <guid>https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/</guid>
+      <description>Syllabus #  The course topics planned for this section are (in rough order):
+Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, &amp;ldquo;Frozen-In&amp;rdquo; Flux)
+Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces
+One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q
+Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/example/hidden/</link>
+      <title>Plasma Models</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/example/hidden/</guid>
-      <description>This page is hidden in menu #  Quondam non pater est dignior ille Eurotas #  Latent te facies #  Lorem markdownum arma ignoscas vocavit quoque ille texit mandata mentis ultimus, frementes, qui in vel. Hippotades Peleus pennas conscia cuiquam Caeneus quas.
- Pater demittere evincitque reddunt Maxime adhuc pressit huc Danaas quid freta Soror ego Luctus linguam saxa ultroque prior Tatiumque inquit Saepe liquitur subita superata dederat Anius sudor  Cum honorum Latona #  O fallor in sustinui iussorum equidem.</description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/</guid>
+      <description>Plasma Models #  Working towards MHD #  Let&amp;rsquo;s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we&amp;rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.
+The most important question to ask ourselves is &amp;ldquo;when is this model going to be useful?&amp;rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid.</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/buttons/</link>
+      <title>Plasma Fluid Model</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/buttons/</guid>
-      <description>Buttons #  Buttons are styled links that can lead to local page or external link.
-Example #  {{&amp;lt; button relref=&amp;#34;/&amp;#34; [class=&amp;#34;...&amp;#34;] &amp;gt;}}Get Home{{&amp;lt; /button &amp;gt;}} {{&amp;lt; button href=&amp;#34;https://github.com/alex-shpak/hugo-book&amp;#34; &amp;gt;}}Contribute{{&amp;lt; /button &amp;gt;}}  Get Home  Contribute  </description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/</guid>
+      <description>Plasma Fluid Model #  We take velocity moments of each of the pieces of the kinetic model:
+ distribution function,   \( f_\alpha \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false }], macros: { &#34;\\lcm&#34;: &#34;</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/columns/</link>
+      <title>Two-Fluid Plasma Model</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/columns/</guid>
-      <description>Columns #  Columns help organize shorter pieces of content horizontally for readability.
-{{&amp;lt; columns &amp;gt;}} &amp;lt;!-- begin columns block --&amp;gt; # Left Content Lorem markdownum insigne... &amp;lt;---&amp;gt; &amp;lt;!-- magic separator, between columns --&amp;gt; # Mid Content Lorem markdownum insigne... &amp;lt;---&amp;gt; &amp;lt;!-- magic separator, between columns --&amp;gt; # Right Content Lorem markdownum insigne... {{&amp;lt; /columns &amp;gt;}} Example #  Left Content Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat stringit, frustra Saturnius uteroque inter!</description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/</guid>
+      <description>Two-Fluid Plasma Model (ions-electrons) #  Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let&amp;rsquo;s start with a cold plasma approximation (   \( p = 0 \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/details/</link>
+      <title>Magnetohydrodynamic (MHD) Model</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/details/</guid>
-      <description>Details #  Details shortcode is a helper for details html5 element. It is going to replace expand shortcode.
-Example #  {{&amp;lt; details &amp;#34;Title&amp;#34; [open] &amp;gt;}} ## Markdown content Lorem markdownum insigne... {{&amp;lt; /details &amp;gt;}} {{&amp;lt; details title=&amp;#34;Title&amp;#34; open=true &amp;gt;}} ## Markdown content Lorem markdownum insigne... {{&amp;lt; /details &amp;gt;}} Title Markdown content Lorem markdownum insigne&amp;hellip;   </description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/</guid>
+      <description>Magnetohydrodynamic (MHD) Model #  Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD).
+First, define the MHD variables:
+  \[\text{mass density:} \qquad \rho = n_i m_i &amp;#43; n_i m_e\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/expand/</link>
+      <title>Boundary Conditions</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/expand/</guid>
-      <description>Expand #  Expand shortcode can help to decrease clutter on screen by hiding part of text. Expand content by clicking on it.
-Example #  Default #  {{&amp;lt; expand &amp;gt;}} ## Markdown content Lorem markdownum insigne... {{&amp;lt; /expand &amp;gt;}}   Expand ↕  Markdown content Lorem markdownum insigne&amp;hellip;    With Custom Label #  {{&amp;lt; expand &amp;#34;Custom Label&amp;#34; &amp;#34;...&amp;#34; &amp;gt;}} ## Markdown content Lorem markdownum insigne.</description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/</guid>
+      <description>Boundary Conditions #  Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region.
+Perfectly Conducting Wall #  For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/hints/</link>
+      <title>Equilibrium for Fusion</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/hints/</guid>
-      <description>Hints #  Hint shortcode can be used as hint/alerts/notification block.
-There are 3 colors to choose: info, warning and danger.
-{{&amp;lt; hint [info|warning|danger] &amp;gt;}} **Markdown content** Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa {{&amp;lt; /hint &amp;gt;}} Example #  Markdown content
-Lorem markdownum insigne. Olympo signis Delphis! Retexi Nereius nova develat stringit, frustra Saturnius uteroque inter! Oculis non ritibus Telethusa  Markdown content</description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/</guid>
+      <description>Equilibrium for Fusion (   \( \beta \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false }], macros: { &#34;\\lcm&#34;: &#34;\\mathop{\\mathrm{lcm}}&#34;, &#34;\\sen&#34;: &#39;\\text{sen}\\,&#39;, &#34;\\dd&#34;: &#34;\\mathop{\\mathrm{d} #1}&#34;, &#34;\\abs&#34;: &#34;\\lvert #1 \\rvert&#34;, &#34;\\dd&#34;: &#34;\\text{d}&#34;, &#34;\\cross&#34;: &#34;\\times&#34;, &#34;\\pdv&#34;: &#34;\\frac{\\partial #1}{\\partial #2}&#34;</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/katex/</link>
+      <title>1-D Equilibria</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/katex/</guid>
-      <description>KaTeX #  KaTeX shortcode let you render math typesetting in markdown document. See KaTeX
-Example #  {{&amp;lt; katex [display] [class=&amp;#34;text-center&amp;#34;] &amp;gt;}} f(x) = \int_{-\infty}^\infty\hat f(\xi)\,e^{2 \pi i \xi x}\,d\xi {{&amp;lt; /katex &amp;gt;}}     document.addEventListener(&#34;DOMContentLoaded&#34;, function() { renderMathInElement(document.body, { delimiters: [ {left: &#34;$$&#34;, right: &#34;$$&#34;, display: true}, {left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true}, {left: &#34;$&#34;, right: &#34;$&#34;, display: false}, {left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false} ], macros: {  &amp;quot;\\lcm&amp;quot;: &amp;quot;\\mathop{\\mathrm{lcm}}&amp;quot;, &amp;quot;\\sen&amp;quot;: &#39;\\text{sen}\\,&#39;, &amp;quot;\\dd&amp;quot;: &amp;quot;\\mathop{\\mathrm{d} #1}&amp;quot;, &amp;quot;\\abs&amp;quot;: &amp;quot;\\lvert #1 \\rvert&amp;quot;, &amp;quot;\\dd&amp;quot;: &amp;quot;\\text{d}&amp;quot;, &amp;quot;\\cross&amp;quot; : &amp;quot;\\times&amp;quot;, &amp;quot;\\pdv&amp;quot;: &amp;quot;\\frac{\\partial #1}{\\partial #2}&amp;quot;, &amp;quot;\curl&amp;quot;: &amp;quot;\\nabla \\cross #1&amp;quot;, &amp;quot;\\vu&amp;quot;: &amp;quot;\\mathbf{\\hat{#1}}&amp;quot;, &amp;quot;\\dot&amp;quot;: &amp;quot;\\boldsymbol\\cdot&amp;quot;, &amp;quot;\\vdot&amp;quot; : &amp;quot;\\dot&amp;quot;, &amp;quot;\\div&amp;quot;: &amp;quot;\\nabla \\dot #1&amp;quot;, &amp;quot;\\grad&amp;quot; : &amp;quot;\\nabla&amp;quot;, &amp;quot;\\dv&amp;quot; : &amp;quot;\\frac{d\\,#1}{d\\,#2}&amp;quot;, &amp;quot;\\gr&amp;quot; : &amp;quot;\\gamma&amp;quot;, &amp;quot;\\vec&amp;quot; : &amp;quot;\\mathbf{#1}&amp;quot;, &amp;quot;\\tagl&amp;quot; : &amp;quot;\\tag{#1}&amp;quot;, &amp;quot;\\eqref&amp;quot; : &amp;quot;(#1)&amp;quot;, &amp;quot;\\laplacian&amp;quot; : &amp;quot;\\mathbf{\\nabla ^2}&amp;quot;, &amp;quot;\\label&amp;quot; : &amp;quot;{}&amp;quot;, &amp;quot;\\negadelta&amp;quot; : &amp;quot;\\nabla&amp;quot;, }  }); }); [f(x) = \int_{-\infty}^\infty\hat f(\xi),e^{2 \pi i \xi x},d\xi]</description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/</guid>
+      <description>1-Dimensional Equilibria #  The   \( \theta \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false }], macros: { &#34;\\lcm&#34;: &#34;\\mathop{\\mathrm{lcm}}&#34;, &#34;\\sen&#34;: &#39;\\text{sen}\\,&#39;, &#34;\\dd&#34;: &#34;\\mathop{\\mathrm{d} #1}&#34;, &#34;\\abs&#34;: &#34;\\lvert #1 \\rvert&#34;, &#34;\\dd&#34;: &#34;\\text{d}&#34;, &#34;\\cross&#34;: &#34;\\times&#34;, &#34;\\pdv&#34;: &#34;\\frac{\\partial #1}{\\partial #2}&#34;</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/mermaid/</link>
+      <title>2D Equilibria</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/mermaid/</guid>
-      <description>Mermaid Chart #   Mermaid is library for generating svg charts and diagrams from text.
-Example #  {{&amp;lt; mermaid [class=&amp;#34;text-center&amp;#34;]&amp;gt;}} sequenceDiagram Alice-&amp;gt;&amp;gt;Bob: Hello Bob, how are you? alt is sick Bob-&amp;gt;&amp;gt;Alice: Not so good :( else is well Bob-&amp;gt;&amp;gt;Alice: Feeling fresh like a daisy end opt Extra response Bob-&amp;gt;&amp;gt;Alice: Thanks for asking end {{&amp;lt; /mermaid &amp;gt;}}    mermaid.initialize({ &#34;flowchart&#34;: { &#34;useMaxWidth&#34;:true }, &#34;theme&#34;: &#34;default&#34; } ) sequenceDiagram Alice-Bob: Hello Bob, how are you?</description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/</guid>
+      <description>2D Equilibria #  Let&amp;rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:
+  \[\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\ = \grad p = \dv{p}{r}\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/section/page1/</link>
+      <title>Equilibrium of 3D Configurations</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/section/page1/</guid>
-      <description>Page 1 #  </description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/</guid>
+      <description>Equilibrium of 3D Configurations #  In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated.
+Some general features of 3D equilibria are:
+ No net toroidal current. This means that they tend to be steady-state configurations.</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/section/page2/</link>
+      <title>MHD Stability</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/section/page2/</guid>
-      <description>Page 2 #  </description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/</guid>
+      <description>MHD Stability #  Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear.</description>
     </item>
     
     <item>
-      <title></title>
-      <link>https://peppyhare.github.io/r/docs/shortcodes/tabs/</link>
+      <title>Formulary</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/formulary/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/shortcodes/tabs/</guid>
-      <description>Tabs #  Tabs let you organize content by context, for example installation instructions for each supported platform.
-{{&amp;lt; tabs &amp;#34;uniqueid&amp;#34; &amp;gt;}} {{&amp;lt; tab &amp;#34;MacOS&amp;#34; &amp;gt;}} # MacOS Content {{&amp;lt; /tab &amp;gt;}} {{&amp;lt; tab &amp;#34;Linux&amp;#34; &amp;gt;}} # Linux Content {{&amp;lt; /tab &amp;gt;}} {{&amp;lt; tab &amp;#34;Windows&amp;#34; &amp;gt;}} # Windows Content {{&amp;lt; /tab &amp;gt;}} {{&amp;lt; /tabs &amp;gt;}} Example #  MacOS MacOS This is tab MacOS content.
-Lorem markdownum insigne. Olympo signis Delphis!</description>
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/formulary/</guid>
+      <description>\( \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false }], macros: { &#34;\\lcm&#34;: &#34;\\mathop{\\mathrm{lcm}}&#34;, &#34;\\sen&#34;: &#39;\\text{sen}\\,&#39;, &#34;\\dd&#34;: &#34;\\mathop{\\mathrm{d} #1}&#34;, &#34;\\abs&#34;: &#34;\\lvert #1 \\rvert&#34;, &#34;\\dd&#34;: &#34;\\text{d}&#34;, &#34;\\cross&#34;: &#34;\\times&#34;, &#34;\\pdv&#34;: &#34;\\frac{\\partial #1}{\\partial #2}&#34;, &#34;\\curl&#34;: &#34;\\nabla \\cross #1&#34;</description>
     </item>
     
     <item>
       <title>Crews (2018)</title>
-      <link>https://peppyhare.github.io/r/docs/notes/crews2018/</link>
+      <link>https://peppyhare.github.io/r/notes/working/crews2018/</link>
       <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
       
-      <guid>https://peppyhare.github.io/r/docs/notes/crews2018/</guid>
+      <guid>https://peppyhare.github.io/r/notes/working/crews2018/</guid>
       <description>Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities #  Shielded potential of a test electron:
-  document.addEventListener(&#34;DOMContentLoaded&#34;, function() { renderMathInElement(document.body, { delimiters: [ {left: &#34;$$&#34;, right: &#34;$$&#34;, display: true}, {left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true}, {left: &#34;$&#34;, right: &#34;$&#34;, display: false}, {left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false} ], macros: { &#34;\\lcm&#34;: &#34;\\mathop{\\mathrm{lcm}}&#34;, &#34;\\sen&#34;: &#39;\\text{sen}\\,&#39;, &#34;\\dd&#34;: &#34;\\mathop{\\mathrm{d} #1}&#34;, &#34;\\abs&#34;: &#34;\\lvert #1 \\rvert&#34;</description>
+  \[\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D}\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;</description>
     </item>
     
   </channel>
-</rss>
\ No newline at end of file
+</rss>
diff --git a/docs/notes/UWAA558/01-syllabus/index.html b/docs/notes/UWAA558/01-syllabus/index.html
new file mode 100644
index 00000000..be82253f
--- /dev/null
+++ b/docs/notes/UWAA558/01-syllabus/index.html
@@ -0,0 +1,716 @@
+<!DOCTYPE html>
+<html lang="en" dir=>
+
+<head>
+  <meta charset="UTF-8">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+<meta name="description" content="Syllabus #  The course topics planned for this section are (in rough order):
+Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, &ldquo;Frozen-In&rdquo; Flux)
+Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces
+One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q
+Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)">
+<meta name="theme-color" content="#FFFFFF"><meta property="og:title" content="Syllabus" />
+<meta property="og:description" content="Syllabus #  The course topics planned for this section are (in rough order):
+Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, &ldquo;Frozen-In&rdquo; Flux)
+Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces
+One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q
+Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)" />
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
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+  <article class="markdown"><h2 id="syllabus">
+  Syllabus
+  <a class="anchor" href="#syllabus">#</a>
+</h2>
+<p>The course topics planned for this section are (in rough order):</p>
+<p>Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, &ldquo;Frozen-In&rdquo; Flux)</p>
+<p>Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces</p>
+<p>One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q</p>
+<p>Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)</p>
+<p>MHD Stability, General Concepts, Linearized MHD, Exponential (Linear) Stability, Force Operator and Properties, Variational Formulation, Energy Principle, Intuitive Form of delta W, Classification of Instabilities (internal/external, pressure-drive/current-driven, kink/interchange/ballooning)</p>
+<p>Stability of One-Dimensional Equilibria, Modal Analysis, Rayleigh-Taylor, Theta-Pinch, Z-Pinch (Kadomtsev Condition), Screw-Pinch (Kruskal-Shafranov Condition, Suydam Criterion), RFP, &ldquo;Straight&rdquo; Tokamak</p>
+<p>Stability of Two-Dimensional Equilibria, Tokamak, Mercier Criterion, Elmo Bumpy Torus</p>
+<p>Resistive (Tearing) Instabilities, Stability of Non-static Equilibria, Nonlinear Stability Effects</p>
+<h2 id="course-motivation">
+  Course Motivation
+  <a class="anchor" href="#course-motivation">#</a>
+</h2>
+<p>Plasma phenomena tend to be hard to treat because of the span of relevant scales. You have ions, electrons, and photons interacting through electromagnetic interactions. There is a tremendous variation in mass across species, which leads to a large span of both spatial and temporal scales. The species can interact through both short scale collisions and long range interactions through EM forces. In contrast, in normal gas dynamics you may consider only the short-scale interactions. As a consequence, we can describe dispersive plasma waves.</p>
+<p>For comparison, remember in gas dynamics, the speed of sound is</p>
+
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \[\frac{\omega}{k} = v_s = \pdv{\omega}{k} \qquad \text{(gas)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Here the phase velocity <span>
+  \( \frac{\omega}{k} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
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+        
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+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is equal to the phase velocity <span>
+  \( \pdv{\omega}{k} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
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+        
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
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+        "\\grad": "\\nabla",
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+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. In a plasma, we can have non-linear dispersion relations in which the phase and group velocity are different.</p>
+<span>
+  \[\frac{\omega}{k} = v(\omega, k) \neq \pdv{\omega}{k} \qquad \text{(plasma)} \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
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+        
+        "\\dd": "\\text{d}",
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The number of particles we typically deal with in a laboratory plasma is roughly on the order of a mole of particles</p>
+<span>
+  \[\text{particles} \sim O(10^{23})\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+      macros: {
+        
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+        
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>With long-range interactions, we have a combinatorial explosion of interacting particles! It is not possible to track individual particles at such a scale, so we need much simpler plasma descriptions. These are the plasma models we will discuss in the next chapter. When simplifying our models, we need to pay careful attention to the simplifications we are making because in general, inaccurate physics lead to incorrect conclusions.</p>
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diff --git a/docs/notes/UWAA558/02-plasma-models/index.html b/docs/notes/UWAA558/02-plasma-models/index.html
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+<meta name="description" content="Plasma Models #  Working towards MHD #  Let&rsquo;s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we&rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.
+The most important question to ask ourselves is &ldquo;when is this model going to be useful?&rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid.">
+<meta name="theme-color" content="#FFFFFF"><meta property="og:title" content="Plasma Models" />
+<meta property="og:description" content="Plasma Models #  Working towards MHD #  Let&rsquo;s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we&rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.
+The most important question to ask ourselves is &ldquo;when is this model going to be useful?&rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid." />
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+    
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+      </header>
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+  <article class="markdown"><h1 id="plasma-models">
+  Plasma Models
+  <a class="anchor" href="#plasma-models">#</a>
+</h1>
+<h2 id="working-towards-mhd">
+  Working towards MHD
+  <a class="anchor" href="#working-towards-mhd">#</a>
+</h2>
+<p>Let&rsquo;s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we&rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.</p>
+<p>The most important question to ask ourselves is &ldquo;when is this model going to be useful?&rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid. This means that we can define a fluid element (some lil' box of plasma) and define the physical properties (mass, density, magnetization, etc.) of the element. We need to make some assumptions about scale in order to do this. In terms of spatial scales, we abstract properties below a discrete scale 
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
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+  \( a_0 \)
+</span>
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+</script>  into the properties of a fluid element</p>
+<span>
+  \[\frac{a_0}{L} \rightarrow 0 \qquad a_0 = \text{discrete scale} \qquad L = \text{spatial scale of interest}\]
+</span>
+<script>
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+</script>
+<p>Length scales smaller than the discrete scale will not be properly captured by the model, so scales like the particle radius will be meaningless in our fluid model.</p>
+<h2 id="plasma-definition">
+  Plasma Definition
+  <a class="anchor" href="#plasma-definition">#</a>
+</h2>
+<p>A plasma is a quasi-neutral gas of charged and neutral particles which exhibit <em>collective behavior</em>. The particles (electrons, ions, neutrals) interact through EM fields and collisions.</p>
+<p>Mathematically you would think that plasmas could be treated as individual particles. Doing so gives an N-body problem with classical interactions through the Lorentz force (Coulomb interaction removed by assumption of quasi-neutrality) and binary collisions. Each particle <span>
+  \( i \)
+</span>
+<script>
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
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+    });
+  };
+</script>  has a well-defined mass <span>
+  \( m_i \)
+</span>
+<script>
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+        "\\gr": "\\gamma",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  and charge <span>
+  \( q_i \)
+</span>
+<script>
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+        
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+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
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+    });
+  };
+</script>  which do not change in time. The governing equations are</p>
+<p><span>
+  \[\dv{\vec v_i}{t} = \frac{q_i}{m_i} (\vec E &#43; \vec v_i \cross \vec B) &#43; \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} (\vec r_i - \vec r_j) \right]\]
+</span>
+<script>
+  function loadKatex() {
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+<span>
+  \[\dv{\vec r_i}{t} = \vec v_i\]
+</span>
+<script>
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+</script></p>
+<p>The fields E and B are described by the Maxwell equations</p>
+<p><span>
+  \[\pdv{B}{t} = - \curl E\]
+</span>
+<script>
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+</script>
+<span>
+  \[\frac{1}{c^2} \pdv{E}{t} = \curl B - \mu_0 \sum_i q_i v_i \delta (r - r_i)\]
+</span>
+<script>
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+</script>
+<span>
+  \[\div B = 0\]
+</span>
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+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\epsilon_0 \div E = \sum_i q_i \delta (r - r_i)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<h2 id="klimontovich-equation">
+  Klimontovich Equation
+  <a class="anchor" href="#klimontovich-equation">#</a>
+</h2>
+<p>Re-writing the force relations as a statement of conservation in phase space, we get Klimontovich equation for species <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\dv{N_\alpha}{t} = 0 = \pdv{N_\alpha}{t} &#43; \pdv{}{q} \cdot (\dot{q} N_\alpha) &#43; \pdv{}{p} \cdot (\dot{p}N_\alpha)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The particle phase space is defined by
+<span>
+  \[N_\alpha(p, q) = \sum_i \delta(p - p_i) \delta(q - q_i)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where <span>
+  \( p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( q \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> are generalized momentum and position coordinates. The resulting <span>
+  \( N(p) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> looks very spiky, with nonzero values only at the exact values inhabited by particles. Unfortunately, that means that only the tools of discrete mathematics are applicable to the distribution, and we&rsquo;re forbidden from our favorite tool (calculus). To make the analysis possible, we can smooth over the discreteness by performing an ensemble average of the Klimontovich equation. This gives us a statistical description using smooth distribution functions:</p>
+<span>
+  \[f(x, v, t) \qquad \frac{(\text{no. of particles})}{(\text{unit distance})^3(\text{unit velocity})^3}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>That is, <span>
+  \( f(x, v, t) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the number of particles at position <span>
+  \( x \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> with velocity <span>
+  \( v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> at time <span>
+  \( t \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. We also work with normalized distributions which give the probability of finding a particle.</p>
+<p>By our definition, we can integrate to get the total number of particles at time <span>
+  \( t \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<span>
+  \[\int \int \dd x \dd v f(x, v, t) = N(t)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The ensemble averaging process works well if the the number of particles is very large</p>
+<span>
+  \[N \gg 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p align="center"> <img alt="Figure 12.1" src="/r/img/12.1.png" /> </p>
+<p>See a kinetic theory text (e.g. Krall and Trivelpiece) for a full description of the ensemble averaging process.</p>
+<p>Now we can write our Klimontovich equation in terms of continuous quantities</p>
+<span>
+  \[\pdv{f}{t} &#43; \left[ \pdv{}{q} \cdot (\dot{q} f) &#43; \pdv{}{p} \cdot (\dot{p}f) \right] = \text{(cross terms)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
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+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The collision terms now have an infinite number of cross terms. We call this the BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) hierarchy.</p>
+<p>Expressing as the Boltzmann equation (generally called the Boltzmann-Maxwell equation, since the solution requires solving for the electromagnetic fields of the Maxwell equations)</p>
+<span>
+  \[\pdv{f_\alpha}{t} &#43; \vec v \cdot \pdv{f_\alpha}{\vec x} &#43; \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
+        "\\dd": "\\text{d}",
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+        "\\vdot": "\\dot",
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+        "\\gr": "\\gamma",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We leave the collision term in. A lot of the work of kinetic theory is coming up with an applicable form of the collision operator which is appropriate but still simple enough to solve. We often write the collision operator as the product of binary collisions</p>
+<span>
+  \[ \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_\beta C_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Notice that the terms <span>
+  \( \vec v \cdot \pdv{f_\alpha}{\vec x} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> are advection equations, advecting in <span>
+  \( \vec x \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> respectively. If we ignore the collision term (set the RHS to zero) we have the Vlasov equation.</p>
+<p>Note that the fields E and B at any location are generated from the charges and currents of the entire plasma volume, including externally applied fields. As a result, there&rsquo;s an inherent integrating process taking into account the sources across the whole volume that leads to long-range smoothly varying forces. This is in contrast to the collisional effects, which by their nature lead to very short range abrupt forces. It makes sense to make a distinction between the long-range electromagnetic forces and the short range collisional forces.</p>
+<p>Because the Boltzmann-Maxwell model is inherently 6-dimensional, it is a very challenging model to implement. The B-M model provides a complete description, but it is often too detailed to solve.</p>
+<p>If we solve the Vlasov equation for two parallel opposite beams, for example, we see that the</p>
+<p>As it turns out, the integral, centroid, and variance are all that are required to fully describe a Maxwellian distribution</p>
+<span>
+  \[f_{M, \alpha} (\vec v) = n_0 \left( \frac{ m_\alpha }{2 \pi T_\alpha} \right)^{3/2} \text{exp}\left[ - \frac{ \frac{1}{2} m_\alpha (v - v_\alpha)^2}{T_\alpha} \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>In other words, we can write the Maxwellian distribution as <span>
+  \( f_M(n_0, \vec v_\alpha, T_\alpha ; \vec v) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. We care about Maxwellian distributions so much in plasma physics because it is the solution to the Boltzmann equation for <span>
+  \( \left. \pdv{f_\alpha}{t} \right|_{coll} = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> (Vlasov equation). That&rsquo;s not saying that there are no collisions, it is saying that there are so many collisions that the effect is isotropic and the overall force is zero.</p>
+<p>Another feature of the Maxwellian distribution is <span>
+  \( \vec v \cdot \pdv{f_\alpha}{\vec x} = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. This is a famous result called the Boltzmann H-theorem, and says that any initial distribution will relax (and very quickly) to a Maxwellian distribution.</p>
+<p>By replacing our velocity distribution with the associated Maxwell distribution, we arrive at a Plasma Fluid Model</p>
+</article>
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+    <a href="https://peppyhare.github.io/r/notes/working/crews2018/" class="">Crews (2018)</a>
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+      
+  <article class="markdown"><h1 id="plasma-fluid-model">
+  Plasma Fluid Model
+  <a class="anchor" href="#plasma-fluid-model">#</a>
+</h1>
+<p>We take velocity moments of each of the pieces of the kinetic model:</p>
+<ul>
+<li>distribution function, 
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \( f_\alpha \)
+</span>
+<script>
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+  };
+</script> <span>
+  \( \rightarrow \)
+</span>
+<script>
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+  };
+</script> fluid variables</li>
+<li>Boltzmann Equation <span>
+  \( \rightarrow \)
+</span>
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+  };
+</script> governing equations describing the evolution of the fluid variables.</li>
+</ul>
+<p>Starting with the zeroth moment (integral) of the distribution function:</p>
+<span>
+  \[\int f_\alpha (\vec x, \vec v, t) \dd \vec v = n_\alpha(\vec x, t)\]
+</span>
+<script>
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+    });
+  };
+</script>
+<p>1st Moment (momentum):</p>
+<span>
+  \[m_\alpha \int \vec v \cdot  f_\alpha(\vec x, \vec v, t) \dd \vec v = \vec p_\alpha (\vec x, t) = m_\alpha n_\alpha \vec v_\alpha\]
+</span>
+<script>
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+</script>
+<p>which is to say that the velocity is the 1st moment divided by the zeroth moment</p>
+<span>
+  \[v_\alpha = \frac{\int \vec v f_\alpha}{\int f_\alpha}\]
+</span>
+<script>
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+</script>
+<p>2nd Moment:</p>
+<span>
+  \[\int \vec v \vec v f_\alpha (\vec x, \vec v, t) \dd \vec v = \vec E_\alpha(\vec x, t) (\text{energy tensor})\]
+</span>
+<script>
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+    });
+  };
+</script>
+<p>We can simplify the 2nd moment by taking a reduced 2nd moment. This means that we&rsquo;re going to insert a dot product</p>
+<span>
+  \[\int \vec v \cdot \vec v f_\alpha (\vec x, \vec v, t) \dd \vec v\]
+</span>
+<script>
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Before moving forward, we want to define a &ldquo;random&rdquo; velocity. Note that</p>
+<span>
+  \[\int \vec v f_\alpha - \vec v_\alpha \int f_\alpha = 0 \rightarrow \int (\vec v - \vec v_\alpha) f_\alpha \dd \vec v = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can define a random velocity <span>
+  \( \vec w = \vec v - \vec v_\alpha \)
+</span>
+<script>
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+  };
+</script>. It is random in the sense that it is a fluctuation about the mean velocity, and when we integrate it we get zero. We can use this to define the energy tensor using the mean velocity to get a meaningful result. The pressure tensor is the second moment, using the random velocity</p>
+<span>
+  \[\vec P_\alpha = m_\alpha \int \vec w \vec w f_\alpha \dd \vec v = P_\alpha \overline{I} &#43; \overline{\Pi}_\alpha\]
+</span>
+<script>
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
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+    });
+  };
+</script>
+<p>where we&rsquo;ve decomposed the pressure into an isotropic value <span>
+  \( P_\alpha \)
+</span>
+<script>
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+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and what&rsquo;s called the Braginskii stress tensor <span>
+  \( \overline{\Pi}_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The average isotropic pressure is given by the reduced 2nd moment:</p>
+<span>
+  \[P_\alpha = n_\alpha T_\alpha = \frac{1}{3} \int m_\alpha \vec w \cdot \vec w f_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where the factor of <span>
+  \( 1/3 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> comes from the number of degrees of freedom in our system. It is related to the thermodynamic factor <span>
+  \( \gamma \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> where <span>
+  \( \gamma = \frac{DOF &#43; 2}{DOF} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Now we can define the temperature <span>
+  \( T_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as</p>
+<span>
+  \[T_\alpha (\vec x, t) = \frac{1}{DOF} \frac{\int m_\alpha \vec w \cdot \vec w f_\alpha \dd \vec v}{\int f_\alpha \dd \vec v}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Now that we&rsquo;ve got <span>
+  \( n_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( v_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, and <span>
+  \( T_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> we have what we need to define a Maxwellian distribution. Higher moments would be required to describe non-Maxwellian distribution functions. For example, the 3rd moment is called the skewness of the distribution. The 4th moment is the kurtosis. So on and so forth. These give a measure of degree of departure from a Maxwellian distribution, in which case it is often more useful to talk about the excess kurtosis, where the excess kurtosis of a Maxwellian is defined to be zero. You can continue to calculate the moment expansion, and in general it requires an infinite number of moments to describe an arbitrary distribution function. Because of the Boltzmann H-theorem and the tendency of plasmas to quickly relax to Maxwellian, we can usually get away with using just the first three moments.</p>
+<p>Now, what are the governing equations? We get these by taking moments of the Boltzmann equation. Let&rsquo;s proceed carefully in sections, so we&rsquo;ll integrate each piece of the BE in terms.</p>
+<h3 id="0th-moment-of-boltzmann-equation-conservation">
+  0th Moment of Boltzmann Equation (Conservation)
+  <a class="anchor" href="#0th-moment-of-boltzmann-equation-conservation">#</a>
+</h3>
+<span>
+  \[\int \pdv{f_\alpha}{t} \dd \vec v &#43; \int \vec v \cdot \pdv{f_\alpha}{x} \dd v &#43; \int \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \left. \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>First, we take <span>
+  \( \int \pdv{f_\alpha}{t} \dd \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Because t and v are both independent variables, with an argument of sufficient smoothness we can reverse the order of integration and differentiation</p>
+<span>
+  \[\int \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t}\int f_\alpha \dd v = \pdv{n_\alpha}{t}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>For <span>
+  \(  \int \vec v \cdot \pdv{f_\alpha}{x} \dd v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> we can perform an integration by parts</p>
+<span>
+  \[ \int \vec v \cdot \pdv{f_\alpha}{x} \dd v = \int \pdv{}{\vec x} \cdot ( \vec v f_\alpha) \dd \vec v - \int f_\alpha \pdv{}{\vec x} \cdot \vec v \dd \vec v \\
+ = \int \pdv{}{\vec x} \cdot ( \vec v f_\alpha) \dd \vec v  \\
+ = \pdv{}{\vec x} \cdot \int \vec v f_\alpha \dd \vec v \\
+ = \pdv{}{\vec x} \cdot (n_\alpha \vec v_\alpha) = \div (n_\alpha \vec v_\alpha)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Once again, we&rsquo;ve switched the order of integration of <span>
+  \( x \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  and <span>
+  \( v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, which we can only do because we have specified that <span>
+  \( f \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is a distribution function, and as such meets the criterion of sufficient smoothness.</p>
+<p>For the last part, we can write it as a surface integral</p>
+<span>
+  \[\int \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \oint f_\alpha \frac{q_\alpha}{m_\alpha} ( \vec E &#43; \vec v \cross \vec B) \cdot \dd S_v - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot (\vec E &#43; \vec v \cross \vec B) \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>For <span>
+  \( f_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to be well-defined, we require <span>
+  \( \lim_{v \rightarrow \infty} v^3 f_\alpha = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> so the surface term vanishes and we&rsquo;re left with</p>
+<span>
+  \[\int \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot (\vec E &#43; \vec v \cross \vec B) \dd \vec v \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Distributing the divergence through,</p>
+<p><span>
+  \[\pdv{}{\vec v} \cdot \vec E = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+because <span>
+  \( \vec E \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is independent of <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p><span>
+  \[\pdv{}{\vec v} \cdot ( \vec v \cross \vec B) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+because <span>
+  \( \vec v \cross \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is always orthogonal to <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>We&rsquo;ve been working through this in normal vector notation for familiarity, but there is another notation known as Einstein Tensor Notation using the lovely Levi-Civita symbol <span>
+  \( \epsilon_{ijk} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<span>
+  \[\epsilon_{ijk} = 1 \qquad \text{even permutations of i j k} \\
+ = -1 \qquad \text{odd permutations of i j k} \\
+= 0 \qquad \text{any repeated indexes}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can write out vector products as products of indices and operators, and any repeated indices are implicitly summed:</p>
+<span>
+  \[\vec v \cross \vec B = \epsilon_{ijk} v_j B_k = \sum_{jk} \epsilon_{ijk} v_j B_k\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where <span>
+  \( \epsilon_{ijk} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is defined to be 1 for even permutations of ijk, -1 for negative permutations if ijk, and 0 for any repeated indices.</p>
+<p>We can now write the derivative with respect to <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as</p>
+<span>
+  \[\pdv{}{\vec v} \cdot \vec v \cross \vec B = \partial _{v_i} \epsilon_{ijk} v_j B_k \\
+= \epsilon_{ijk} (\partial_{v_i} v_j) B_k\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We see that we&rsquo;re taking the derivative of the j-th component of velocity with respect to the i-th component of velocity, and <em>that&rsquo;s</em> how we can most easily point out that the quantity is zero without relying on properties of 3-vector products.</p>
+<p>We&rsquo;ll also want to use the divergence in Einstein tensor notation</p>
+<span>
+  \[\div \vec A = \partial_i A_j \delta_{ij} = \partial_i A_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Finally, we come back to the collision term in the zeroth moment of the B-M equation</p>
+<span>
+  \[\sum_\beta \int C_{\alpha \beta} \dd \vec v = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can say the collision term is zero by making a physical argument, rather than a mathematical one. We assert that collisions cannot create or destroy particles. This assumption is now baked into our equations going forward, but is not always true! Ionization, recombination, fusion reactions all create/destroy species.</p>
+<p>Finally, we&rsquo;ve got</p>
+<span>
+  \[\pdv{n_\alpha}{t} &#43; \div ( n_\alpha \vec v_\alpha) = 0 \qquad \text{continuity equation}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>So by taking the 0th moment of the Boltzmann Equation, we&rsquo;ve arrived at the continuity equation for <span>
+  \( n_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> by introducing <span>
+  \( \vec v_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<h3 id="1st-moment-of-boltzmann-equation">
+  1st Moment of Boltzmann Equation
+  <a class="anchor" href="#1st-moment-of-boltzmann-equation">#</a>
+</h3>
+<span>
+  \[\int \vec v \pdv{f_\alpha}{t} \dd \vec v &#43; \int \vec v \vec v  \pdv{f_\alpha}{\vec x} \dd \vec v &#43; \int \vec v \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \left. \vec v \pdv{f_\alpha}{t} \right|_{coll}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Term 1:</p>
+<span>
+  \[\int \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v f_\alpha \dd \vec v - \int \pdv{\vec v}{t} f_\alpha \dd \vec v \\
+= \pdv{}{t} \left( n_\alpha \vec v_\alpha \right)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Before we get to the second term, let&rsquo;s have an aside about dyad math. An outer product <span>
+  \( \vec A \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> gives a second-rank tensor. In Cartesian coordinates, it looks like</p>
+<span>
+  \[\vec A \vec B = \begin{bmatrix}
+A_x B_x &amp; A_x B_y &amp; A_x B_z \\
+A_y B_x &amp; A_y B_y &amp; A_y B_z \\
+A_z B_x &amp; A_z B_y &amp; A_z B_z \\
+ \end{bmatrix}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>A useful property to do with dot products:</p>
+<span>
+  \[\vec A \vec B \cdot \vec C = \vec A ( \vec B \cdot \vec C ) = (\vec A \vec B) \cdot \vec C\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[( \vec V \cdot \grad) \vec B = \vec V \cdot ( \grad \vec B)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Back to term 2:</p>
+<span>
+  \[\int \vec v \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v = \int \vec v \cdot \pdv{}{\vec x} ( \vec v f_\alpha) \dd \vec v - \int \vec v f_\alpha \pdv{}{\vec x} \cdot \vec v \dd \vec v \\
+ = \int \pdv{}{\vec x} \cdot ( \vec v \vec v f_\alpha) \dd \vec v - \int \vec v f_\alpha \cdot \pdv{\vec v}{\vec x} \dd \vec v \\
+ = \pdv{}{\vec x} \cdot \int \vec v \vec v f_\alpha \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Re-expanding in terms of random velocity <span>
+  \( \vec v = \vec v_\alpha &#43; \vec w \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[= \pdv{}{\vec x} \cdot \int (\vec v_\alpha &#43; \vec w) (\vec v_\alpha &#43; \vec w) f_\alpha \dd \vec v \\
+= \pdv{}{\vec x} \cdot \int (\vec v_\alpha \vec v_\alpha &#43; \vec v_\alpha \vec w &#43; \vec w \vec v_\alpha  &#43; \vec w \vec w)  f_\alpha \dd \vec v \\
+= \pdv{}{\vec x} \cdot \left[ \vec v_\alpha \vec v_\alpha \int f_\alpha \dd \vec v &#43; \vec v_\alpha \int \vec w f_\alpha \dd \vec v &#43; \left( \int \vec w f_\alpha \dd \vec v \right) \vec v_\alpha &#43; \int \vec w \vec w f_\alpha f_\alpha \dd \vec v \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The middle two terms are zero by the very definition of <span>
+  \( \vec v_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The random velocity <span>
+  \( \vec w \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is defined such that the integral of the random velocity over all phase space is zero.</p>
+<span>
+  \[= \pdv{}{\vec x} \cdot \left( \vec v_\alpha \vec v_\alpha n_\alpha \right) &#43; \pdv{}{\vec x} \cdot \frac{ \vec P_\alpha}{m_\alpha} \\
+= \div (n_\alpha \vec v_\alpha \vec v_\alpha) &#43; \frac{1}{m_\alpha} \div \vec P_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Substituting in the pressure tensor,</p>
+<span>
+  \[= \div ( n_\alpha \vec v_\alpha \vec v_\alpha ) &#43; \frac{1}{m_\alpha} \left( \grad P_\alpha  &#43; \div \vec \Pi _\alpha \right)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Moving on to term 3:</p>
+<span>
+  \[\int \vec v \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \int \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[( \vec E &#43; \vec v \cross \vec B) f_\alpha \right] \vec v \dd \vec v - \int \frac{q_\alpha}{m_\alpha} f_\alpha \pdv{}{\vec v} \cdot \left[ \vec E &#43; \vec v \cross \vec B \right] \vec v \dd \vec v \\
+= \int \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[ f_\alpha (\vec E &#43; \vec v \cross \vec B) \vec v \right] \dd \vec v - \int \frac{q\alpha}{m\alpha} f ( \vec E &#43; \vec v\alpha \cross \vec B) \cdot \pdv{\vec v}{\vec v} \dd \vec v \\
+= \frac{q\alpha}{m\alpha} \oint f\alpha ( \vec E &#43; \vec v\alpha \cross \vec B) \vec v \cdot \dd \vec S_v - \frac{q\alpha}{m\alpha} \left[ \vec E \int f \dd \vec v\alpha - \vec B \cross \int \vec v f\alpha \dd \vec v \right] \\
+= - \frac{q\alpha}{m\alpha} n\alpha (\vec E &#43; v\alpha \cross \vec B)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Finally, the 4th term gives</p>
+<span>
+  \[ \int \left. \vec v \pdv{f_\alpha}{t} \right|_{coll} = \sum_\beta \int \vec v C_{\alpha \beta} \dd \vec v \\
+= \int \vec v C_{\alpha \alpha} \dd \vec v &#43; \sum_{\beta \neq \alpha} \int \vec v C_{\alpha \beta} \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Collisions of like particles <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> do not result in a net change of momentum of species <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, so all we have left is the change in momentum due to collisions of unlike particles</p>
+<span>
+  \[= \sum_{\beta \neq \alpha} \left[ \vec v_{\alpha} \int C_{\alpha \beta} \dd \vec v &#43; \int \vec w C_{\alpha \beta} \dd \vec v \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>For the same reason as before, we assert that collisions between particles <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  and <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  do not lead to the creation or destruction of any species, so the 0th moment <span>
+  \( \int C_{\alpha \beta} \dd \vec v = 0\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. This leads to the conclusion that only random motion contributes to momentum transfer, not <span>
+  \( \vec v_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Viscosity and friction are good examples of similar physical processes where bulk velocity does not transfer momentum, but random motion does.</p>
+<p>The momentum transfer from <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  to <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> must equal transfer from <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Momentum is globally conserved.</p>
+<span>
+  \[\int \vec w C_{\alpha \beta} \dd \vec v = - \int \vec w C_{\beta \alpha} \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Now we can finally write out the full momentum equation</p>
+<span>
+  \[\pdv{}{t} (n_\alpha \vec v_\alpha ) &#43; \div (n_\alpha \vec v_\alpha \vec v_\alpha) &#43; \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E &#43; \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The collision term is often represented as the momentum transfer vector <span>
+  \( \vec R_{\alpha \beta} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> from <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p><span>
+  \[\vec R_{\alpha \beta} \equiv m_\alpha \int \vec w C_{\alpha \beta} \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\pdv{}{t} (m_\alpha n_\alpha v_\alpha) &#43; \div (m_\alpha n_\alpha \vec v_\alpha \vec v_\alpha) &#43; \grad \vec P_{\alpha} &#43; \div \vec \Pi_{\alpha} - q_\alpha n_\alpha (\vec E &#43; \vec v _\alpha \cross \vec B) = \sum_{\beta} R_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>Once again, we&rsquo;ve written the above in a conservation law form. The terms that aren&rsquo;t strictly conservation terms are the source term <span>
+  \( q_\alpha n_\alpha (\vec E &#43; \vec v _\alpha \cross \vec B) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and sink term <span>
+  \( \sum_{\beta} R_{\alpha \beta} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>Introducing the mass density</p>
+<span>
+  \[\rho_\alpha = m_\alpha n_\alpha \qquad \text{mass density}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p><span>
+  \[\pdv{}{t} (\rho_\alpha \vec v _\alpha) &#43; \div (\rho_\alpha \vec v_\alpha v_\alpha &#43; \ldots \\
+= \rho_\alpha \pdv{\vec v_\alpha}{t} &#43; \vec v_\alpha \left( \pdv{\rho_\alpha}{t} &#43; \div \rho_\alpha \vec v_\alpha \right) &#43; \rho_\alpha \vec v_\alpha \cdot \grad \vec v_\alpha &#43; \ldots\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+The term in parentheses is just the continuity equation, which is zero, and what&rsquo;s left is a more usual form of the momentum equation</p>
+<span>
+  \[\text{Momentum Equation:}\\
+\rho_\alpha \left(\pdv{\vec v_\alpha}{t} &#43; \vec v_\alpha \cdot \grad \vec v_\alpha \right) &#43; \grad \vec P_\alpha &#43; \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E &#43; \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>So by taking the 1st moment of the BE, we arrive at the momentum equation for <span>
+  \( \vec v_\alpha \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and we have introduced <span>
+  \( \vec P_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<h3 id="momentum-transfer-collisions">
+  Momentum transfer (collisions)
+  <a class="anchor" href="#momentum-transfer-collisions">#</a>
+</h3>
+<p>Looking back at <span>
+  \( \vec R_{\alpha \beta} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, it&rsquo;s worth noting that the actual situation is complicated by the potential for multi-body collisions. While it is true that the probability of, e.g. three-body collisions is small, but there are many three-body collisional processes which are extremely important in plasma physics. Three-body recombination is the primary loss term in the ionization balance, for example. For now the simplest model is to only include binary collisions that result in small angel deflections. If we suppose we only have two species with distributions that look like the following:</p>
+<p align="center"> <img alt="Figure 12.2" src="/r/img/12.2.png" /> </p>
+<p>then we expect collisions between species to tend to drag the distributions towards each other. We should expect the collision-based momentum transfer to be proportional to the distance between the distributions, i.e. the total current density <span>
+  \( \vec j \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>For an ion-electron plasma, the collision momentum transfer vector (to ions from electrons) is</p>
+<span>
+  \[\vec R_{ie} = - n_e e \eta \vec j\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where</p>
+<span>
+  \[\eta = \frac{m_e \nu_{ei}}{n e^2} \qquad \text{resistivity}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\vec j = n e (\vec v_i - \vec v_e) \qquad \text {current density for Z = 1}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
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+        "\\gr": "\\gamma",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h3 id="2nd-moment-of-boltzmann-equation">
+  2nd Moment of Boltzmann Equation
+  <a class="anchor" href="#2nd-moment-of-boltzmann-equation">#</a>
+</h3>
+<p>Finally we&rsquo;re on to our last expansion term. In general, we would say that the second moment would be</p>
+<span>
+  \[\int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\gr": "\\gamma",
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+      }
+    });
+  };
+</script>
+<p>Usually we think of the second moment as giving us energy, a scalar, rather than a tensor equation that we have here. Since the pressure tensor is symmetric, the energy equation will contain 6 independent quantities. Together with the continuity and velocity equations, we have 10 quantities in total, so we call this expansion a 10-moment expansion.</p>
+<p>Instead of the full tensor equation, we&rsquo;re actually going to compute the <em>reduced 2nd moment</em> by contracting <span>
+  \( \vec v \cdot \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\int \vec v \cdot \vec v \pdv{f_\alpha}{t} \dd v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>So, let&rsquo;s get after it:</p>
+<span>
+  \[\int v^2 \pdv{f_\alpha}{t} \dd \vec v &#43; \int v^2 \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v &#43; \int v^2 \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v = \left. \int v^2 \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Simplifying&hellip; Term 1:</p>
+<span>
+  \[\int v^2 \pdv{f_\alpha}{t} \dd \vec v = \int \pdv{}{t} (v^2 f_\alpha) \dd \vec v - \int \pdv{v^2}{t} f_\alpha \dd \vec v \\
+= \pdv{}{t} \int (\vec v_\alpha &#43; \vec w) \cdot (\vec v_\alpha &#43; \vec w) f_\alpha \dd \vec v \\
+= \pdv{}{t} \left[ \int v_\alpha ^2 f_\alpha \dd \vec v &#43; 2 \int \vec v_\alpha \cdot \vec w f_\alpha \dd \vec v &#43; \int w^2 f_\alpha \dd \vec v \right] \\
+= \pdv{}{t} \left[ v_\alpha ^2 \int f_\alpha \dd \vec v &#43; 2 \vec v_\alpha \cdot \int \vec w f_\alpha \dd \vec v &#43; 3 \frac{n_\alpha T_\alpha}{m_\alpha} \right] \\
+= \pdv{}{t} \left[ n_\alpha v_\alpha ^2 &#43; 3 \frac{n_\alpha T_\alpha}{m_\alpha} \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
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+        "\\gr": "\\gamma",
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+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Term 2:</p>
+<span>
+  \[\int v^2 \vec v \cdot \pdv{f_\alpha}{\vec x} \dd \vec v = \int \pdv{}{\vec x} \cdot (v^2 \vec v f_\alpha) \dd \vec v - \int f_\alpha \pdv{}{\vec x} \cdot (v^2 \vec v) \dd \vec v \\
+= \pdv{}{\vec x} \cdot \int ( \vec v_\alpha &#43; \vec w) \cdot (\vec v_\alpha &#43; \vec w) \vec w f_\alpha \dd \vec v \\
+= \pdv{}{\vec x} \cdot \int (v_\alpha ^2 &#43; 2 \vec v_\alpha \cdot \vec w &#43; w^2) \vec v f_\alpha \dd \vec v \\
+= \pdv{}{\vec x} \cdot \left[ \int v_\alpha ^2 (\vec v_\alpha &#43; \vec w) f_\alpha \dd \vec v &#43; \int 2 \vec v_\alpha \cdot \vec w (\vec v_\alpha &#43; \vec w) f_\alpha \dd \vec v &#43; \int w^2 (\vec v_\alpha &#43; \vec w) f_\alpha \dd \vec v \right] \\
+= \pdv{}{\vec x} \cdot \left[ v_\alpha ^2 \vec v_\alpha \int f_\alpha \dd \vec v &#43; v_\alpha ^2 \int \vec w f_\alpha \dd \vec v &#43; 2 \vec v_\alpha \cdot \left(\int \vec w f_\alpha \dd \vec v \right) \vec v_\alpha \\
+\qquad &#43; 2 \vec v_\alpha \cdot \int \vec w \vec w f_\alpha \dd \vec v &#43; \vec v_\alpha \int w^2 f_\alpha \dd \vec v &#43; \int \vec w w^2 f_\alpha \dd \vec v \right] \\
+= \pdv{}{\vec x} \cdot (n_\alpha v_\alpha ^2 \vec v_\alpha) &#43; \pdv{}{\vec x} \cdot 2 \vec v_\alpha \cdot \frac{\vec P_\alpha}{m_\alpha} &#43; \pdv{}{\vec x} \cdot \left(\frac{3 n_\alpha T_\alpha \vec v_\alpha}{m_\alpha} \right) &#43; \pdv{}{\vec x} \cdot \left( \frac{2 \vec h_\alpha}{m_\alpha} \right)\]
+</span>
+<script>
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+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>In keeping with the progression so far, the very last term would be something like a full 3rd moment <span>
+  \( \int \vec w \vec w \vec w f_\alpha \dd \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Instead, we have a &ldquo;contracted 3rd moment&rdquo; of the distribution <span>
+  \( \vec h_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> which is actually the <em>heat flux</em>, or the random energy flux of random energy.</p>
+<p>Term 3:</p>
+<span>
+  \[\int v^2 \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} \dd \vec v \\
+ = \oint f_\alpha \frac{q_\alpha}{m_\alpha} v^2 ( \vec E &#43; \vec v \cross \vec B) \cdot \dd \vec S_v - \int f_\alpha \frac{q_\alpha}{m_\alpha} \pdv{}{\vec v} \cdot \left[ v^2 ( \vec E &#43; \vec v \cross \vec B) \right] \dd \vec v \\
+= - \frac{q_\alpha}{m_\alpha} \int f_\alpha \left[ v^2 \left(\pdv{}{\vec v} \cdot \vec E \right) &#43; \left(\vec E \cdot \pdv{}{\vec v} v^2 \right) \right] \dd \vec v  \\
+\qquad - \frac{q_\alpha}{m_\alpha}\int f_\alpha \left[v^2 \pdv{}{\vec v} \cdot (\vec v \cross \vec B) &#43; \vec v \cross \vec B \cdot \pdv{}{\vec v} v^2 \right] \dd \vec v \\
+= - \frac{q_\alpha}{m_\alpha} \int f_\alpha (2 \vec E \cdot \vec v) \dd \vec v \\
+= - \frac{2 q_\alpha}{m_\alpha} n_\alpha \vec v_\alpha \cdot \vec E\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We&rsquo;ve ended up with the work done on a fluid by the electric field. As expected, there is no work term associated with the bulk magnetic field.</p>
+<p>Term 4:</p>
+<span>
+  \[\left. \int v^2 \pdv{f_\alpha}{t} \right|_{coll} \dd \vec v = \int v^2 C_{\alpha \alpha} \dd \vec v &#43; \sum _{\beta \neq \alpha} \int v^2 C_{\alpha \beta} \dd \vec v \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can make an energy conservation argument to get rid of the first term. The first term goes to 0 since collisions of like particles results in no net energy exchange.</p>
+<span>
+  \[= \sum_{\beta \neq \alpha} \int ( \vec v_\alpha &#43; \vec w) \cdot(\vec v_\alpha &#43; \vec w) C_{\alpha \beta} \dd \vec v \\
+= \sum_{\beta \neq \alpha}\left( v_\alpha ^2 \int C_{\alpha \beta} \dd \vec v &#43; 2 \vec v_\alpha \cdot \int \vec w C_{\alpha \beta} \dd \vec v &#43; \int w^2 C_{\alpha \beta} \dd \vec v \right)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Since collisions (we assume) don&rsquo;t create or destroy particles, the first term <span>
+  \( v_\alpha ^2 \int C_{\alpha \beta} \dd \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> goes away. The second term <span>
+  \( \vec v_\alpha \cdot \int \vec w C_{\alpha \beta} \dd \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> we interpret as a frictional force due to the relative motion between <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  and <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[= \sum_{\beta \neq \alpha} 2 \frac{\vec v_\alpha \cdot \vec R_{\alpha \beta}}{m_\alpha} &#43; 2 \frac{Q_{\alpha \beta}}{m_\alpha}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where <span>
+  \( Q_{\alpha \beta} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the heat exchange term (heat generation term in Braginskii). It describes the heat exchange by random collisions of unlike particles, analogous to viscous heating.</p>
+<span>
+  \[Q_{\alpha \beta} = \frac{1}{2} \int m_\alpha w^2 C_{\alpha \beta} \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we were to draw distribution functions for species <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  and <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> (as shown), even though they have the same centroid they will interact by viscous heating. Species <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> will heat up and <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> will cool</p>
+<p align="center"> <img alt="Figure 12.3" src="/r/img/12.3.png" /> </p>
+<p>If we multiply by <span>
+  \( \frac{1}{2} m_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> then the energy equation becomes</p>
+<span>
+  \[\text{Energy Equation: } \\ \qquad \pdv{}{t} \left( \frac{3}{2} n_\alpha T_\alpha &#43; \frac{1}{2} \rho_\alpha v_\alpha ^2 \right) &#43; \\
+ \div \left(\frac{3}{2} n_\alpha T_\alpha &#43; \frac{1}{2} \rho_\alpha v_\alpha ^2 \right) \vec v_\alpha &#43; \div (\vec v_\alpha \cdot \vec P_\alpha) &#43; \div \vec h_\alpha - q_\alpha n_\alpha \vec v_\alpha \cdot \vec E \\ =  \sum_{\beta \neq \alpha} (v_\alpha \cdot \vec R_{\alpha \beta} &#43; Q_{\alpha \beta})\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Notice that the total energy appears</p>
+<span>
+  \[E_\alpha = \frac{3}{2} n_\alpha T_\alpha &#43; \frac{1}{2} \rho_\alpha v_\alpha ^2\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The factor of <span>
+  \( \frac{3}{2} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as usual comes from <span>
+  \( \gamma \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{3}{2} = \frac{1}{\gamma - 1} = \frac{1}{5/3 - 1}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where</p>
+<span>
+  \[\gamma = \frac{DOF &#43; 2}{DOF} = \frac{5}{3}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can remove the kinetic portion of the energy equation by subtracting the product of the momentum equation with <span>
+  \( \vec v_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<span>
+  \[\rho_\alpha \vec v_\alpha \cdot \pdv{v_\alpha}{t} = \frac{1}{2} \pdv{}{t} (\rho_\alpha v_\alpha ^2)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p><span>
+  \[\pdv{}{t} \left( \frac{3}{2} n_\alpha T_\alpha \right) &#43; \div \left(\frac{3}{2} n_\alpha T_\alpha \vec v_\alpha \right) &#43; \div \left( \vec v_\alpha \cdot \vec P_\alpha \right) - \vec v_\alpha \cdot (\div \vec P_\alpha) &#43; \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} &#43; \vec v_\alpha \cdot \div T_\alpha \right) &#43; \frac{3}{2} T_\alpha \left[ \pdv{n_\alpha}{t} &#43; \div (n_\alpha \vec v_\alpha) \right] &#43; \vec P_\alpha \cdot \cdot \grad \vec v_\alpha &#43; \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} &#43; \vec v_\alpha \cdot \div T_\alpha \right) &#43; \vec P_\alpha \cdot \cdot \grad \vec v_\alpha &#43; \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>Where <span>
+  \( \vec P_\alpha \cdot \cdot \grad \vec v_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is a tensor contraction in two indices. It is a generalization of the dot product, i.e.
+<span>
+  \[\vec P_\alpha \cdot \cdot \grad \vec v_\alpha = \delta_{ik} \delta_{jl} P_{ij} \partial_k v_l \\
+= P_{xx} \partial_x v_x &#43; P_{xy} \partial_y v_x &#43; \ldots\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} &#43; \vec v_\alpha \cdot \grad T_\alpha \right) &#43; P_\alpha \div \vec v_\alpha &#43; \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha &#43; \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Now the continuity, momentum, and energy equations describe the evolution of each fluid <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The only velocity information we have retained in taking moments is the centroid <span>
+  \( \vec v_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and the width <span>
+  \( T_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. This is the only information we will include in the 5-moment model (5N-moment plasma fluid model). The higher moments (describing skewness, kurtosis) are not evolved with the 5-moment model. There are fluid models that do evolve those moments, but we won&rsquo;t touch on those here.</p>
+<h3 id="closure-problem">
+  Closure Problem
+  <a class="anchor" href="#closure-problem">#</a>
+</h3>
+<p>We still need to &ldquo;close&rdquo; the fluid model (i.e. solving the <strong>Closure Problem</strong>). As we have been finding out, each additional moment of the Boltzmann Equation introduces the next-higher moment. In calculating the 0th moment, we introduce the 1st moment, etc. We need to address the closure problem by relating higher moment variables to lower moment variables. Applying a closure scheme (relating variables that are not evolved directly) is usually equivalent to making a statement on heat flow.</p>
+<p>In our model, we are not evolving the heat flux <span>
+  \( \vec h_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, so we need to relate it to one of the other variables that we do evolve directly. We usually do that by writing down a conductivity relation</p>
+<span>
+  \[\vec h_\alpha = - \kappa \grad T_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>for some conductivity <span>
+  \( \kappa \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>We&rsquo;re also only evolving the scalar pressure, so we need to relate the stress tensor <span>
+  \( \vec \Pi_\alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to the other variables. We can do that with a viscosity relation</p>
+<span>
+  \[\vec \Pi_\alpha = \nu \grad \vec v_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>for some viscosity <span>
+  \( \nu \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. We call the introduced quantities <span>
+  \( \kappa \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  and <span>
+  \( \nu \)
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> the <strong>transport coefficients</strong>. They are usually derived by performing an expansion of the equations we&rsquo;ve already discussed. They are what lead to &ldquo;nondimensional&rdquo; numbers that characterize the flow in our system.</p>
+<p>We can also achieve closure by constraining portions of the system. In an isothermal system</p>
+<p><span>
+  \[\text{isothermal: } \quad T = \text{const.}, p \propto n\]
+</span>
+<script>
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{adiabatic: } \quad  p \propto n^\gamma\]
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{force-free, cold: } \quad p = 0\]
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
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+      }
+    });
+  };
+</script></p>
+<p>The expressions that lead us to a closure scheme are the <strong>equations of state</strong>. They are not time-dependent, they simply relate fluid parameters to other properties of the system.</p>
+<p>Maxwell&rsquo;s equations to couple the electromagnetic terms to the fluid variables.
+<span>
+  \[\pdv{\vec B}{t} = - \curl \vec E\]
+</span>
+<script>
+  function loadKatex() {
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+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+      }
+    });
+  };
+</script>
+<span>
+  \[\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_{alpha} q_\alpha n_\alpha \vec v_\alpha \]
+</span>
+<script>
+  function loadKatex() {
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+</script>
+<span>
+  \[\epsilon_0 \div \vec E = \sum_\alpha q_\alpha n_\alpha \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+      }
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+</script>
+<span>
+  \[\div \vec B = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>The fluid variables provide the source terms for Ampere&rsquo;s law and Gauss' law, and we&rsquo;ve already seen how the electromagnetic fields appear in the momentum equations as source terms for the fluid forces.</p>
+<h3 id="stopping-after-the-2nd-moment-5n-moment-plasma-fluid-model">
+  Stopping after the 2nd moment (5N-Moment Plasma Fluid Model)
+  <a class="anchor" href="#stopping-after-the-2nd-moment-5n-moment-plasma-fluid-model">#</a>
+</h3>
+<p>What assumptions have we made by stopping here? It&rsquo;s important to know what they are and to recognize what they mean for the fluid model.</p>
+<ul>
+<li>Each species <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is well-represented by a Maxwellian with a small perturbation. The pressure tensor is not strictly diagonal and the heat flux is not zero, so the small perturbations are what lead to the transport coefficients. This is due to the process by which we obtain the transport coefficients, called the Chapman-Enskog expansion.</li>
+<li>Kinetic effects (stream instabilities, counter-flow instabilities, etc.) are not captured.</li>
+</ul>
+<p>The variables of the 5-moment model that we <em>do</em> evolve are
+<span>
+  \[n, v_x, v_y, v_z, T\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>We can take additional moments. The 10N-moment equation evolves, in addition to the quantities in the 5N-moment model, we evolve all of the independent terms of the pressure tensor
+<span>
+  \[n, v_x, v_y, v_z, P_{xx}, P_{xy}, P_{xz} , P_{yy} , P_{yz} , P_{zz} \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>The 13-N moment model has everything from the 10N-moment model plus the heat flux tensor <span>
+  \( h_x, h_y, h_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
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+<meta property="og:description" content="Two-Fluid Plasma Model (ions-electrons) #  Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let&rsquo;s start with a cold plasma approximation (   \( p = 0 \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;" />
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+          
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+  
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class=" active">Two-Fluid Plasma Model</a>
+  
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+      
+        <li>
+          
+  
+  
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/" class="">Boundary Conditions</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class="">Equilibrium for Fusion</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/" class="">Equilibrium of 3D Configurations</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/" class="">MHD Stability</a>
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+  <article class="markdown"><h1 id="two-fluid-plasma-model-ions-electrons">
+  Two-Fluid Plasma Model (ions-electrons)
+  <a class="anchor" href="#two-fluid-plasma-model-ions-electrons">#</a>
+</h1>
+<p>Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let&rsquo;s start with a cold plasma approximation (
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \( p = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>) and neglect collisions. The momentum equation reduces to</p>
+<span>
+  \[m_\alpha \left( \pdv{\vec v_\alpha}{t} &#43; \vec v_\alpha \cdot \grad \vec v_\alpha \right) - q_\alpha (\vec E &#43; \vec v_\alpha \cross \vec B) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>From here on out we can avoid some clutter (and wrist strain) by dropping the <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> subscripts and acknowledging that we have sets of equations for ions and electrons. Apply a perturbation to an equilibrium <span>
+  \( g = g_0 &#43; g_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[m \left(\pdv{\vec v_0}{t} &#43; \vec v_0 \cdot \grad \vec v_0 &#43; \pdv{\vec v_0}{t} &#43; \vec v_1 \cdot \grad \vec v_0 &#43; \vec v_0 \cdot \grad \vec v_1 &#43; \vec v_1 \cdot \grad \vec v_1 \right) \\
+\qquad - q (E_0 &#43; E_1) - q (\vec v_0 \cross \vec B_0 &#43; \vec v_1 \cross \vec B_0 &#43; \vec v_0 \cross \vec B_1 &#43; \vec v_1 \cross \vec B_1) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can drop some terms because equilibrium has to satisfy the original equation. We can balance all of the subscript-0 terms and sum them to get zero.</p>
+<span>
+  \[m \left( \pdv{\vec v_1}{t} &#43; \vec v_1 \cdot \grad \vec v_0 &#43; \vec v_0 \cdot \grad \vec v_1 &#43; \vec v_1 \cdot \grad \vec v_1 \right) &#43; q (- E_1 - \vec v_1 \cross \vec B_0 - \vec v_0 \cross \vec B_1 - \vec v_1 \cross \vec B_1)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Let&rsquo;s now make the assumption that the perturbation is small, that is <span>
+  \( g_1 \ll g_0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. That means that nonlinear products of perturbation terms are negligible (linearization process).</p>
+<span>
+  \[m \left( \pdv{\vec v_1}{t} &#43; \vec v_0 \cdot \grad \vec v_1 &#43; \vec v_1 \cdot \grad \vec v_0 \right) - q \vec E_1 - q (\vec v_1 \cross \vec B_0 &#43; \vec v_0 \cross \vec B_1) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Now, assume that the equilibrium is a <strong>static equilibrium</strong>, that is <span>
+  \( \vec v_0 = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. If we decompose into components that are parallel and perpendicular to the equilibrium magnetic field <span>
+  \( \vec B_0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, then</p>
+<p><span>
+  \[\pdv{v_{1, \parallel}}{t} - \frac{q}{m} E_{1, \parallel} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\pdv{\vec v_{1, \perp}}{t} - \frac{q}{m} \left(\vec E_{1, \perp} &#43; B_0 \vec v_{1, \perp} \cross \vu z \right) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>The parallel component <span>
+  \( E_{1, \parallel} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> will lead us to the ordinary wave (<strong>O-wave</strong>). Consideration of the more general case with perpendicular components will lead to the <strong>X-wave</strong>.</p>
+<p>The plasma velocity is related to the fields through the current density (Maxwell equations). Faraday&rsquo;s law gives</p>
+<p><span>
+  \[\pdv{\vec B}{t} = - \curl \vec E\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\rightarrow \curl \pdv{\vec B}{t} = - \curl \curl \vec E = - \grad (\div \vec E) &#43; \nabla ^2 \vec E \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>Ampere&rsquo;s law gives</p>
+<p><span>
+  \[\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_{\alpha} q_\alpha n_\alpha \vec v_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\epsilon_0 \pdv{^2 \vec E }{t ^2} = \frac{1}{\mu_0} \curl \pdv{\vec B}{t} \\
+= \frac{1}{\mu_0} \left[ \nabla ^2 \vec E - \grad (\div \vec E) \right] - \sum_\alpha q_\alpha \pdv{}{t} (n_\alpha \vec v_\alpha) \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>Since this is a linear system, assume that the perturbed quantities have a wave-like structure. That is, the perturbed quantities <span>
+  \( g_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> are proportional to <span>
+  \( e^{i(\omega t &#43; \vec k \cdot \vec r)} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. This lets us transform the spatial and temporal derivatives into factors of <span>
+  \( \omega \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \vec k \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[- \epsilon_0 \omega ^2 \vec E_1 = - \frac{1}{\mu_0} \left[k^2 \vec E_1 - \vec k (\vec k \cdot \vec E_1) \right] &#43; i \omega e n_0 \vec v_1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Let&rsquo;s now consider only high frequency oscillations, assuming that only the electrons respond and the ions remain stationary. There&rsquo;s nothing particularly complicated about including the ion response, this just lets us drop the <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> subscripts and focus on a single set of equations.</p>
+<span>
+  \[- i \omega \frac{e n_0}{\epsilon_0} \vec v_1 = (\omega ^2 - c^2 k^2) \vec E_1 &#43; c^2 \vec k (\vec k \cdot \vec E_1)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Now let&rsquo;s apply the perturbed form to the linearized momentum equation</p>
+<span>
+  \[\pdv{v_{1, \parallel}}{t} - \frac{q}{m} E_{1, \parallel} = 0 \\
+\rightarrow i \omega v_{1, \parallel} = - \frac{e}{m} E_{1, \parallel} \\
+\rightarrow i \omega \frac{e n_0}{\epsilon_0} v_{1, \parallel} = \frac{e^2 n_0}{\epsilon_0 m} E_{1, \parallel}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Combine the momentum equation and the Maxwell equations to eliminate <span>
+  \( \vec E_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \vec v_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{e^2 n_0}{\epsilon_0 m} E_{1, \parallel} = ( \omega ^2 - c^2 k^2 ) E_{1, \parallel} &#43; c^2 k_{\parallel} \vec (k \cdot \vec E_1)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Consider different possibilities for the <span>
+  \( \vec k \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> vector. If it is along the magnetic field <span>
+  \( \vec k = k_{\parallel} \vu{e}_\parallel \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> (longitudinal wave) then</p>
+<span>
+  \[\frac{e^2 n_0}{\epsilon_0 m} = \omega ^2 - c^2 k^2 &#43; c^2 k^2 = \omega_{pe}^2\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>For <span>
+  \( \vec k = k_{\perp} \vu e_\perp \)
+</span>
+<script>
+  function loadKatex() {
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+      }
+    });
+  };
+</script> (transverse wave) then we get the dispersion relation for the O-wave</p>
+<span>
+  \[\text{dispersion relation for O-waves:} \qquad \omega^2 - c^2 k^2 = \omega_p ^2\]
+</span>
+<script>
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+</script>
+<p>The electric field is in the same direction as the magnetic field <span>
+  \( (\vec E_1 = \vec E_{1, \parallel}) \)
+</span>
+<script>
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+</script>, which means the O-wave is linearly polarized. At large <span>
+  \( k \)
+</span>
+<script>
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+    });
+  };
+</script> we just have regular light waves, but as we turn the frequency downwards we see a cut-off at the plasma frequency:</p>
+<p align="center"> <img alt="Figure 12.4" src="/r/img/12.4.png" /> </p>
+<p>It turns out that the dispersion relation for the X-wave has the same cut-off, but also has another branch with a resonance</p>
+<p align="center"> <img alt="Figure 12.5" src="/r/img/12.5.png" /> </p>
+<p>The two-fluid plasma model is highly reduced from the full kinetic model, but it is still too complete to be useful when studying gross plasma behavior. Further reductions of the model are possible by making asymptotic assumptions:</p>
+<h2 id="low-frequency-asymptotic-assumption">
+  Low-frequency Asymptotic Assumption
+  <a class="anchor" href="#low-frequency-asymptotic-assumption">#</a>
+</h2>
+<ul>
+<li>Eliminate high frequency, short wavelength phenomena by using pre-Maxwell field equations. Formally, this is <span>
+  \( \epsilon_0 \rightarrow 0 \)
+</span>
+<script>
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+      }
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+  };
+</script>.</li>
+</ul>
+<p>The direct consequences of the low-frequency approximation are</p>
+<p><span>
+  \[c^2 = \frac{1}{\epsilon_0 \mu_0} \qquad c \rightarrow \infty\]
+</span>
+<script>
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+</script>
+<span>
+  \[\omega_p ^2 = \frac{n e^2}{\epsilon_0 m} \qquad \omega_p \rightarrow \infty\]
+</span>
+<script>
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+</script>
+<span>
+  \[\lambda_D = \frac{v_T}{\omega_p} \rightarrow 0\]
+</span>
+<script>
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+</script></p>
+<p>This means that all phenomena will have <span>
+  \( \omega \ll \omega_p \)
+</span>
+<script>
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+      }
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+  };
+</script>, limiting the frequencies we can resolve to the ion plasma frequency. The characteristic speeds will be limited by the speed of light</p>
+<span>
+  \[\frac{\omega}{k} \ll c\]
+</span>
+<script>
+  function loadKatex() {
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+  };
+</script>
+<p>and all characteristic lengths will be much greater than the Debye length</p>
+<span>
+  \[x_0 \gg \lambda_D\]
+</span>
+<script>
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+  };
+</script>
+<p>Looking at Gauss' law,
+<span>
+  \[\epsilon_0 \div \vec E = \sum_\alpha q_\alpha n_\alpha \rightarrow \sum_\alpha q_\alpha n_\alpha = 0\]
+</span>
+<script>
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+  };
+</script></p>
+<p>so we now have charge neutrality everywhere in the domain. For H plasma, locally we have <span>
+  \( n_e = n_i \)
+</span>
+<script>
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+  };
+</script> everywhere.</p>
+<p>Looking at Ampere&rsquo;s law,</p>
+<span>
+  \[\epsilon_0 \pdv{\vec E}{t} = \frac{1}{\mu_0} \curl \vec B - \sum_\alpha q_\alpha n_\alpha \vec v_\alpha = 0 \\
+\rightarrow \vec j = \frac{1}{\mu_0} \curl \vec B\]
+</span>
+<script>
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+</script>
+<p>Things we do <strong>not</strong> get from this approximation are <span>
+  \( \vec E = 0 \)
+</span>
+<script>
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+</script> or <span>
+  \( \pdv{\vec E}{t} = 0 \)
+</span>
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. It does mean that plasma dynamics occur on a sufficiently large spatial scale that charge separation is small, and they occur on a sufficiently long temporal scale that electrons respond quickly.</p>
+<h2 id="tiny-electron-asymptotic-assumption">
+  Tiny electron asymptotic assumption
+  <a class="anchor" href="#tiny-electron-asymptotic-assumption">#</a>
+</h2>
+<p>2nd approximation: neglect electron inertia in the momentum equation. Formally, we let the electron mass <span>
+  \( m_e \rightarrow 0 \)
+</span>
+<script>
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+      }
+    });
+  };
+</script></p>
+<p><span>
+  \[\omega_{pe}^2 = \frac{n e^2}{\epsilon_0 m_e} \rightarrow \infty\]
+</span>
+<script>
+  function loadKatex() {
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+      }
+    });
+  };
+</script>
+<span>
+  \[\omega_{c, e} = \frac{e B}{m_e} \rightarrow \infty\]
+</span>
+<script>
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+</script>
+<span>
+  \[v_{T, e} \rightarrow \infty\]
+</span>
+<script>
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+    });
+  };
+</script>
+The Larmor radius goes to zero
+<span>
+  \[r_{l, e} = \frac{v_{T, e}}{\omega_{c, e}} \rightarrow 0\]
+</span>
+<script>
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+</script>
+Importantly, as the gyroradius <span>
+  \( r_{l, e} \)
+</span>
+<script>
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+</script> goes to 0 (because the thermal velocity goes as <span>
+  \( \sqrt{m_e} \)
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and the cyclotron frequency goes as <span>
+  \( m_e \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>), this means that the electrons are tied to the magnetic field.
+The skin depth is also small.
+<span>
+  \[\delta_e = \frac{c}{\omega_{p, e}} \rightarrow 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>So all phenomena that we capture must have <span>
+  \( \omega \ll \omega_{p, e} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\negadelta": "\\nabla",
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+    });
+  };
+</script>, <span>
+  \( \omega \ll \omega_{c, e} \)
+</span>
+<script>
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+        "\\div": "\\nabla \\cdot #1",
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+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
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+  };
+</script>, and <span>
+  \( x_0 \gg r_{L, e} \)
+</span>
+<script>
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
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+    });
+  };
+</script>, <span>
+  \( x_0 \gg \delta_e \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> .</p>
+<p>The electron momentum equation becomes</p>
+<span>
+  \[\grad P_e &#43; \div \vec \Pi _e &#43; e n_e (\vec E &#43; \vec v \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}\]
+</span>
+<script>
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+</script>
+<p>The momentum equation is now a state equation, not an evolution equation. It now simply relates the dynamical variables to each other at any point in time.</p>
+<p>Now, note that along magnetic field lines electrons can travel long distances at very fast (finite) speeds which can produce low frequency, long wavelength phenomena. Neglecting electron inertia implies that electrons respond instantaneously, meaning we cannot capture these modes. An example of such a phenomena is drift waves.</p>
+<p>The characteristic speeds <span>
+  \( c \)
+</span>
+<script>
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+  };
+</script>  and <span>
+  \( v_{T, e} \)
+</span>
+<script>
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> have disappeared from the model. Remaining is <span>
+  \( v_{T, i} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. This means that the ion dynamics dictate the plasma evolution.</p>
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+First, define the MHD variables:
+  \[\text{mass density:} \qquad \rho = n_i m_i &#43; n_i m_e\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;">
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+<meta property="og:description" content="Magnetohydrodynamic (MHD) Model #  Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD).
+First, define the MHD variables:
+  \[\text{mass density:} \qquad \rho = n_i m_i &#43; n_i m_e\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;" />
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class=" active">Magnetohydrodynamic (MHD) Model</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/" class="">Boundary Conditions</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class="">Equilibrium for Fusion</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/" class="">1-D Equilibria</a>
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+  <article class="markdown"><h1 id="magnetohydrodynamic-mhd-model">
+  Magnetohydrodynamic (MHD) Model
+  <a class="anchor" href="#magnetohydrodynamic-mhd-model">#</a>
+</h1>
+<p>Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD).</p>
+<p>First, define the MHD variables:</p>
+
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \[\text{mass density:} \qquad \rho = n_i m_i &#43; n_i m_e\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{fluid velocity:} \qquad \vec v = \frac{n_i m_i \vec v_i &#43; n_e m_e \vec v_e}{n_i m_i &#43; n_e m_e}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{current density:} \qquad \vec j = q_i n_i \vec v_i - e n_e \vec v_e  \rightarrow \vec v_e = \frac{q_i n_i \vec v_i}{e n_e} - \vec j / e n_e\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
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+        
+        "\\dd": "\\text{d}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{total pressure:} \qquad p = p_i &#43; p_e = n_e T_e &#43; n_i T_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{total temperature:} \qquad T = \frac{n_i T_i &#43; n_e T_e}{(n_i &#43; n_e)/2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Now let&rsquo;s begin applying asymptotic approximations. For the mass density, applying the first approx (charge neutrality) we have</p>
+<span>
+  \[\rho \approx n (m_i &#43; m_e)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Using approx 2 (vanishing electron mass)</p>
+<p><span>
+  \[\rho \approx n m_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where
+<span>
+  \[n = n_i \qquad n_e = Z n\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>The center-of-mass velocity (charge neutrality) gives</p>
+<span>
+  \[\vec v \approx \frac{m_i \vec v_i &#43; m_e Z \vec v_e}{m_i &#43; Z m_e}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>with small electron mass approximation:</p>
+<span>
+  \[\vec v \approx \vec v_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The current density is (charge neutrality approx)</p>
+<span>
+  \[\vec j \approx Z e n (\vec v_i - \vec v_e) \qquad \vec v_e = \vec v_i - \vec j / Z e n\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The pressure and total temperature are (with charge neutrality)</p>
+<span>
+  \[P \approx n (T_i &#43; Z T_e)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[T \approx \frac{T_i &#43; Z T_e}{(1 &#43; Z)/2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h3 id="mhd-momentum-equation">
+  MHD Momentum Equation
+  <a class="anchor" href="#mhd-momentum-equation">#</a>
+</h3>
+<p>Now we combine the two-fluid equations with these asymptotic approximations to obtain the governing equations for the MHD variables. Multiplying the ion continuity equation by the ion mass gives</p>
+<span>
+  \[\pdv{}{t} (m_i n_i) &#43; \div (m_i n_i \vec v_i) = 0 \\
+\rightarrow \pdv{\rho}{t} &#43; \div (\rho \vec v) = 0 \quad \text{(MHD continuity eq.)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we multiply the two-fluid continuity equations by the charge and sum them, we get</p>
+<span>
+  \[\pdv{}{t}(q_i n_i - e n_e) &#43; \div (q_i n_i \vec v_i - e n_e \vec v_e) = 0 \\
+\rightarrow \div \vec j = 0 \quad \text{(no accumulation of charge)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we add the electron and ion momentum equations and apply the small electron mass approximation, and recognizing that <span>
+  \( \vec R_{ei} = - \vec R_{ie} = n e \eta \vec j \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p><span>
+  \[\rho_i \left(\pdv{\vec i}{t} &#43; \vec v_i \cdot \grad \vec v_i \right) &#43; \grad (P_i &#43; P_e) &#43; \div (\vec \Pi _i &#43; \vec \Pi_e) \\
+\qquad  - (q_i n_i - e n_e) \vec E - (q_i n_i \vec v_i - e n_e \vec v_e) \cross \vec B = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{MHD Momentum Equation} \\
+\rightarrow \rho \left(\pdv{\vec v}{t} &#43; \vec v \cdot \grad \vec v \right) &#43; \grad p - \vec j \cross \vec B = - \div (\vec \Pi _i &#43; \vec \Pi _e)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
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+        "\\gr": "\\gamma",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>!!! info &ldquo;MHD Momentum Equation&rdquo;</p>
+<pre><code><span>
+  \[    \rho \left(\pdv{\vec v}{t} &#43; \vec v \cdot \grad \vec v \right) &#43; \grad p - \vec j \cross \vec B = - \div (\vec \Pi _i &#43; \vec \Pi _e)
+    \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+</code></pre>
+<p>If we take the electron momentum equation, apply the small electron mass asymptotic approximation and introduce the current density, then we have</p>
+<h3 id="mhd-ohms-law">
+  MHD Ohm&rsquo;s Law
+  <a class="anchor" href="#mhd-ohms-law">#</a>
+</h3>
+<span>
+  \[e n_e (\vec E &#43; \vec v_i \cross \vec B - \vec j \cross \vec B / Z e n) = - \grad P_e - \div \vec \Pi_e &#43; \vec R_{ei}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Substituting <span>
+  \( \vec R_{ei} = e n_e \eta \vec j \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{MHD Generalized Ohms Law}\\
+\vec E &#43; \vec v \cross \vec B = \eta \vec j &#43; \frac{1}{Z e n} \left(\vec j \cross \vec B - \grad p_e - \div \vec \Pi_e \right)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>!!! info &ldquo;MHD Generalized Ohms Law&rdquo;</p>
+<pre><code><span>
+  \[    \vec E &#43; \vec v \cross \vec B = \eta \vec j &#43; \frac{1}{Z e n} \left(\vec j \cross \vec B - \grad p_e - \div \vec \Pi_e \right)
+    \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+</code></pre>
+<h3 id="mhd-energy-equation">
+  MHD Energy Equation
+  <a class="anchor" href="#mhd-energy-equation">#</a>
+</h3>
+<p>The energy equation is found by adding the ion and electron energies, but first we need to manipulate them into a common form. The ion energy equation is</p>
+<span>
+  \[\frac{1}{\gamma - 1} \left[ \pdv{}{t} (n_i T_i) &#43; \div (n_i T_i \vec v_i) \right] &#43; n_i T_i \div \vec v_i = RHS_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where <span>
+  \( RHS_i \equiv - \vec \Pi_i \cdot \cdot \grad \vec v_i - \div \vec h_i &#43; Q_{ie} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{1}{\gamma - 1} \left( \pdv{p_i}{t} &#43; \vec v_i \cdot \grad p_i \right) &#43; \frac{\gamma}{\gamma - 1} p_i (\div \vec v_i) = RHS_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Multiply by <span>
+  \( \frac{\gamma - 1}{n_i ^\gamma} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and define the total derivative with respect to the ion velocity as</p>
+<span>
+  \[\dv{}{t} \equiv \pdv{}{t} &#43; \vec v_i \cdot \grad\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\rightarrow \dv{}{t} \left( \frac{p_i}{n_i ^\gamma} \right) - p_i \dv{}{t} \left( \frac{1}{n_i ^\gamma} \right) &#43; \frac{\gamma p_i}{n_i ^\gamma} \div \vec v_i = \frac{\gamma - 1}{n_i ^\gamma} RHS_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Simplify further by recognizing that carrying out the total derivative gives</p>
+<span>
+  \[\dv{}{t} \left( \frac{p_i}{n_i ^\gamma} \right) - \frac{\gamma p_i}{n_i ^{\gamma &#43; 1}} \dv{}{t} n_i &#43; \frac{\gamma p_i}{n_i ^\gamma} \div \vec v_i = \frac{\gamma - 1}{n_i ^\gamma} RHS_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The continuity equation says
+<span>
+  \[\dv{}{t} n_i = - n_i \div \vec v_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+Multiply by <span>
+  \( m_i ^{- \gamma} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and use <span>
+  \( \rho = n m_i \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and the resulting ion energy equation is</p>
+<span>
+  \[\dv{}{t} \left( \frac{p_i}{\rho ^\gamma} \right) = \frac{\gamma - 1}{\rho ^\gamma} RHS_i\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We want to get the electron energy equation in the same form. The steps are very similar:</p>
+<span>
+  \[\dv{}{t} \left( \frac{p_e}{n_e ^\gamma} \right) &#43; \vec v_e \cdot \grad \left( \frac{p_e}{n_e ^\gamma} \right) = \frac{\gamma - 1}{n_e ^\gamma} RHS_e\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Since we&rsquo;ve defined our center-of-mass reference frame to be that of the ions, we can not use the same total derivative</p>
+<span>
+  \[\pdv{}{t} &#43; \vec v_e \cdot \grad \neq \dv{}{t}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We now apply the charge neutrality approximation</p>
+<span>
+  \[n_e = Z n \qquad \vec v_e = \vec v_i - \frac{1}{Z e n} \vec j\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\rightarrow \dv{}{t} \left( \frac{p_e}{Z^\gamma n^\gamma} \right) &#43; \vec v_i \cdot \grad \left( \frac{p_e}{Z^\gamma n^\gamma} \right) = \frac{1}{Z e n} \vec j \cdot \grad \frac{p_e}{Z^\gamma n^\gamma} &#43; \frac{\gamma - 1}{Z^\gamma n^\gamma} RHS_e\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Multiply by <span>
+  \( (Z / m_i)^\gamma \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and the result is</p>
+<span>
+  \[\dv{}{t} \frac{p_e}{\rho ^\gamma} = \frac{1}{Z e n} \vec j \cdot \grad \frac{p_e}{\rho^\gamma} &#43; \frac{\gamma - 1}{\rho^\gamma} RHS_e\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Finally we can add the ion energy equation to the electron energy equation to get</p>
+<span>
+  \[\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \\
+\frac{\gamma - 1}{\rho^\gamma} \left[ Q_{ie} &#43; Q_{ei} - \div (\vec h_i &#43; \vec h_e) - \vec \pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e \right] &#43; \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>!!! info &ldquo;MHD Energy Equation&rdquo;</p>
+<pre><code><span>
+  \[    \dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \frac{\gamma - 1}{\rho^\gamma} \left[ Q_{ie} &#43; Q_{ei} - \div (\vec h_i &#43; \vec h_e) - \vec \pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e \right] &#43; \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}
+    \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+</code></pre>
+<p>Obviously, we&rsquo;ve retained a number of terms that are specific to the behavior of the electrons. It is possible to incorporate the electron behavior by using a single-fluid MHD model with two temperatures <span>
+  \( T_i \neq T_e \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. One can imagine a hierarchy of models, in which the most simplified is the single-fluid MHD model in which you evolve <span>
+  \( \rho \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, and <span>
+  \( T \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Moving up a level, you have a MHD model with two temperatures in which you evolve <span>
+  \( \rho \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( T_i \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, and <span>
+  \( T_e \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Upwards from there you move back into the realm of multi-fluid models.</p>
+<p>Now to relate the fields back to source terms. The low-frequency Maxwell&rsquo;s equations are</p>
+<p><span>
+  \[\pdv{\vec B}{t} = - \curl \vec E\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\vec j = \frac{1}{\mu_0} \curl \vec B\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\div \vec B = \div \vec E = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
+        "\\dd": "\\text{d}",
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>Like in the multi-fluid plasma model, we still need to close the system by expressing some of our variables using equations of state (<span>
+  \( \vec h \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( \vec \Pi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>).</p>
+<p>To simplify further, we can make some assumptions about heat flow</p>
+<p><span>
+  \[\text{isothermal} \qquad \rightarrow p \propto n \qquad T = \text{const.} \qquad \gamma = 1\]
+</span>
+<script>
+  function loadKatex() {
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+        
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+        
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+        
+        "\\dd": "\\text{d}",
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\gr": "\\gamma",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{adiabatic} \qquad \rightarrow p \propto n^\gamma\]
+</span>
+<script>
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\text{cols plasma / force-free} \qquad \rightarrow p = \text{const.} \qquad \gamma = 0\]
+</span>
+<script>
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+      }
+    });
+  };
+</script></p>
+<h2 id="ideal-mhd-model">
+  Ideal MHD Model
+  <a class="anchor" href="#ideal-mhd-model">#</a>
+</h2>
+<p>The extended MHD equations are simpler than the two-fluid model, but they can still be quite complicated. We can often still get useful analysis from further reductions. The ideal MHD model is such a reduction that we can get by dropping (with justification) several terms from the extended model. We justify the simplifications by comparing the magnitude of the neglected terms to the terms that are retained.</p>
+<p>Recall the characteristic speed is <span>
+  \( v_{T, i} \)
+</span>
+<script>
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+        
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. If we say that the characteristic length plasma length is <span>
+  \( L \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, then we can define characteristic time <span>
+  \( \tau = L / v_{T, i} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>The derivation of the two-fluid plasma model assumed a Maxwellian velocity distribution. We need the velocity distribution to thermalize, reach local thermodynamic equilibrium, and become Maxwellian. This means that we need many collisions, in fact so many collisions occurring frequently enough that we can ignore collisional effects. There must then be many collisions during the characteristic time <span>
+  \( \tau \)
+</span>
+<script>
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>For ions to be thermalized,</p>
+<span>
+  \[\frac{\tau_{ii}}{\tau} \ll 1\]
+</span>
+<script>
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>And similarly for electrons</p>
+<span>
+  \[\frac{\tau_{e}}{\tau} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The continuity equation remains unchanged from the extended MHD model</p>
+<p>!!! info &ldquo;Ideal MHD Continuity Equation&rdquo;</p>
+<pre><code><span>
+  \[    \pdv{\rho}{t} &#43; \div (\rho \vec v) = 0
+    \]
+</span>
+<script>
+  function loadKatex() {
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>    
+</code></pre>
+<p>The momentum equation is</p>
+<p><span>
+  \[\rho \left( \pdv{\vec v}{t} &#43; \vec v \cdot \grad \vec v \right) &#43; \grad p - \vec j \cross \vec B = - \div (\vec \Pi_e &#43; \vec \Pi_e)\]
+</span>
+<script>
+  function loadKatex() {
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+Drop the anisotropic pressure</p>
+<p>!!! info &ldquo;Ideal MHD Momentum Equation&rdquo;</p>
+<pre><code><span>
+  \[    \rho \left( \pdv{\vec v}{t} &#43; \vec v \cdot \grad \vec v \right) &#43; \grad p - \vec j \cross \vec B = 0
+    \]
+</span>
+<script>
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+</code></pre>
+<p>The generalized Ohm&rsquo;s law is</p>
+<span>
+  \[\vec E &#43; \vec v \cross \vec B = \eta \vec j &#43; \frac{1}{Z e n} \left( \vec j \cross B - \grad p_e - \div \vec \Pi_e \right)\]
+</span>
+<script>
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+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We&rsquo;re going to drop the entire right hand side</p>
+<p>!!! info &ldquo;Ideal MHD Generalized Ohm&rsquo;s Law&rdquo;</p>
+<pre><code><span>
+  \[    \vec E &#43; \vec v \cross \vec B = 0
+    \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+</code></pre>
+<p>For the energy equation we have</p>
+<span>
+  \[\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = \frac{\gamma - 1}{\rho^\gamma} [Q_{ie} &#43; Q_{ei} - \div (\vec h_i &#43; \vec h_e) - \vec \Pi_i \cdot \cdot \grad \vec v_i - \vec \Pi_e \cdot \cdot \grad \vec v_e] &#43; \frac{\vec j}{Z e n} \cdot \grad \frac{p_e}{\rho^\gamma}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We neglect the entire right-hand side</p>
+<p>!!! info &ldquo;Ideal MHD Energy Equation&rdquo;</p>
+<pre><code><span>
+  \[    \pdv{\rho}{t} &#43; \div (p \vec v) = (1 - \gamma) p \div \vec v 
+    \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+</code></pre>
+<h3 id="collisionpressure-terms">
+  Collision/Pressure terms
+  <a class="anchor" href="#collisionpressure-terms">#</a>
+</h3>
+<p>If we assume that the ions and electrons are in thermal equilibrium<span>
+  \( T_i = T_e \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, we can relate the collision times</p>
+<span>
+  \[\tau_{ee} : \tau_{ii} : \tau_{ei} = 1 : \left( \frac{m_i}{m_e} \right) ^{1/2} : \frac{m_i}{m_e}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The collision times are specifically collisional relaxation times of the Boltzmann equation</p>
+<span>
+  \[\left. \pdv{f}{t} \right|_{coll} = \frac{f - f_{\text{Maxwellian}}}{\tau_{\alpha \beta}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>For electrons, the thermalization condition is much stricter for the ions</p>
+<span>
+  \[\frac{\tau_{ee}}{\tau} = \left( \frac{m_e}{m_i} \right)^{1/2} \frac{\tau_{ii}}{\tau} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Neglect the anisotropic pressure tensor in the momentum and generalized Ohm&rsquo;s law, <span>
+  \( \div \vec \Pi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. <span>
+  \( \vec \Pi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is primarily the shear stress tensor. The ion thermal speed gives us a characteristic velocity for the plasma, so we use it to characterize the shear stress</p>
+<span>
+  \[\vec \Pi_{i, max} \sim 2 \mu \left( \pdv{u}{x} - \frac{1}{3} \div \vec v \right) \sim \mu \frac{v_{T, i}}{L}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Standard treatments of the viscosity (Braginskii, etc.) show that viscosity scales with the number density, temperature, and collision time</p>
+<span>
+  \[\mu \sim n T_i \tau_{ii} \sim p_i \tau_{ii}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\Pi_{i, max} \sim p_i \frac{\tau_{ii} v_{T, i}}{L}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The specific term we want to get rid of is <span>
+  \( \div \vec \Pi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, so let&rsquo;s compare it to a term we want to keep <span>
+  \( \grad P \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{\div \vec \Pi}{\grad p} \sim \frac{p_i \tau_{ii} v_{T, i} / L^2}{p_i / L} \sim \frac{\tau_{ii}}{\tau}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>So, to neglect the anisotropic pressure term in the momentum equation, once again we require</p>
+<span>
+  \[\frac{\tau_{ii}}{\tau} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>In other words, as long as the plasma is collision-dominated, we can drop the ion anisotropic pressure term. What about associated the electron term? If you can assume <span>
+  \( T_i \approx T_e \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Then <span>
+  \( p_i \approx p_e \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> for a neutral plasma, and</p>
+<span>
+  \[\frac{\div \vec \Pi_e}{\grad p} \sim \frac{p_i \tau_{ee} v_{T, i} / L^2}{p_i / L} \sim \frac{\tau_{ee}}{\tau} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h3 id="magnetic-terms">
+  Magnetic terms
+  <a class="anchor" href="#magnetic-terms">#</a>
+</h3>
+<p>In the generalized Ohm&rsquo;s law, the diamagnetic drift term is</p>
+<span>
+  \[\frac{\grad p_e}{Z e n} \sim \frac{n T_e / L}{Z en} \sim \frac{T_i / L}{Ze} \sim \frac{m_i v_{T, i}^2}{L Z e}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Compare <span>
+  \( \grad p_e / Z e n \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to a term that we&rsquo;re going to keep, which is the dynamo term <span>
+  \( \vec v \cross \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{ \grad p_e / Z en}{|\vec v \cross \vec B|} \sim \frac{m_i v_{T, i}^2}{LZe}{v_{T, i} B} \sim \frac{m_i}{ZeB}\frac{v_{T, i}}{L} = \frac{v_{T, i}}{\omega_{c, i}} \frac{1}{L} \sim \frac{r_{L, i}}{L} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>So to neglect the diamagnetic drift term, we need the plasma to be well-magnetized. This means the Larmor radius must be much less than the plasma characteristic length <span>
+  \( r_{L, i} \ll L \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>Now what can we do with the Hall term <span>
+  \( \frac{\vec j \cross \vec B}{Zen} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. For a static plasma (or one with subsonic flows):</p>
+<span>
+  \[\rho (\pdv{\vec v}{t} &#43; \vec v \cdot \grad \vec v) - \grad p - \vec j \cross \vec B = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>so by &ldquo;subsonic&rdquo; we mean that the static pressure is much larger than the dynamic pressure and we can discard the <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
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+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> terms.</p>
+<span>
+  \[\vec j \cross \vec B \approx \grad p\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Comparing the Hall term to the dynamo term <span>
+  \( \vec v \cross \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> also gives the same requirement</p>
+<span>
+  \[\frac{r_{L, i}}{L} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
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+        "\\dd": "\\mathop{\\mathrm{d} #1}",
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>How well do some real plasmas hold up to the requirements of ideal MHD? Consider field nulls in a Z-pinch or an FRC, and weakly magnetized plasmas (such as those in a Hall thruster). Clearly, the magnetization requirement does not hold up at all points in space, and ideal MHD does not necessarily apply across the whole domain.</p>
+<h3 id="resistivity">
+  Resistivity
+  <a class="anchor" href="#resistivity">#</a>
+</h3>
+<p>We neglect resistivity and the resistive electric field <span>
+  \( \eta \vec j \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> in the generalized Ohm&rsquo;s law</p>
+<span>
+  \[\eta \vec j \sim \eta \frac{\grad p}{B} \sim \frac{ \eta n T}{LB}\]
+</span>
+<script>
+  function loadKatex() {
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\eta = \frac{m_e \nu_{ei}}{n e^2} = \frac{m_e}{n e^2 \tau_{ei}} = \frac{m_e ^2/ m_i}{n e^2 \tau_{ee}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\rightarrow \eta \vec j \sim \frac{m_e ^2}{e^2 L B} \frac{v_{T, i} ^2}{\tau_{ee}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>How does it compare to <span>
+  \( \vec v \cross \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+      macros: {
+        
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+        
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+        
+        "\\dd": "\\text{d}",
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{\eta \vec j}{|\vec v \cross \vec B|} \sim \left(\frac{m_e}{m_i} \right)^2 \frac{m_i}{e ^2 B^2} \frac{v_{T, i} ^2}{L^2} \frac{L}{v_{T, i} \tau_{ee}} \sim \left( \frac{m_e}{m_i}\right)^2 \frac{v_{T, i} ^2}{\omega_{c, i} ^2} \frac{1}{L^2} \sim \left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^{2} \frac{\tau}{\tau_{ii}} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>This now places a lower limit on <span>
+  \( \tau_{ii} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>; to neglect resistivity <span>
+  \( \tau_{ii} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
+        "\\dd": "\\text{d}",
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> cannot be too low</p>
+<span>
+  \[\left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^{2} \ll  \frac{\tau_{ii}}{\tau} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
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+        
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
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+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h3 id="heating-sources">
+  Heating sources
+  <a class="anchor" href="#heating-sources">#</a>
+</h3>
+<p>Neglect the collisional heating sources <span>
+  \( Q_{ei} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( Q_{ie} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> in the energy equation. We do that by assuming that they are equal and opposite, which only happens when the temperatures are equal. In other words, we are again assuming local thermodynamic equilibrium <em>between</em> electrons and ions.</p>
+<span>
+  \[1 \gg \frac{\tau_{ei}}{\tau} = \left( \frac{m_i}{m_e} \right)^{1/2} \frac{\tau_{ii}}{\tau} \rightarrow \frac{\tau_{ii}}{\tau} \ll \left( \frac{m_e}{m_i} \right) ^{1/2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>This is a much more restrictive condition than ion collisionality. Alternatively, we could track two temperature independently.</p>
+<p>We also neglect the heat flux terms <span>
+  \( \div (\vec h_i &#43; \vec h_e) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
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+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> in the energy equation. Consider parallel (to the magnetic field) heat conduction which dominates since <span>
+  \( \kappa_\perp \ll \kappa_\parallel \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        "\\sen": '\\text{sen}\\,',
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+        "\\div": "\\nabla \\cdot #1",
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+        "\\gr": "\\gamma",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, so</p>
+<span>
+  \[\div \vec h \approx \grad_\parallel \cdot (\kappa_\parallel \grad_\parallel T)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\grad_\parallel \cdot \left[ ( \kappa_{\parallel, i} &#43; \kappa_{\parallel, e}) \grad_\parallel T \right]\]
+</span>
+<script>
+  function loadKatex() {
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+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\kappa_{\parallel, e} \sim \frac{n T_e}{m_e} \tau_{ee} \qquad \text{ and } \qquad \kappa_{\parallel, i} \sim \frac{n T_i}{m_i} \tau_{ii}\]
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\frac{\kappa_{\parallel, e}}{\kappa_{\parallel, i}} \sim \frac{\tau_{ee}}{\tau_{ii}} \frac{m_i}{m_e} \sim \left( \frac{m_i}{m_e} \right) ^{1/2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Compare the thermal conductivity to the rate of pressure change <span>
+  \( \pdv{p}{t} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{\grad_\parallel \cdot (\kappa_{\parallel, e} \grad_\parallel T)}{\pdv{p}{t}} \sim \frac{n T^2 \tau_{ee}/m_e L^2}{nT/\tau} \sim \left( \frac{m_i}{m_e} \right)^{1/2} \frac{\tau_{ii}}{\tau} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We&rsquo;re back to the same requirement that we have ion-electron local thermodynamic equilibrium.</p>
+<h3 id="anisotropic-pressure-terms-in-energy-equation">
+  Anisotropic pressure terms in energy equation
+  <a class="anchor" href="#anisotropic-pressure-terms-in-energy-equation">#</a>
+</h3>
+<p>Neglect the anisotropic pressure terms in the energy equation. That is, <span>
+  \( \vec \Pi_i \cdot \cdot \grad \vec v_i \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \vec \Pi_e \cdot \cdot \grad \vec v_e \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.  Skipping ahead, the result is</p>
+<p><span>
+  \[\frac{\tau_{ii}}{\tau} \ll 1 \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\left( \frac{m_e}{m_i} \right)^{1/2} \frac{\tau_{ii}}{\tau} \frac{r_{L, i}}{L} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<h3 id="electron-convection-term">
+  Electron convection term
+  <a class="anchor" href="#electron-convection-term">#</a>
+</h3>
+<p>Lastly, we neglect the electron convection term in the energy equation <span>
+  \( \frac{\vec j}{Zen} \cdot \grad \frac{P_e}{\rho^\gamma} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The result is</p>
+<span>
+  \[\frac{r_{L, i}}{L} \ll 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h2 id="conservation-law-form-of-mhd">
+  Conservation Law Form of MHD
+  <a class="anchor" href="#conservation-law-form-of-mhd">#</a>
+</h2>
+<span>
+  \[\pdv{}{t} q &#43; \div \vec f = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can express momentum in conservation law form:</p>
+<span>
+  \[\pdv{}{t} (\rho \vec v) &#43; \div \left[ \rho \vec v \vec v - \frac{ \vec B \vec B}{\mu_0} &#43; (p &#43; \frac{B^2}{2 \mu_0})\vec 1 \right] = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The conservation law form for the magnetic field looks like</p>
+<span>
+  \[\pdv{ \vec B}{t} &#43; \div \left[ \vec v \vec B - \vec B \vec v \right ] = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>And of course the energy equation for</p>
+<span>
+  \[E = \frac{1}{\gamma - 1} p &#43; \frac{1}{2} \rho v^2 &#43; \frac{ B^2}{2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\pdv{E}{t} &#43; \div \left[ \left( E &#43; p &#43; \frac{B^2}{2 \mu_0} \right) - \left( \frac{ \vec B \cdot \vec v}{\mu_0} \right) \vec B \right] = 0\]
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
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+        "\\gr": "\\gamma",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Conservation law forms are particularly useful when considering equilibrium steady-state force balance. This means that in steady-state equilibrium we have</p>
+<span>
+  \[\div \left[ \rho \vec v \vec v - \frac{ \vec B \vec B}{\mu_0} &#43; \left( p &#43; \frac{B^2}{2 \mu_0} \right) \vec 1 \right] = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can use this relationship and integrate over various volumes to determine the relationship between the various force balance terms</p>
+<h3 id="examples-of-equilibrium-plasma-confinement">
+  Examples of equilibrium plasma confinement
+  <a class="anchor" href="#examples-of-equilibrium-plasma-confinement">#</a>
+</h3>
+<p>For a plasma in static equilibrium <span>
+  \( \vec v = 0 \rightarrow \vec j \cross \vec B = \grad p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[ \div \vec T = 0 = \div \left( \right) 
+ \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\vec j \cross \vec B = \frac{1}{\mu_0} (\curl \vec B) \cross \vec B = \frac{1}{\mu_0} (\vec B \cdot \grad \vec B - \frac{1}{2} \grad B^2) \\
+= \frac{1}{\mu_0} ( B^2 \vu e_B \cdot \grad \vu e_B &#43; \frac{1}{2} \vu e_B \vu e_B \cdot \grad B^2 - \frac{\grad B^2}{2} )\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where <span>
+  \( \vu e_B = \vec B / B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p><span>
+  \( \vu e_B \cdot \grad \vu e_B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the curvature of <span>
+  \( \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Write it like a curvature</p>
+<span>
+  \[\vec K = - \vu r / R_c\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p><span>
+  \[\vu e_B \vu e_B \cdot \grad B^2\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+is gradient of <span>
+  \( B^2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> that is parallel to B. Multiply by <span>
+  \( e_B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> gives the component of gradient along <span>
+  \( e_B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The difference between that and <span>
+  \( \grad B^2 / 2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> gives you the perpendicular gradient</p>
+<span>
+  \[\vec j \cross \vec B = \frac{1}{\mu_0} (B^2 \vec \kappa - \frac{1}{2} \grad _\perp B^2) = \grad p = \grad_\perp p\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>identify <span>
+  \( B^2 \vec \kappa \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is magnetic tension resulting from having a bent magnetic field line. <span>
+  \( \frac{1}{2} \grad_\perp B^2  \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is magnetic pressure. They have to balance the plasma pressure at equilibrium.</p>
+<p>For example, consider a cylindrical plasma that&rsquo;s in equilibrium with a helical magnetic field
+<span>
+  \[\vec B = B_\theta (r) \vu \theta &#43; B_z (r) \vu z\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>How is plasma pressure profile determined by the different components of the magnetic field? If we want to maximize the amount of pressure we confine, what should be maximized/minimized?</p>
+<span>
+  \[\frac{B^2}{\mu_0} \vu \kappa = \grad _\perp ( p &#43; \frac{B^2}{2 \mu_0} ) \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p><span>
+  \( B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is straight and has no curvature, so the only magnetic tension comes from <span>
+  \( B_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, so the magnetic tension from <span>
+  \( B_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> must balance the total pressure.</p>
+<p align="center"> <img alt="Figure 12.6" src="/r/img/12.6.png" /> </p>
+<p>The role of <span>
+  \( B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is displacing plasma pressure. The utility in defining <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as</p>
+<span>
+  \[\beta = \frac{\text{plasma pressure}}{\text{magnetic pressure}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h3 id="conditions-of-ideal-mhd-validity">
+  Conditions of Ideal MHD Validity
+  <a class="anchor" href="#conditions-of-ideal-mhd-validity">#</a>
+</h3>
+<p>The conditions for ideal MHD to be valid are</p>
+<ol>
+<li>High Ion Collisionality: <span>
+  \( \frac{\tau_{ii}}{\tau} \ll 1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></li>
+<li>Small ion Larmor radius: <span>
+  \( \frac{r_{L, i}}{L} \ll 1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></li>
+<li>Low resistivity: <span>
+  \( \left(\frac{m_e}{m_i} \right)^{3/2} \left( \frac{r_{L, i}}{L} \right)^2 \ll 1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></li>
+</ol>
+<p>For a given plasma in force balance, we can relate the plasma pressure to the magnetic pressure</p>
+<span>
+  \[\beta = \frac{n T}{B^2 / 2 \mu_0} = 4 \times 10^{-16} \frac{n_{cm^{-3}} T_{keV}}{B^2 _{T}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Ion collision time is (Spitzer collisionality)</p>
+<p><span>
+  \[\tau_{ii} = 2.09 \times 10^{7} \frac{T_{eV} ^{3/2} \mu ^{1/2}}{\ln \Lambda n_{cm ^{-3}}} \left[\text{s}\right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where <span>
+  \( \mu \equiv m_i / m_p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>Putting the conditions for ideal MHD in terms of <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  and <span>
+  \( \tau_{ii} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+</script>,</p>
+<ol>
+<li>High collisionality:
+<span>
+  \[\frac{\tau_{ii}}{\tau} = 2.14 \times 10^{12} \frac{T^2}{n L} \ll 1 \]
+</span>
+<script>
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+  };
+</script>
+<span>
+  \[\rightarrow T_{eV} \ll 6.8 \times 10^{-7} L_{cm} ^{1/2} n_{cm^{-3}} ^{1/2}\]
+</span>
+<script>
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+</script></li>
+<li>Small gyroradius:
+<span>
+  \[\frac{r_{L, i}}{L} = 2.3 \times 10^7 \frac{1}{Z L} \sqrt{ \frac{\mu B}{n}} \ll 1\]
+</span>
+<script>
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+</script>
+<span>
+  \[\rightarrow n_{cm^{-3}} \gg 5.3 \times 10^{14} \frac{\mu B_{T}}{Z^2 L_{cm} ^2}\]
+</span>
+<script>
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+</script></li>
+<li>Low resistivity:
+<span>
+  \[\left( \frac{m_e}{m_i} \right) ^{3/2} \left( \frac{r_{L, i}}{L} \right) ^2 \frac{\tau}{\tau_{ii}} = 5.65 \frac{\mu B}{Z^2 L T^2}\]
+</span>
+<script>
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+</script>
+<span>
+  \[\rightarrow T_{eV} \gg 2.4 \frac{\mu ^{3/2} \beta^{1/2}}{Z L _{cm} ^{1.2}}\]
+</span>
+<script>
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+</script></li>
+</ol>
+<p align="center"> <img alt="Figure 12.7" src="/r/img/12.7.png" /> </p>
+<h2 id="perpendicular-mhd">
+  Perpendicular MHD
+  <a class="anchor" href="#perpendicular-mhd">#</a>
+</h2>
+<p>For most configurations for magnetic fusion confinement, we are able to satisfy 2. and 3. but often have densities much too low to meet the high collisionality constraint. However, in practice ideal MHD does accurately model macroscopic behavior of many plasmas. At the same time, magnetized / fusion plasmas are often largely collisionless. We can understand why by re-writing the collisionality requirement as</p>
+<span>
+  \[1 \gg \frac{\tau_{ii}}{\tau} = \frac{\tau_{ii} v_{T, i}}{\tau v_{T, i}} \sim \frac{\lambda}{L}\]
+</span>
+<script>
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+  };
+</script>
+<p>where <span>
+  \( \lambda \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\negadelta": "\\nabla",
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+  };
+</script> is the mean free path. The ratio <span>
+  \( \lambda / L \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+    });
+  };
+</script> is also known as the <strong>Knudsen number</strong>. In magnetized plasmas, the mean free path is often very long, but the path between collisions can cover a great distance only by following magnetic field lines. Motion <span>
+  \( \perp \)
+</span>
+<script>
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+    });
+  };
+</script> to the magnetic field is constrained by <span>
+  \( r_{L, i} \)
+</span>
+<script>
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+  };
+</script>. This suggests an approach wherein we divide the plasma model into a 1-D kinetic model to be solved along the magnetic field and a 2-D MHD model <span>
+  \( \perp \)
+</span>
+<script>
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+</script> to the magnetic field.</p>
+<p>We consider diffusivity (terms like viscosity, conductivity) parallel and perpendicular to <span>
+  \( \vec B \)
+</span>
+<script>
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+</script>
+<span>
+  \[k_\parallel \sim \frac{\lambda ^2}{\tau_{ii}}\]
+</span>
+<script>
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+</script>
+<span>
+  \[k_\perp \sim \frac{r_{L, i} ^2}{\tau_{ii}}\]
+</span>
+<script>
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+</script>
+<span>
+  \[\frac{k_\parallel}{k_\perp} \sim \left( \frac{\lambda}{r_{L, i}} \right)^2 \sim (\omega_{c, i} \tau_{ii} )^2\]
+</span>
+<script>
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>This changes one of our conditions for validity. Specifically, we can write</p>
+<span>
+  \[\frac{r_{L, i}}{L} \frac{1}{ \omega_{c, i} \tau_{i i}} = 254 \frac{\mu \beta}{Z^2 L_{cm} T_{eV} ^2}\]
+</span>
+<script>
+  function loadKatex() {
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>or</p>
+<span>
+  \[T_{eV} \gg 16 \frac{\mu ^{1/2} \beta^{1/2}}{Z L^{1/2}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>This is now only slightly more restrictive than the low-resistivity condition. In fact, most of the plasmas that we looked at before (large tokamaks &amp; toruses, propulsion systems) which had temperatures above the validity range now fall comfortably within the region of validity. The perpendicular MHD model is equivalent to a collisionless model, giving a much wider applicability than the collisional MHD model.</p>
+<p>What about the parallel component? Our collisionality condition still isn&rsquo;t valid parallel to the field. Basically, we ignore the parallel component, which is the same as assuming that <span>
+  \( \rho \)
+</span>
+<script>
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( T \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> are constant along magnetic field lines, with <span>
+  \( \div \vec v = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> . Let&rsquo;s write out the expressions for collision-less MHD and for ideal MHD and compare:</p>
+<table>
+<thead>
+<tr>
+<th></th>
+<th>Collisionless MHD</th>
+<th>Ideal MHD</th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td>Continuity</td>
+<td><span>
+  \( \pdv{\rho}{t} = 0 \)
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+<td><span>
+  \( \pdv{\rho}{t} = - \rho \div \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+</tr>
+<tr>
+<td>Momentum</td>
+<td><span>
+  \( \rho \dv{\vec v_\perp}{t} = \vec j \cross \vec B - \grad_\perp p \)
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
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+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
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+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+      }
+    });
+  };
+</script></td>
+<td><span>
+  \( \rho \dv{\vec v}{t} = \vec j \cross \vec B - \grad p \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+</tr>
+<tr>
+<td>Parallel constraint</td>
+<td><span>
+  \( \vec B \cdot \grad \frac{v_\parallel}{B} = - \div \vec v _\perp \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\grad": "\\nabla",
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+        "\\gr": "\\gamma",
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+      }
+    });
+  };
+</script></td>
+<td></td>
+</tr>
+<tr>
+<td>Energy</td>
+<td><span>
+  \( \dv{p}{t} = 0 \)
+</span>
+<script>
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+        "\\dd": "\\text{d}",
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+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
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+    });
+  };
+</script></td>
+<td><span>
+  \( \dv{p}{t} = - \gamma p \div \vec v \)
+</span>
+<script>
+  function loadKatex() {
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+</tr>
+</tbody>
+</table>
+<p>Collisionless MHD reproduces many of the effects of ideal MHD but has a wider region of validity. Corollary: ideal MHD is accurate beyond its region of validity, unless results lead to parallel gradients. For example, we know that MHD is not valid when representing confinement of a plasma confined in a magnetic mirror, which is an inherently kinetic phenomenon. But ideal MHD <em>can</em> generate parallel gradients within its region of validity, and we need to be careful. Ideal MHD does not require different models <span>
+  \( \parallel \)
+</span>
+<script>
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+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
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+    });
+  };
+</script> and <span>
+  \( \perp \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to the magnetic field, and is therefore preferred. We will continue to use ideal MHD outside of its region of validity.</p>
+</article>
+ 
+      <script>window.onload = function() {
+    window.scrollTo(0,document.body.scrollHeight);
+};</script>
+
+
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+Perfectly Conducting Wall #  For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:">
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+  <article class="markdown"><h1 id="boundary-conditions">
+  Boundary Conditions
+  <a class="anchor" href="#boundary-conditions">#</a>
+</h1>
+<p>Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region.</p>
+<h3 id="perfectly-conducting-wall">
+  Perfectly Conducting Wall
+  <a class="anchor" href="#perfectly-conducting-wall">#</a>
+</h3>
+<p>For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:</p>
+
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \[\left. \vec E_t \right|_{wall} = 0 \quad \rightarrow \quad \left. \vu n \cross \vec E \right| _{wall} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Applying Faraday&rsquo;s law at the wall,</p>
+<p><span>
+  \[\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = \left. - \vu n \cdot \curl \vec E \right|_{wall} = \left. \div (\vu n \cross \vec E) \right| _{wall} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\pdv{}{t} \vu n \cdot \vec B |_{wall} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+If initially there is no normal magnetic field, then
+<span>
+  \[\vu n \cdot \vec B|_{wall} = 0 \quad \text{if initially true}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>And of course, for an impermeable wall,</p>
+<span>
+  \[\vu n \cdot \vec v |_{wall} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Is this a sufficient set of boundary conditions? Think back to the governing equations in conservation form
+<span>
+  \[\pdv{}{t} \vec Q &#43; \div \vec F = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+The boundary conditions come into play when defining <span>
+  \( \vec F \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> at the boundary. In particular, we need to know what <span>
+  \( \dd \vec S \cdot \vec F |_{wall} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is. In our governing equations, this will involve <span>
+  \( \vec E \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, and <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<h3 id="insulating-boundary">
+  Insulating Boundary
+  <a class="anchor" href="#insulating-boundary">#</a>
+</h3>
+<p>As a slight modification, an insulating boundary can have a tangential electric field. Consider a simple geometry of parallel electrodes with an insulating wall between them.</p>
+<p align="center"> <img alt="Figure 12.8" src="/r/img/12.8.png" /> </p>
+<p>From Ohm&rsquo;s law
+<span>
+  \[\vec E &#43; \vec v \cross \vec B = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+so the only way an electric field tangential to the wall can exist is if <span>
+  \( \vu n \cdot \vec v \neq 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>For either a perfectly conducting or an insulating boundary, the other variables are arbitrary: <span>
+  \( \rho \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> , <span>
+  \( p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( \vec v_t \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( \vec B_t \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<h3 id="vacuum-region">
+  Vacuum Region
+  <a class="anchor" href="#vacuum-region">#</a>
+</h3>
+<p>The plasma (radius <span>
+  \( R_p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>)  is supported by a region of vacuum out to a perfectly conducting wall at some radius <span>
+  \( R_w \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. We assume that there is no plasma in the vacuum region. The governing equations in vacuum are just Maxwell&rsquo;s equations</p>
+<span>
+  \[\curl \vec B_{vac} = 0 \qquad \text{and} \qquad \div \vec B_{vac} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>At the wall,
+<span>
+  \[\vu n \cross \vec E |_{wall} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>What happens at the plasma-vacuum interface? We need to specify jump conditions and continuity conditions. Let&rsquo;s use square brackets to signify a jump:</p>
+<span>
+  \[\left[ X \right] = \left. X \right|_{R_p &#43; dr} - \left. X \right|_{R_p - dr}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The normal magnetic field has to be continuous.</p>
+<span>
+  \[[\vu n \cdot \vec B]_{R_p} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The tangential magnetic field jump is given by the surface current density at the jump</p>
+<span>
+  \[\left[ \vu n \cross \vec B \right] _{R_p} = \mu_0 \vec K \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Integrating <span>
+  \( \grad_\perp (p &#43; \frac{B^2}{2 \mu_0}) = \frac{B^2}{\mu_0} \vec \kappa \)
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> over a differential volume across the surface gives</p>
+<span>
+  \[\left[ p &#43; \frac{B^2}{2 \mu_0} \right] _{R_p} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The plasma shape is determined self-consistently by the wall shape and surface current. This is a free-boundary problem. Another option is to specify the plasma shape, and then determine the required wall shape. This is a fixed-boundary problem.</p>
+<p>The most realistic case includes externally applied magnetic fields coming from source coils, perhaps computed by Biot-Savart law. The vacuum magnetic field is then <span>
+  \( \vec B_{vac} = \vec B_{ext} &#43; \vec B_{plasma} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The crazy coil shapes in the stellarator design come from the 3D geometry computations solving this problem.</p>
+<h3 id="conservation-of-magnetic-flux-frozen-in-flux">
+  Conservation of Magnetic Flux (&ldquo;Frozen-In&rdquo; Flux)
+  <a class="anchor" href="#conservation-of-magnetic-flux-frozen-in-flux">#</a>
+</h3>
+<p>Locally, <span>
+  \( \vec E &#43; \vec v \cross \vec B = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> with Faraday&rsquo;s law
+<span>
+  \[\pdv{B}{t} = - \curl \vec E = - B \div \vec v &#43; \vec B \cdot \grad \vec v - \vec v \cdot \grad B\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+From the continuity equation,
+<span>
+  \[\pdv{\rho}{t} &#43; \vec v \cdot \grad \rho = - \rho \div \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+Combining we find that
+<span>
+  \[\dv{\vec B}{t} = \frac{\vec B}{\rho} \dv{\rho}{t} &#43; \vec B \cdot \grad \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
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+        "\\tagl": "\\tag{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>This says that the field and plasma density move together. Locally, if the magnetic field increases then mass density increases, such that the ratio <span>
+  \( \vec B / \rho \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> remains constant. In the direction parallel to the magnetic field we have a term that involves field line twisting, which is a bit more complicated, but in the perpendicular direction
+<span>
+  \[\dv{}{t} \left( \frac{\vec B}{\rho} \right) _\perp = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>If we consider globally the magnetic flux through a moving surface S at velocity <span>
+  \( \vec u \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The magnetic flux penetrating the surface is
+<span>
+  \[\Psi = \int \vec B \cdot \dd \vec S\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+or
+<span>
+  \[\dv{\Psi}{t} = \int \dv{\vec B}{t} \cdot \vu n \dd S\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        
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+        
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+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[= \int \pdv{\vec B}{t}\cdot \vu n \dd S &#43; \oint \vec B \cross \vec u \dd \vec l\]
+</span>
+<script>
+  function loadKatex() {
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+Using Faraday&rsquo;s law</p>
+<p><span>
+  \[\dv{\Psi}{t} = \int - \curl \vec E \cdot \vu n \dd S &#43; \oint \vec B \cross \vec u \cdot \dd \vec l\]
+</span>
+<script>
+  function loadKatex() {
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+        
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+<span>
+  \[= \oint (- \vec E &#43; \vec B \cross \vec u) \cdot \dd \vec l\]
+</span>
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+Using the electric field from Ohm&rsquo;s law
+<span>
+  \[\dv{\Psi}{t} = \oint(\vec v - \vec u) \cross \vec B \cdot \dd \vec l\]
+</span>
+<script>
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+This tells us that if the surface moves with the plasma <span>
+  \( \vec u = \vec v \)
+</span>
+<script>
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> then
+<span>
+  \[\dv{\Psi}{t} = 0\]
+</span>
+<script>
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+</script>
+the flux through the surface is constant, and the flux is a constant of the topology. This is a direct consequence of ideal MHD. If we add even a small amount of resistivity, we dramatically alter the results in a process called &ldquo;tearing&rdquo; where the magnetic field &ldquo;tears&rdquo; and reconnects with itself.</p>
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+    <a href="https://peppyhare.github.io/r/notes/working/crews2018/" class="">Crews (2018)</a>
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+          
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+  
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
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+          
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+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class=" active">Equilibrium for Fusion</a>
+  
+
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+  
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/" class="">1-D Equilibria</a>
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+  <article class="markdown"><h1 id="equilibrium-for-fusion-hahahugoshortcode-s0-hbhb">
+  Equilibrium for Fusion (
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+  };
+</script>)
+  <a class="anchor" href="#equilibrium-for-fusion-hahahugoshortcode-s0-hbhb">#</a>
+</h1>
+<p>For a fusion device we would like to determine a magnetic configuration that confines plasma while it fuses. At fusion temperatures, the power required to maintain the equilibrium will be substantial. For a device to be useful, the power required to sustain the equilibrium must be less than the power released from fusion. Important loss terms for a confined plasma are transport (thermal conduction primarily) and radiation terms. The scaling factors are <span>
+  \( P_{Brem} \sim n^2 T^{1/2} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+</script> and <span>
+  \( P_{cycl} \sim n^2 T^2 \)
+</span>
+<script>
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+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> for radiation, and <span>
+  \( P_L \sim \frac{3nT}{\tau_E} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> for thermal losses.</p>
+<p>We know that the fusion source term will primarily come from the DT fusion reaction</p>
+<span>
+  \[\text{D} &#43; \text{T} \rightarrow \text{He}^4 (3.5\, MeV) &#43; \text{n} (14.1\, MeV)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The primary fusion reaction releases an <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-particle and a high-energy neutron. The concept of ignition is that the neutron leaves the plasma, and the <span>
+  \( \alpha \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> (with energy <span>
+  \( E_\alpha = 3.5 MeV) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> remains to heat the plasma.</p>
+<span>
+  \[P_\alpha = \frac{1}{4} n^2 \langle \sigma v \rangle E_\alpha \qquad \text{(assuming} \quad n_D = n_T = n/2 \text{)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[P_\alpha &gt; P_L \quad \rightarrow \quad n \tau_E &gt; \frac{12 T}{E_\alpha \langle \sigma v \rangle}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>To sustain fusion, we set the fusion heating term above the thermal loss term. The reaction cross-section <span>
+  \( \sigma \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> can be maximized to give the Lawson criterion</p>
+<span>
+  \[n \tau_E &gt; 10^{14} s / cm^3\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The Lawson criterion only applies at fusion temperatures, but it is a useful parameter even outside of ignition since it gives a ratio of fusion power to lost power</p>
+<table>
+<thead>
+<tr>
+<th><span>
+  \( T_i (keV) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></th>
+<th><span>
+  \( \langle \sigma v \rangle (cm^3 / s) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></th>
+<th>Required <span>
+  \( n \tau_E (s / cm^3) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></th>
+</tr>
+</thead>
+<tbody>
+<tr>
+<td>1</td>
+<td><span>
+  \( 7 \cdot 10^{-21} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+<td><span>
+  \( 5 \cdot 10^{17} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+</tr>
+<tr>
+<td>5</td>
+<td><span>
+  \( 1.4 \cdot 10^{-17} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+<td><span>
+  \( 1.2 \cdot 10^{15} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+</tr>
+<tr>
+<td>20</td>
+<td><span>
+  \( 4.3 \cdot 10^{-16} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+<td><span>
+  \( 1.6 \cdot 10^{14} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+</tr>
+<tr>
+<td>60</td>
+<td><span>
+  \( 8.7 \cdot 10^{-16} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+<td><span>
+  \( 2.4 \cdot 10^{14} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></td>
+</tr>
+</tbody>
+</table>
+<p>We can see that the required <span>
+  \( n \tau_E \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> actually has a minimum around <span>
+  \( 20 keV \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> (at least, as far as the data in the table goes). Even though the maximum cross-section is at a much higher temperature, what we&rsquo;re really concerned with is the ratio of the fusion source term to the thermal loss term, which is linear in temperature.</p>
+<p>MHD equilibrium does not place a limit on the density <span>
+  \( n \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Instead, it places a limit on <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> in order to achieve equilibrium force-balance <span>
+  \( (\beta = 1) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\beta = \frac{n (T_e &#43; T_i)}{B^2 / 2 \mu_0} \rightarrow n = \frac{ \beta B^2}{4 \mu_0 T}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>In this form, we can more clearly see what our options are to achieve MHD equilibrium. Some devices (large-scale tokamaks) are able to achieve the requisite confinement time at a low <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> by making use of very strong magnetic fields. Other devices are able to make use of more modest magnetic fields by working at a higher <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>Therefore,</p>
+<span>
+  \[\tau_E &gt; \frac{1}{\beta B^2} \frac{48 \mu_0}{E_\alpha} \frac{T^2}{\langle \sigma v \rangle}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The term <span>
+  \( \frac{T^2}{\langle \sigma v \rangle} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> has a minimum at <span>
+  \( 10-20 keV \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. At 15 keV and a magnetic field of <span>
+  \( 5T \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> (many actual components cannot reasonably exceed such magnetic fields) then</p>
+<span>
+  \[\tau_E &gt; \frac{0.1}{\beta} \text{s}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>For a large-scale toroidal device with <span>
+  \( \beta = 1\% \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, the confinement time <span>
+  \( \tau_E &gt; 10s \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. If we consider a common diffusivity (how fast energy will leave due to thermal conductivity) <span>
+  \( D_E \approx 1 m^2 / s\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, so for a characteristic radius <span>
+  \( a \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\tau_E \approx \frac{a^2}{4 D_E} \rightarrow a &gt; 6.3 \text{m}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>This gives you a sense of why low-<span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> devices need to be so large. Instead, if we consider <span>
+  \( \beta \sim 50\% \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( \tau_E &gt; 0.2 \text{s} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and</p>
+<span>
+  \[\beta \sim 50\% \rightarrow a &gt; 0.9 \text{m}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>When you consider that the cost of a device (to first order) scales with the volume of the device, achieving a high <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is very important for fusion equilibrium. However, when we consider MHD stability we are generally forced into lower <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to avoid destructive instabilities. Configuration optimization is the process of balancing this trade-off.</p>
+<h2 id="virial-theorem">
+  Virial Theorem
+  <a class="anchor" href="#virial-theorem">#</a>
+</h2>
+<p>Application of the virial theorem to energy balance for the stress tensor <span>
+  \( \vec T \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> tells us that MHD equilibria must be supported by externally supplied currents.  Many times you&rsquo;ll hear of theoretical designs for compact toroid devices which can maintain stability under their own currents, but they are the MHD stability equivalent of a perpetual motion machine. A compact toroid cannot exist unsupported.</p>
+<p>Writing static equilibrium:</p>
+<span>
+  \[\div \left[ - \frac{ \vec B \vec B}{\mu_0} &#43; \left( p &#43; \frac{B^2}{2 \mu_0}\right) \vec I \right] = \div \vec T = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
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+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we define the direction of the magnetic field to be <span>
+  \( \vu e _B = \vu z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> then</p>
+<p><span>
+  \[\vec T = p_\perp ( \vec I - \vu e_B \vu e_B ) &#43; p_\parallel \vu e_B \vu e_B \\
+= \begin{bmatrix} p_\perp &amp; 0 &amp; 0 \\ 0 &amp; p_\perp &amp; 0 \\ 0 &amp; 0 &amp; p_\parallel \end{bmatrix}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where</p>
+<span>
+  \[p_\perp = p &#43; \frac{B^2}{2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>and
+<span>
+  \[p_\parallel = p - \frac{B^2}{2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>A gradient vector identity gives</p>
+<span>
+  \[\div (\vec r \cdot \vec T) = \vec r \cdot ( \div \vec T) &#43; \vec T \cdot \cdot \grad \vec r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Integrating this expression over a volume and assuming that the volume contains a confined MHD equilibrium that is self-contained and self-supported:</p>
+<p align="center"> <img alt="Figure 12.10" src="/r/img/12.10.png" /> </p>
+<span>
+  \[\int_V \div ( \vec r \cdot \vec T) \dd V = \int_V (\vec r \cdot \overbrace{\cancel{(\div \vec T)}}^{\text{MHD equil.}} &#43; \vec T \cdot \cdot \grad \vec r) \dd V\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\grad \vec r = \vec I\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>so
+<span>
+  \[\vec T \cdot \cdot \grad \vec r = p_\perp &#43; p_\perp &#43; p_\parallel \\
+= 3p &#43; \frac{B^2}{2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\int_V (3p &#43; \frac{B^2}{2 \mu_0} ) \dd V = \int _V \div ( \vec r \cdot \vec T) \dd V = \oint _S (\vec r \cdot \vec T) \cdot \vu n \dd S \\
+= \oint _S  \left[ \vec r \cdot \vec I p_\perp &#43; \vec r \cdot \vu e_B \vu e_B (p_\parallel - p_\perp) \right]\cdot \vu n \dd S \\
+=\oint \left[ \left( \cancel{p} &#43; \frac{B^2}{2 \mu_0} \right) \vu r \cdot \vu n - \frac{B^2}{\mu_0} (\vec r \cdot \vu e_B)(\vu e_B \cdot \vu n) \right] \dd S\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Beyond where the plasma is contained, the pressure does not contribute <span>
+  \( p = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. If all current sources are contained in the configuration, the magnetic field <span>
+  \( \sim 1/r^3 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> for a dipole, <span>
+  \( \sim 1/r^4 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> for a quadrupole, etc. Therefore the right-hand side will fall off like</p>
+<p><span>
+  \[RHS \propto \oint_S B^2 r \dd S \propto \left( \frac{1}{r^3} \right) ^2 r r^2 \propto \frac{1}{r^3} \text{(dipole)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+so <span>
+  \( RHS \rightarrow 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as <span>
+  \( r \rightarrow \infty \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. But what about the left-hand side? Both of the terms in the volume integral are positive definite, so the LHS must be positive finite and the equality can&rsquo;t possibly hold. The assumption that the plasma is self-contained must be invalid. This tells us that we must have external currents.</p>
+<h2 id="magnetic-flux-surfaces">
+  Magnetic Flux Surfaces
+  <a class="anchor" href="#magnetic-flux-surfaces">#</a>
+</h2>
+<p>The vast difference in thermal conductivity parallel and perpendicular to the magnetic field in a plasma confinement configuration leads to an avoidance of any open field lines. Magnetic equilibria are generally toroidal to eliminate end losses from open configurations. In general fusion confinement devices, magnetic field lines lie on a set of closed nested toroidal surfaces. This means that we can no longer describe any equilibria in a solely 1D geometry. The minor radius is no longer the only important scale length.</p>
+<p>From <span>
+  \( \vec j \cross \vec B = \grad p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, we know that the pressure gradient is perpendicular to <span>
+  \( \vec j \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, and therefore both <span>
+  \( \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \vec j \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> lie on surfaces of uniform pressure. We call these toroidal surfaces either magnetic surfaces or <strong>flux surfaces</strong>. We can use these surfaces to build a 1-dimensional description.</p>
+<p align="center"> <img alt="Figure 12.11" src="/r/img/12.11.png" /> </p>
+<p>As a brief aside, some geometrical vocabulary will be useful when describing toroidal geometry. A toroid is any surface of revolution with a hole in the middle. A torus is the particular case of a toroid in which the revolved figure is a circle.</p>
+<p>We will define our global toroidal coordinate system to consist of the major axis <span>
+  \( (z) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, the distance from the major axis <span>
+  \( (R) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, and the azimuthal angle around the major axis <span>
+  \( (\phi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p align="center"> <img alt="Figure 12.13" src="/r/img/12.13.png" /> </p>
+<p>We will also make use of a poloidal coordinate system measured by minor radius (distance from the minor axis) <span>
+  \( r \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and the poloidal angle from the minor axis <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. We will generally refer to a point on the torus relative to the major axis <span>
+  \( (R, z, \phi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, or relative to the minor axis <span>
+  \( (r, \theta, \phi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, or in spherical coordinates <span>
+  \( (R, \theta, \phi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The major radius <span>
+  \( R_0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the distance from the major axis to the minor axis. The minor radius <span>
+  \( a \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the characteristic distance from the minor axis to the exterior of the revolved figure. Usually we will find symmetry under <span>
+  \( \phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>The aspect ratio of a torus is the ratio of the major radius to the minor radius.</p>
+<span>
+  \[A = \frac{R_0}{a}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>When we talk about a &ldquo;toroidal surface,&rdquo; we mean a cross-section of the toroidal rotation. When we talk about a &ldquo;poloidal surface&rdquo; we mean a surface which is coplanar with the minor axis:</p>
+<p align="center"> <img alt="Figure 12.14" src="/r/img/12.14.png" /> </p>
+<p>The poloidal flux is determined by the size of the poloidal surface and the poloidal magnetic field:</p>
+<span>
+  \[\Psi _p = \int_{S_p} \vec B \cdot \dd \vec S\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>and the toroidal flux is determined by the size of a toroidal surface and the toroidal magnetic field:</p>
+<span>
+  \[\Psi_t = \int_{S_t} \vec B \cdot \dd \vec S\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Considering the poloidal flux, we can see that if we expand the size of the surface towards the minor radius, the flux will increase until eventually we come to a point where the flux begins to decrease. The position of this maximum is called the <strong>magnetic axis</strong>, which does not necessarily correspond to the minor axis. In fact, it is generally displaced from the minor axis.</p>
+<p align="center"> <img alt="Figure 12.15" src="/r/img/12.15.png" /> </p>
+<p>To refer back to something more familiar, we&rsquo;ll define the same terms for a cylindrical geometry <span>
+  \((r, \theta, z)  \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. An axial surface corresponds with a toroidal surface, and an azimuthal surface corresponds with a poloidal surface:</p>
+<p align="center"> <img alt="Figure 12.16" src="/r/img/12.16.png" /> </p>
+<p>If we consider the trajectory of a single field line, what sorts of surfaces will it trace out? What surface will contain the field line? As it turns out, there are three options:</p>
+<ol>
+<li>Rational surface - the field line closes on itself, and it does so after a finite number of revolutions. One way to quickly visualize such a surface is to draw a Poincaré puncture plot. Choose a toroidal plane and plot a point wherever the field line punctures the surface. A Poincaré puncture plot of a rational surface contains a finite number of points and no continuous curves.</li>
+<li>Ergodic surface - the field line completely covers an entire surface, which is to say the field line punctures any toroidal surface an infinite number of times. In other words, it never closes on itself and defines an irrational curve.</li>
+<li>Stochastic region - In this case, there is no definite surface and the field line fills a volume.</li>
+</ol>
+<p align="center"> <img alt="Figure 12.17" src="/r/img/12.17.png" /> </p>
+<p>Generally rational surfaces and ergodic surfaces are largely equivalent, but by introducing a small amount of resistivity a rational surface can lead to magnetic islands. One can imagine the addition of resistivity equivalent to allowing a small degree of motion of the magnetic field lines. In an ergodic surface, a flux surface is defined by a single (irrational) field line. If it moves toward itself in one location it will necessarily move away from itself in another location. But in a rational surface, different field lines can lie on the same constant pressure surface and will tend to move towards each other. By concentrating into magnetic islands, the flux surfaces are now more closely spaced, and the pressure gradient increases (a bad thing!)</p>
+<p align="center"> <img alt="Figure 12.18" src="/r/img/12.18.png" /> </p>
+<p><strong>Surface quantities</strong>: Since pressure, current (not current density!), and <em>flux</em> (not field!) are constant along a flux surface, it is convenient to use flux <span>
+  \( \Psi_p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as a coordinate. A particular poloidal flux itself uniquely determines a poloidal surface with constant pressure and current. The flux surface quantities are <span>
+  \( p, \, \Psi_p, \, \Psi_t, \, I_p = \int_{S_p} \vec j \cdot \dd \vec S, \, I_t = \int_{S_t} \vec j \cdot \dd \vec S\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>Surface quantities are not independent. The poloidal current <span>
+  \( I_p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> affects the toroidal field <span>
+  \( B_t \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and toroidal flux <span>
+  \( \Psi_t \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The toroidal current affects the poloidal field <span>
+  \( B_p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and poloidal flux <span>
+  \( \Psi_p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<span>
+  \[B_t = \vec B \cdot \vu \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\vec B_p = B_\theta \vu \theta &#43; B_z \vu z \\
+= B_r \vu R &#43; B_z \vu z\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h2 id="toroidal-force-balance">
+  Toroidal Force Balance
+  <a class="anchor" href="#toroidal-force-balance">#</a>
+</h2>
+<p>Toroidal equilibria solves the end losses of linear equilibrium, but generates a new force which must be balanced. This is a result of the virial theorem.</p>
+<h3 id="poloidal-fields-tire-tube-pressure-force">
+  Poloidal Fields (Tire Tube Pressure Force)
+  <a class="anchor" href="#poloidal-fields-tire-tube-pressure-force">#</a>
+</h3>
+<p>If you think of a flexible bike tire being inflated, as the pressure within the inner tube increases, the major radius will increase! Why is this? The pressure within the tire is isotropic. As we pump up the tire, the pressure increase causes a force imbalance with the atmospheric pressure that causes the tube to expand. To simplify, we can consider a square tube with inner &ldquo;radius&rdquo; <span>
+  \( a \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The radial force scales with the pressure over the outer and inner surfaces</p>
+<span>
+  \[F_r = p S_{outer} - p S_{inner} \\
+\sim p (R_0 &#43; a) a - p (R_0 - a) a \sim 2 a^2 p\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>!!! error &ldquo;Missing content here on how to balance radial forces for purely poloidal fields (axisymmetric) with a conducting wall&rdquo;</p>
+<p>Alternatively, we can use vertical external field coils to increase the field strength near the outer wall. They will also tend to center the plasma between the coils, so they have the advantage of preventing the plasma from drifting upwards or downwards</p>
+<p align="center"> <img alt="Figure 12.19" src="/r/img/12.19.png" /> </p>
+<h3 id="toroidal-fields">
+  Toroidal Fields
+  <a class="anchor" href="#toroidal-fields">#</a>
+</h3>
+<p>Now let&rsquo;s consider driving a toroidal field in the plasma using a central coil on the major axis. Driving a toroidal field <span>
+  \( B_\phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> will also create a poloidal current <span>
+  \( j_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. We know that the driven field will decay as <span>
+  \( 1/R \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> through the plasma. In actuality, the relationship is not perfectly <span>
+  \( 1/R \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> because of the generated poloidal current, which will modify the field within the plasma. Depending on the orientation of <span>
+  \( j_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, it could either increase or decrease the poloidal field within the plasma. These are two different operations by which the plasma interacts with the magnetic field. When the induced plasma current tends to increase the field, we have a paramagnetic effect. In the opposite case (decrease) there is a diamagnetic effect. Generally, plasmas are diamagnetic, but there are certain situations where they become paramagnetic.</p>
+<p align="center"> <img alt="Figure 12.20" src="/r/img/12.20.png" /> </p>
+<p>Here again, we have a larger field on the in-bore side than on the out-bore side, so there will be a force imbalance tending to push the plasma towards larger radii. We might consider surrounding the plasma with a conducting wall, as we did previously, but we run into a difficulty determining a current distribution in the wall which would balance the radial effect. Any current distribution must be circular; because of the geometry of a torus such a current distribution would have the same force contribution on both the inside and outside edges. There is no way to stabilize the distribution with purely toroidal fields.</p>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/" class="">MHD Stability</a>
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+  <article class="markdown"><h1 id="1-dimensional-equilibria">
+  1-Dimensional Equilibria
+  <a class="anchor" href="#1-dimensional-equilibria">#</a>
+</h1>
+<h2 id="the-hahahugoshortcode-s0-hbhb-pinch">
+  The 
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch
+  <a class="anchor" href="#the-hahahugoshortcode-s0-hbhb-pinch">#</a>
+</h2>
+<p>In a <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> pinch, we have an applied axial field generated by a driven azimuthal current distribution. The way these usually work is that you begin with a plasma generated by some pre-ionization process and zero field. Then you crank up the current to drive an azimuthal current in the plasma (in the opposite direction as the external current).</p>
+<span>
+  \[\vec j \cross \vec B = \grad p \quad \rightarrow \quad j_\theta B_z = \dv{p}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p><span>
+  \[\dv{p}{r} = - \frac{1}{\mu_0} B_z \dv{B_z}{r} = - \dv{}{r} \left( \frac{B_z ^2}{2 \mu_0} \right)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[p &#43; \frac{B_z ^2}{2 \mu_0} = \text{constant} = \frac{B_0 ^2}{2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p align="center"> <img alt="Figure 12.21" src="/r/img/12.21.png" /> </p>
+<p>At equilibrium, the magnetic pressure balances the plasma pressure. If we say that the pressure is</p>
+<span>
+  \[p = p_0 e^{- r^2 / a^2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>with <span>
+  \( p_0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> the pressure on-axis, then we can solve for the axial field</p>
+<span>
+  \[B_z = B_0 (1 - B_0 e^{-r^2/a^2})^{1/2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can define the peak <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to be the ratio of the on-axis pressure to the maximum magnetic field</p>
+<span>
+  \[\beta_0 = \frac{p_0}{B_0 ^2 / 2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>By definition, the peak <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  will always be <span>
+  \( \leq  1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. We can define <span>
+  \( \langle \beta \rangle \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\langle \beta \rangle = \frac{ \langle p \rangle }{B_a ^2 / 2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where <span>
+  \( B_a \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is a characteristic field value, typically taken to be at the plasma edge.</p>
+<p><span>
+  \[\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \, \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[= \frac{2}{a^2} \int_0 ^a \frac{ rp}{B_0 ^2/2 \mu_0} \dd r = \frac{2}{a^2} \int_0 ^a \left( 1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>In this form, we can see that because <span>
+  \( B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> will everywhere be less than <span>
+  \( B_0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, we can increase <span>
+  \( \langle \beta \rangle \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> by driving <span>
+  \( B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as low as possible. In this particular example, <span>
+  \( \langle \beta \rangle / \beta_0 = 63\% \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<h2 id="z-pinch">
+  Z-pinch
+  <a class="anchor" href="#z-pinch">#</a>
+</h2>
+<p>In the case of a Z-pinch, we only have an applied axial current.</p>
+<span>
+  \[\vec j = j_z (r) \vu z\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>For force balance</p>
+<p><span>
+  \[\vec j \cross \vec B = \grad p \quad \rightarrow \quad - j_z B_\theta = \dv{p}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[j_z = \frac{1}{\mu_0} \frac{1}{r} \dv{}{r} ( r B_\theta) \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[- \dv{p}{r} = \frac{B_\theta}{\mu_0 r} \dv{}{r} ( r B_\theta)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we find it convenient we can separate this into a magnetic pressure term</p>
+<span>
+  \[- \dv{}{r} \left( p &#43; \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h3 id="bennett-profile">
+  Bennett Profile
+  <a class="anchor" href="#bennett-profile">#</a>
+</h3>
+<p>An example of an achievable distribution is the Bennett profile, which has a diffuse form</p>
+<p><span>
+  \[B_\theta = \frac{\mu_0 I}{2 \pi} \frac{r}{r^2 &#43; a^2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[j_z = \frac{I}{\pi} \frac{a^2}{(r^2 &#43; a^2) ^2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[p = \frac{\mu_0 I^2}{8 \pi ^2} \frac{a^2}{(r^2 &#43; a^2)^2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Interestingly, <span>
+  \( j \propto p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. For a uniform temperature, <span>
+  \( j \propto n \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Since current density is the product of <span>
+  \( \vec v \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( n \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, this says that we have a uniform drift velocity and all particles are drifting with the same velocity at all points along the profile. If we consider what the equilibrium profile looks like for a Bennett profile:</p>
+<p align="center"> <img alt="Figure 12.22" src="/r/img/12.22.png" /> </p>
+<p>So for <span>
+  \( r &lt; a \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> we have magnetic tension and pressure which balance the plasma pressure. For <span>
+  \( r \geq a \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> we have magnetic tension which balances both plasma pressure and magnetic pressure.</p>
+<p>The Z-pinch <span>
+  \( \langle \beta \rangle \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\langle \beta \rangle \equiv \frac{ \langle p \rangle}{B_a ^2 / 2 \mu_0} \\
+= \frac{2 \mu_0}{B_a ^2 \pi a^2} \int_0 ^a 2 \pi r p \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we multiply the force balance by <span>
+  \( r^2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and integrate</p>
+<span>
+  \[\int_0 ^a r^2 \dv{p}{r} \dd r &#43; \frac{1}{\mu_0} \int_0 ^a r B_\theta \dv{}{r} (r B_\theta) \dd r = 0 \\
+0 = \left[ r^2 p \right] _0 ^a - \int_0 ^a p \dd (r^2) &#43; \left[ \frac{(r B_\theta)^2}{2 \mu_0} \right] _0 ^a\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we have a discrete pinch such that <span>
+  \( p(a) = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> then the first term vanishes.</p>
+<span>
+  \[\int_0 ^a 2 r p \dd r = \frac{(a B_a)^2}{2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we substitute our definition of <span>
+  \( \langle \beta \rangle \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, we find <span>
+  \( \langle \beta \rangle = 1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. For a diffuse pinch in which <span>
+  \( p(a) \neq 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> we end up with <span>
+  \( \langle \beta \rangle \leq 1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and we have a wall-supported plasma. Ideal confinement (<span>
+  \( \langle \beta \rangle = 1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>) is a very nice property and is what makes the Z-pinch configuration so interesting.</p>
+<h2 id="stability-considerations">
+  Stability Considerations
+  <a class="anchor" href="#stability-considerations">#</a>
+</h2>
+<p>Instability results if there exists a plasma displacement that leads to a lower energy state. There are several ways to provide stability in the context of MHD. The two most common are <strong>magnetic shear</strong> and <strong>magnetic well</strong>.</p>
+<h3 id="magnetic-shear">
+  Magnetic Shear
+  <a class="anchor" href="#magnetic-shear">#</a>
+</h3>
+<p>In ideal MHD, magnetic field lines can not break or tear. Let&rsquo;s consider some flux surface containing field lines <span>
+  \( \vec B_3 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Behind it, we have another flux surface containing field lines <span>
+  \( B_2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> which are not parallel to <span>
+  \( \vec B_3 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, and the same for <span>
+  \( \vec B_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p align="center"> <img alt="Figure 12.23" src="/r/img/12.23.png" /> </p>
+<p>Because the field lines are a different angles to each other, these flux surfaces can <em>not</em> interpenetrate. In other words, if the flux surface pressures are <span>
+  \( P_1 &gt; P_2 &gt; P_3 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, we can maintain the pressure gradient and prevent the flux surfaces from moving each other. What prevents the surfaces from achieving a lower energy state is the magnetic shear between flux surfaces.</p>
+<p>Without shear, the surfaces can interpenetrate and exchange positions. In the case of a toroidal geometry, magnetic shear is defined by the rotational transform <span>
+  \( \iota \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, or by the safety factor</p>
+<span>
+  \[q = \frac{2 \pi}{\iota} \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Generally speaking, <span>
+  \( q \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is generally referenced for tokamaks and <span>
+  \( \iota \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is referenced for stellarators. Another way of picturing the safety factor in a toroidal geometry is</p>
+<span>
+  \[q \equiv \frac{\text{no. of windings long way}}{\text{no. of windings short way}} \\
+= \frac{ \dv{\psi_t}{V}}{\dv{\phi_p}{V}} = \dv{\phi_t}{\phi_p} \\
+= \frac{n}{m} = \frac{\text{toroidal transits}}{\text{poloidal transits}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>In a cylindrical (1D) geometry it is just</p>
+<span>
+  \[q = \frac{\text{longitudinal transits}}{\text{azimuthal transits}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Let&rsquo;s calculate the safety factor for a toroidal geometry:</p>
+<p align="center"> <img alt="Figure 12.24" src="/r/img/12.24.png" /> </p>
+<span>
+  \[\dv{\phi_p}{V} = \frac{ B_\theta 2 \pi R \dd r}{2 \pi R_0 2 \pi r \dd r} \\
+= \frac{ B_\theta}{2 \pi r} \frac{R}{R_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\dv{\phi_t}{V} = \frac{B_\phi 2 \pi r \dd r}{2 \pi R_0  2 \pi r \dd r} \\
+= \frac{B_\phi}{2 \pi R_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[q = \frac{r B_\phi}{R B_\theta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>In a cylindrical geometry the analysis is even simpler</p>
+<span>
+  \[q = 2 \pi \frac{r B_z}{L B_\theta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>As a note, it would appear that <span>
+  \( q \rightarrow 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> at the magnetic axis as <span>
+  \( r \rightarrow 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, but in general <span>
+  \( B_\theta \rightarrow 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as well, and the safety factor is generally bounded at <span>
+  \( r \rightarrow 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p><span>
+  \( q \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is a flux surface quantity.</p>
+<p>We care about magnetic shear. How does that relate to the safety factor? Magnetic shear is defined as</p>
+<span>
+  \[s \equiv 2 \frac{\dd q / q}{\dd V /V} = 2 \dv{\ln (q)}{\ln(V)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Even a uniform <span>
+  \( B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> or <span>
+  \( B_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> produces a finite magnetic shear because of the way that <span>
+  \( r \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( B_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> change. The safety factor is often considered synonymous with magnetic shear, and often we don&rsquo;t even compute <span>
+  \( s \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>Shear is generally a stabilizing effect. Interchange between flux surfaces can be prevented/inhibited by shear, or by making it energetically unfavorable. Shear tends to stabilize current-driven instabilities.</p>
+<h3 id="magnetic-well">
+  Magnetic Well
+  <a class="anchor" href="#magnetic-well">#</a>
+</h3>
+<p>As before, we can consider two adjacent flux surfaces <span>
+  \( B_1, P_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( B_2, P_2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. If <span>
+  \( B_2 &gt; B_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( P_2 &gt; P_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, the interchange is energetically favorable. But if <span>
+  \( B_2 &gt; B_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( P_2 &lt; P_1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> then the interchange may be unfavorable without any magnetic shear.</p>
+<p>Consider a plasma confined by an externally applied magnetic field generated by a coil <span>
+  \( I \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p align="center"> <img alt="Figure 12.25" src="/r/img/12.25.png" /> </p>
+<p>On the left side, the magnetic field gradient is in the same direction as the plasma pressure gradient, which is a destabilizing configuration. Flux surfaces are able to interchange easily, and the magnetic field is described as having bad curvature. On the right side, the gradients are in the same direction and the magnetic field has a good curvature.</p>
+<p>We can define the &ldquo;wellness&rdquo; <span>
+  \( W \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> as</p>
+<p><span>
+  \[W \equiv \frac{ \text{total pressure change relative to mag. pressure}}{\text{relative volume change}} \\
+= \frac{\dd \langle p &#43; B^2/2 \mu_0 \rangle / \langle B^2/2 \mu_0 \rangle}{\dd V / V}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where the angle brackets indicate a quantity integrated along a field line
+<span>
+  \[\langle Q \rangle \equiv \frac{\int_0 ^L \frac{ Q \dd l}{|B|}}{\int_0 ^l \frac{\dd l}{|B|}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>For a stabilizing effect, the wellness must be greater than 0. This means that the magnetic pressure must increase faster than the pressure decreases to prevent pressure-driven instabilities.</p>
+<p>Since <span>
+  \( W \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is evaluated along a field line, it is also a surface quantity.</p>
+<h3 id="application-to-1d-equilibria">
+  Application to 1D Equilibria
+  <a class="anchor" href="#application-to-1d-equilibria">#</a>
+</h3>
+<p><span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch: Since <span>
+  \( B_\theta = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( q \rightarrow \infty \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, which really just means <span>
+  \( q \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is not well defined for a <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch. If we consider some small <span>
+  \( \delta B_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, we get a very large <span>
+  \( q \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. From a magnetic shear perspective, a <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch has very large values of shear and very good stability properties.</p>
+<p>The wellness is</p>
+<p><span>
+  \[W = \frac{V}{\langle B^2 \rangle} \dv{}{V} \langle 2 \mu_0 p &#43; B^2 \rangle \\
+ = \frac{\pi r^2 L}{B_z ^2} \frac{1}{2 \pi r L} \dv{}{r} (2 \mu_0 p &#43; B_z ^2) \\
+ = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p &#43; \frac{B_z ^2}{2 \mu_0} \right) = 0 \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+so a <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch has neutral magnetic well.</p>
+<p>Vacuum case:</p>
+<p><span>
+  \[W = \frac{\mu_0 r}{ B_z ^2} \dv{}{r} \left( \frac{ B_z ^2}{2 \mu_0} \right) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+So vacuum magnetic fields also have neutral wellness. This leads to a general result sometimes referred to as &ldquo;a plasma cannot dig its own well.&rdquo; In other words, by introducing plasma to a magnetic configuration, it cannot make the configuration more stable than it was. Plasmas make stability more challenging, not less.</p>
+<p>Z-pinch:</p>
+<p>Since <span>
+  \( B_z = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( q = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and there is no magnetic shear. Even for a small value of <span>
+  \( \delta B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> you still get a small <span>
+  \( q \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The magnetic well properties of a Z-pinch are</p>
+<span>
+  \[W = \frac{V}{\langle B^2 \rangle} \dv{}{V} \langle 2 \mu_0 p &#43; B^2 \rangle \\
+= \frac{\mu_0 r}{B_\theta ^2} \dv{}{r} \left( p &#43; \frac{B_\theta ^2}{2 \mu_0} \right) \\
+= \frac{\mu_0 r}{B_\theta ^2} \left( - \frac{ B_\theta ^2}{ \mu_0 r} \right) = -1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Recall that <span>
+  \( W &gt; 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> for stability, so the Z-pinch has negative magnetic well and provides no pressure stability.</p>
+<p>In summary,</p>
+<ul>
+<li>Both <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>- and Z-pinch have high <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></li>
+<li><span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch is stable</li>
+<li>Z-pinch is unstable</li>
+<li>End losses in a <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> pinch enormous since <span>
+  \( k_\parallel \gg k_\perp \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></li>
+</ul>
+<h2 id="screw-pinch">
+  Screw Pinch
+  <a class="anchor" href="#screw-pinch">#</a>
+</h2>
+<p>A natural extension is to combine a moderate toroidal field and a moderate poloidal field to produce a screw pinch configuration.</p>
+<p align="center"> <img alt="Figure 12.26" src="/r/img/12.26.png" /> </p>
+<p><span>
+  \[\curl \vec B = \mu_0 \vec j\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\rightarrow j_\theta = - \frac{1}{\mu_0 } \dv{B_z}{r} \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[j_z = \frac{1}{\mu_0 r} \dv{}{r} (r B_\theta)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>For static MHD equilibrium</p>
+<p><span>
+  \[\vec j \cross \vec B = \grad p\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\dv{}{r} \left( p &#43; \frac{ B_\theta ^2 &#43; B_z ^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>We can define a toroidal <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> where <span>
+  \( B_0 = B_z (a) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\beta_t = \frac{\langle p \rangle }{B_0 ^2 / 2 \mu_0} \\
+= \frac{2 \mu_0}{ B_0 ^2} \frac{1}{\pi a^2} \int _0 ^a 2 \pi r p \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>and in the poloidal direction with <span>
+  \( B_{\theta, a} = B_\theta (a) =  \frac{\mu_0 I}{2 \pi a}  \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\beta_p = \frac{\langle p \rangle}{B_{\theta, a} ^2 / 2 \mu_0} \\
+= \frac{8 \pi ^2 a^2}{ \mu_0 I_0 ^2} \left( \frac{1}{\pi a^2} \int _0 ^a 2 \pi r p \dd r \right) \\
+= \frac{16 \pi ^2}{\mu_0 I_0 ^2} \int_0 ^a r p \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>To proceed we can multiply the force balance by <span>
+  \( r^2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and integrate</p>
+<span>
+  \[\underbrace{\int_0 ^a r^2 \pdv{p}{r} \dd r}_{(1)} &#43; \underbrace{\int _0 ^a \pdv{}{r} \left( \frac{ B_\theta ^2 &#43; B_z ^2 }{2 \mu_0} \right) \dd r}_{(2)} &#43; \underbrace{\int_0 ^a r^2 \frac{B_\theta ^2}{\mu_0 r} \dd r}_{(3)} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[(1) = \int_0 ^a r^2 \dd p = \left. r^2 p \right|_0 ^a - \int_0 ^a p \dd (r^2) = - \int_0 ^a 2 r p \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[(2) = \int_0 ^a r^2 \dd \left( \frac{B_\theta ^2 &#43; B_z ^2}{2 \mu_0 } \right) \\
+= \left. r^2 \left( \frac{ B_\theta ^2 &#43; B_z ^2}{2 \mu_0 } \right) \right|_0 ^a - \int_0 ^a r \left( \frac{ B_\theta ^2 &#43; B_z ^2}{\mu_0} \right) \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[(3) = \int_0 ^a r \frac{B_\theta ^2}{\mu_0} \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Combining we have</p>
+<span>
+  \[- \int_0 ^a 2 r p \dd r &#43; \frac{a^2 B_{\theta, a}^2}{2 \mu_0} &#43; \overbrace{\frac{a^2 B_0 ^2}{2 \mu_0}}^{B_0 = B_z(r = a)} - \int_0 ^a r \frac{B_z ^2}{\mu_0} \dd r = 0 \\
+- \int_0 ^a 2 r p \dd r &#43; \frac{ \mu_0 I_0 ^2}{8 \pi ^2} &#43; \int_0 ^a r \left( \frac{ B_\theta ^2 - B_z ^2}{\mu_0} \right) \dd r = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Dividing <span>
+  \( \int_0 ^a 2 r p \dd r \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> gives</p>
+<p><span>
+  \[\left[ \frac{16 \pi^2}{\mu_0 I^2} \int_0 ^a r p \dd r \right] ^{-1} &#43; \left[\frac{4 \mu_0}{B_0 ^2 a^2} \int_0 ^a r p \dd r \right] ^{-1} \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r = 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+or</p>
+<p><span>
+  \[\frac{1}{\beta_p} &#43; \frac{\alpha_t}{\beta_t} = 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where
+<span>
+  \[\alpha_t = \frac{2}{a^2} \int_0 ^a \left( 1 - \frac{B_z^2}{B_0 ^2} \right) r \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+is the diamagnetism.</p>
+<span>
+  \[\beta _p = \left( 1 - \frac{\alpha_t}{\beta_t} \right) ^{-1}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we have a diamagnetic current, then <span>
+  \( \alpha &gt; 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. This maximizes confinement, since we have confinement in the azimuthal field, as well as the axial field. The limit where you have a skin current such that <span>
+  \( B_z = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> inside the plasma results in the best confinement and <span>
+  \( \alpha _t = 1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>Looking at the safety factor,</p>
+<span>
+  \[q = \frac{2 \pi r B_z}{L B_\theta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we look at the edge <span>
+  \( r = a \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>,
+<span>
+  \[q_a = \frac{2 \pi a B_0}{L B_{\theta, a}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>As it turns out, this value of the edge safety factor is critically important, and for stability we require that <span>
+  \( q_a &gt; 1 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>The magnetic shear of a screw pinch is</p>
+<span>
+  \[s = 2 \frac{V}{q} \dv{q}{V} \\
+V = \pi r^2 L \\
+\dd V = 2 \pi L r \dd r = \frac{2 V}{r} \dd r \\
+s = \frac{r}{q} \dv{q}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The shear can be adjusted by changing the applied axial field.</p>
+<p>The magnetic well is</p>
+<span>
+  \[W = \frac{V}{B^2} \dv{(2 \mu_0 p &#43; B^2)}{V} \\
+= \frac{\mu_0 r}{B^2} \dv{}{r} \left( p &#43; \frac{B^2}{2 \mu_0} \right) \\
+= \frac{\mu_0 r}{B^2} \left( - \frac{B_\theta ^2}{\mu_0 r} \right) \\
+= - \frac{B_\theta ^2}{B^2} \\
+= - \frac{B_\theta ^2}{B_\theta ^2 &#43; B_z ^2} \\
+= - \left( 1 &#43; \frac{B_z ^2}{B_\theta ^2} \right) ^{-1}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>So the well is always less than zero, but adding <span>
+  \( B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> improves the well.</p>
+<p>By combining the properties of <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch and Z-pinch, we are able to sacrifice some <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>  to achieve better stability properties. Of course, we have not addressed the end losses in any way; to do that, we need to connect the ends.</p>
+</article>
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new file mode 100644
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+<meta name="description" content="2D Equilibria #  Let&rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:
+  \[\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\ = \grad p = \dv{p}{r}\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;">
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+<meta property="og:description" content="2D Equilibria #  Let&rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:
+  \[\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\ = \grad p = \dv{p}{r}\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;" />
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+  <ul>
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+        <li class="book-section-flat" >
+          
+  
+  
+
+  
+    <span>Notes</span>
+  
+
+          
+  <ul>
+    
+      
+        <li class="book-section-flat" >
+          
+  
+  
+
+  
+    <span>Working</span>
+  
+
+          
+  <ul>
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/working/crews2018/" class="">Crews (2018)</a>
+  
+
+        </li>
+      
+    
+  </ul>
+
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+      
+    
+      
+        <li>
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+  
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+    <input type="checkbox" id="section-edf338bf880234e7aba93d04934ebde0" class="toggle" checked />
+    <label for="section-edf338bf880234e7aba93d04934ebde0" class="flex justify-between">
+      <a  class="">MHD Theory</a>
+      <span>▾</span>
+    </label>
+  
+
+          
+  <ul>
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/" class="">Syllabus</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/" class="">Plasma Models</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/" class="">Plasma Fluid Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/" class="">Boundary Conditions</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class="">Equilibrium for Fusion</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/" class="">1-D Equilibria</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/" class=" active">2D Equilibria</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/" class="">Equilibrium of 3D Configurations</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/" class="">MHD Stability</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/formulary/" class="">Formulary</a>
+  
+
+        </li>
+      
+    
+  </ul>
+
+        </li>
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+  </ul>
+
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+      
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+  </ul>
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+<ul>
+  
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+    <a href="https://github.com/Peppyhare/r" target="_blank" rel="noopener">
+        Github
+      </a>
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+    <a href="https://themes.gohugo.io/hugo-book/" target="_blank" rel="noopener">
+        Hugo Themes
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+      <header class="book-header">
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+    <img src="/r/svg/menu.svg" class="book-icon" alt="Menu" />
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+
+  <strong>2D Equilibria</strong>
+
+  <label for="toc-control">
+    
+  </label>
+</div>
+
+
+  
+ 
+      </header>
+
+      
+      
+  <article class="markdown"><h1 id="2d-equilibria">
+  2D Equilibria
+  <a class="anchor" href="#2d-equilibria">#</a>
+</h1>
+<p>Let&rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:</p>
+<p align="center"> <img alt="Figure 12.27" src="/r/img/12.27.png" /> </p>
+
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \[\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\
+= \grad p = \dv{p}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>we move to an axisymmetric 2-dimensional torus, replacing our cylindrical coordinate system with a toroidal one</p>
+<p align="center"> <img alt="Figure 12.28" src="/r/img/12.28.png" /> </p>
+<span>
+  \[\vec j \cross \vec B = \vec j_\theta \cross \vec B_\phi &#43; \vec j_\phi \cross \vec B_\theta = \grad p\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Eventually, the toroidal force balance will lead to the Grad-Shafranov Equation, which tells us how we can solve for a general equilibrium that solves <span>
+  \( \vec j \cross \vec B = \grad p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>Let&rsquo;s consider how we might achieve such a configuration. A toroidal magnetic field can be achieved by driving current through a poloidal coil. A more complicated problem is how to drive toroidal current. In general this is done by means of a transformer, where the plasma itself is the secondary circuit. Driving a time-varying current through the primary induces a toroidal current through the plasma. This is called a transformer drive for current.</p>
+<h2 id="grad-shafranov-equation">
+  Grad-Shafranov equation
+  <a class="anchor" href="#grad-shafranov-equation">#</a>
+</h2>
+<p>Computing <span>
+  \( j_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( B_\phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> can be computationally difficult in a toroidal geometry, so let&rsquo;s do some work towards simplifying our force balance expression. The <strong>toroidal magnetic vector potential</strong> is defined as</p>
+<span>
+  \[\vec B_\theta = \curl \vec A_\phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If we integrate <span>
+  \( B_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> over a poloidal surface, Stokes' theorem gives</p>
+<span>
+  \[\int _{S_p} \curl \vec A_\phi \cdot \dd \vec S = \oint \vec A_\phi \cdot \dd \vec l \\
+= \int _{S_p} B_\theta \cdot \dd \vec S = \Psi _p \]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If the equilibrium is axisymmetric, <span>
+  \( A_\phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> must be uniform along <span>
+  \( \dd l \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, so</p>
+<span>
+  \[A_\phi \vu \phi \cdot \oint \dd \vec l = A_\phi 2 \pi R = \Psi _p \\
+\rightarrow A_\phi = \frac{ \Psi_p}{R} \vu \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where we absorb the factor of <span>
+  \( 2 \pi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> into the poloidal flux <span>
+  \( \Psi _p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. After some manipulation, we can relate <span>
+  \( B_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to the poloidal flux</p>
+<span>
+  \[\vec B_\theta = \curl \vec A_\phi = - \frac{ \vu R}{R} \pdv{\Psi}{z} &#43; \frac{\vu z}{R} \pdv{\Psi}{R}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\mu_0 j_\phi \cross B_\theta = (\curl \vec B_\theta) \cross \vec B_\theta \\
+= \curl \vec B_\theta \cross \left( \grad \Psi \cross \frac{\vu \phi}{R} \right) \\
+= - \left[ \pdv{}{R} \left( \frac{1}{R} \pdv{\Psi}{R} \right) &#43; \frac{1}{R} \pdv{\Psi ^2}{z^2} \right] \cdot \left[\frac{ \grad \Psi}{R} ( \vu \phi \cdot \vu \phi) - \frac{ \phi}{R} \cancel{(\grad \Psi \cdot \vu \phi)} \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>That gives the first component of <span>
+  \( \grad p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, now let&rsquo;s do the other one</p>
+<span>
+  \[\mu_0 \vec j_\theta \cross \vec B_\phi = ( \curl \vec B_\phi) \cross \vec B_\phi \\
+ = \left[ - \vu R \pdv{B_\phi}{z} &#43; \vu z \frac{1}{R} \pdv{}{R} ( R B_\phi) \right] \cross \vec B_\phi \\
+ = - \frac{B_\phi}{R} \left[ \vu R \pdv{}{R} (R B_\phi) &#43; \vu z \pdv{}{z} (R B_\phi) \right] \\
+ = - \frac{B_\phi}{R} \grad (R B_\phi)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Finally, since pressure is a flux surface quantity we can write</p>
+<span>
+  \[\grad p = \dv{p}{\Psi} \grad \Psi = p&#39; \grad \Psi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The toroidal force balance now looks like</p>
+<span>
+  \[\mu_0 p&#39; \grad \Psi = - \frac{1}{R} \left( \pdv{}{R} \frac{1}{R} \pdv{\Psi}{R} &#43; \frac{1}{R} \pdv{^2 \Psi}{z^2} \right) \grad \Psi - \frac{B_\phi}{R} \grad(R B_\phi)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We notice that the only vector quantities here are <span>
+  \( \grad \Psi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \grad (R B_\phi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, so <span>
+  \( \grad (R B_\phi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> must be parallel to <span>
+  \( \grad \Psi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and is a flux surface quantity. We can define our new flux surface quantity as</p>
+<span>
+  \[F(\Phi) \equiv R B_\phi = \frac{\mu_0 I_\theta}{2 \pi} = \frac{\mu_0}{2 \pi} \int_{S_p} \vec j_\theta \cdot \dd \vec S\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\grad F = \dv{F}{\Psi} \grad \Psi = F&#39; \grad \Psi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Now each term in the toroidal force balance has a factor of <span>
+  \( \grad \Psi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> attached. Let&rsquo;s multiply through by <span>
+  \( R^2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and factor out the gradient to arrive at the <strong>Grad-Shafranov equation</strong>:</p>
+<span>
+  \[R^2 \mu_0 p&#39; = - \Delta ^\star \Psi - F F&#39;\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where
+<span>
+  \[\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} &#43; \pdv{^2}{z^2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>To solve the Grad-Shafranov equation, you solve for <span>
+  \( \Psi(R, z) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, which determines <span>
+  \( p(\Psi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( F(\Psi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, which directly gives you <span>
+  \( p(R, z) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( F(R, z) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and completely defines the equilibrium.</p>
+<p>You can solve for the other terms as well. Since <span>
+  \( \vec B_\theta = \frac{\grad \Psi}{R} \cross \vu \phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \Psi \vu \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>and since <span>
+  \( \vec B_\phi = \frac{F}{R} \vu \phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\vec j_\theta = - \frac{1}{\mu_0 R} \grad (R B_\phi) \cross \vu \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>For the G-S equation to be solvable, you need to specify the equilibrium by specifying <span>
+  \( p(\Phi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( F(\Phi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. In practice, this is usually done by making experimental measurements to determine <span>
+  \( p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( F \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. A common code that does this is called EFIT, which takes the boundary conditions of the magnetic field and measurements of temperature, density to perform a least-squares fit to solve the G-S equation.</p>
+<p>In general, the Grad-Shafranov equation leads to a matrix equation</p>
+<span>
+  \[\overline \vec A \vec \Psi &#43; \vec f(\Psi) = \vec g\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Depending on the conditions we place on <span>
+  \( \Psi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, <span>
+  \( \vec f(\Psi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> can be a nonlinear function.</p>
+<h2 id="solutions-to-the-grad-shafranov-equation">
+  Solutions to the Grad-Shafranov equation
+  <a class="anchor" href="#solutions-to-the-grad-shafranov-equation">#</a>
+</h2>
+<p>In the limit that <span>
+  \( \vec j \parallel \vec B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, then <span>
+  \( \vec j \cross \vec B = 0 = \grad p \rightarrow p&#39; = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. These are called force-free states. In the G-S equation, the pressure term vanishes and we&rsquo;re left with</p>
+<span>
+  \[\Delta ^\star \Psi &#43; F F&#39; = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Spheromaks and RFPs are examples of nearly force-free states in which the current is nearly parallel to the magnetic field. Notice that in completely force-free states, <span>
+  \( \langle \beta \rangle = 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>Another interesting limit is the case where <span>
+  \( F F&#39; \gg \Delta ^\star \Psi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. Now we have
+<span>
+  \[\grad p \approx \vec j_\theta \cross \vec B_\phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+which looks like a <span>
+  \( \theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch which has been connected at the ends. Remember from the previous section that we can not maintain radial force balance with purely toroidal fields, so the toroidal current is not zero (hence the <span>
+  \( \approx \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>) but is just high enough to maintain radial force balance. This sort of configuration is called a high-<span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> tokamak.</p>
+<p>The other limit is <span>
+  \( F F&#39; \ll \Delta ^\star \Psi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\grad p \approx \vec j_\phi \cross \vec B_\theta\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+which looks like an end-connected z-pinch. This configuration is usually called an Ohmically heated Tokamak, and the majority of currently operating tokamaks operate this way. As we know, a purely poloidal field has very bad stability properties, so <span>
+  \( \vec B_\phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> needs to be added to provide stability. The toroidal <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is very small
+<span>
+  \[\beta _t \ll 1 \qquad \beta _p \approx 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<h2 id="stability-considerations">
+  Stability Considerations
+  <a class="anchor" href="#stability-considerations">#</a>
+</h2>
+<p>The same stability factors exist in 2D equilibria that we found for 1D equilibria:</p>
+<p>Magnetic shear - the safety factor <span>
+  \( q(\Psi) = \frac{\Delta \phi}{\Delta \theta} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> for <span>
+  \( \Delta \theta = 2 \pi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. We can calculate <span>
+  \( q \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> more easily by integrating along a flux surface in the poloidal plane:</p>
+<span>
+  \[q(\Psi) = \frac{1}{2 \pi} \int_0 ^{\Delta \phi} \dd \phi \\
+ = \frac{1}{2 \pi} \int _0 ^{2 \pi} \dv{\phi}{\theta} \dd \theta \\
+ = \frac{1}{2 \pi} \int_0 ^{2 \pi} \dd \theta \left. \frac{r B_\phi}{R B_\theta} \right|_{\text{along flux surf.}}  \\
+ = \frac{F(\Psi)}{2 \pi} \oint \frac{ \dd l_p}{R^2 B_\theta} \qquad \dd l_p = r \dd \theta\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Magnetic well: similarly we can get the magnetic well factor by integrating around a flux surface in the poloidal plane</p>
+<span>
+  \[\langle Q \rangle = \frac{\oint \frac{Q \dd l_p}{B_\theta}}{\oint \frac{ \dd l_p}{B_\theta}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<h2 id="shafranov-shift">
+  Shafranov Shift
+  <a class="anchor" href="#shafranov-shift">#</a>
+</h2>
+<p>Remember that when we had an equilibrium which had a toroidal current and a corresponding poloidal magnetic field, and a poloidal magnetic field, then radial force balance will tend to shift the configuration outwards away from the major axis and a conducting wall or external coil will be required to maintain the equilibrium. The radial force balance is really achieved by <span>
+  \( \vec j_\phi \cross B_p \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>As we move towards the magnetic axis, <span>
+  \( B_p \rightarrow 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> by definition. With less poloidal field to balance the radial force imbalance, there is more radial expansion. This means that inner portion of the plasma (inner flux surfaces) must shift radially further to achieve radial force balance.</p>
+<p align="center"> <img alt="Figure 12.29" src="/r/img/12.29.png" /> </p>
+<p>The shift increases with plasma pressure. This effect is further enhanced if we have low poloidal fields, for example in the high-<span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> tokamak configurations.</p>
+<p>Low aspect ratios also enhance the effect. Recall that the radial force imbalance increases with smaller aspect ratio, leading to a larger shift.</p>
+</article>
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+Some general features of 3D equilibria are:
+ No net toroidal current. This means that they tend to be steady-state configurations.">
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+Some general features of 3D equilibria are:
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+  <article class="markdown"><h1 id="equilibrium-of-3d-configurations">
+  Equilibrium of 3D Configurations
+  <a class="anchor" href="#equilibrium-of-3d-configurations">#</a>
+</h1>
+<p>In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated.</p>
+<p>Some general features of 3D equilibria are:</p>
+<ol>
+<li>No net toroidal current. This means that they tend to be steady-state configurations.</li>
+<li>Radial confinement is accomplished by toroidal fields, as in the end-connected 
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
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+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>-pinch. As we saw, toroidal fields cannot provide radial confinement in a purely axisymmetric configuration, but radial variation with <span>
+  \( \phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> <em>can</em> provide confinement.</li>
+<li>Toroidal effect (radial force balance) is generated by helical magnetic fields. You can do an expansion of the magnetic field into a toroidal component, and a helical component that traces out a twisted shape as you move around the torus. These twisted shapes are what lead to radial confinement.</li>
+</ol>
+<h2 id="elmo-bumpy-torus-ebt">
+  ELMO Bumpy Torus (EBT)
+  <a class="anchor" href="#elmo-bumpy-torus-ebt">#</a>
+</h2>
+<p>In contrast to most other 3D configurations, even though the EBT is a 3D equilibrium, it has no helical windings.</p>
+<p align="center"> <img alt="Figure 12.30" src="/r/img/12.30.png" /> </p>
+<p>Since there are no helical windings, we have to provide radial stability in another way. In the EBT configuration, you also drive hot poloidal electron rings (driven by electron cyclotron resonance) to provide both stability and heating.</p>
+<h2 id="stellarator">
+  Stellarator
+  <a class="anchor" href="#stellarator">#</a>
+</h2>
+<p>The stellarator configuration is composed of a number of helical current lines (generated by helical coils with <em>alternating</em> currents), and a net toroidal field driven by poloidal coils. The direction of the currents alternate, for a total of <span>
+  \( 2l \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> current lines.</p>
+<p align="center"> <img alt="Figure 12.31" src="/r/img/12.31.png" /> </p>
+<p>The result is a net magnetic field with a ratio such that <span>
+  \( B_{\text{helical}} \gg B_\phi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
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+        "\\cross": "\\times",
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>Stellarators raise some very complicated engineering challenges both in the design and construction of the complicated geometry. It is also very difficult to maintain no net current within the plasma, especially during start-up. As you add plasma, you raise from zero <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        "\\sen": '\\text{sen}\\,',
+        
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+        
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
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+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> to a finite <span>
+  \( \beta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, introducing things like bootstrap currents that need to be balanced.</p>
+<h2 id="torsatron">
+  Torsatron
+  <a class="anchor" href="#torsatron">#</a>
+</h2>
+<p>Similar to a stellarator, the torsatron does not have alternating currents. All of the helical current lines are in the same direction. There are also no toroidal field coils.</p>
+<p>The engineering is slightly simpler, but it is slightly less efficient at generating the helical magnetic field.</p>
+<p>The flux surfaces in stellarators and torsatrons have geometrical cross-sections depending on the number <span>
+  \( l \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> of helical current lines. About the current lines, the flux surfaces are nearly circular. The flux surfaces within the plasma volume are determined by the separatrix of the helical coil fields.</p>
+<p align="center"> <img alt="Figure 12.32" src="/r/img/12.32.png" /> </p>
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+  MHD Stability
+  <a class="anchor" href="#mhd-stability">#</a>
+</h1>
+<p>Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear. We can define the initial deviation from equilibrium to be a linear phenomenon. As usual, we perform this linearization by letting 
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \( Q(r, t) = Q_0 &#43; Q_1(r, t) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> with <span>
+  \( Q_1 \)
+</span>
+<script>
+  function loadKatex() {
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+        "\\curl": "\\nabla \\cross #1",
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+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> being a small first-order perturbation. Since the equilibrium is both time and space independent, the general form of the perturbation is</p>
+<span>
+  \[Q_1(r, t) = \vu Q_1 e^{-i (\omega t - \vec k \cdot \vec r)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p><span>
+  \[\grad p_0 = \vec j_0 \cross \vec B_0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
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+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
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+        "\\eqref": "(#1)",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[p = p_0 &#43; p_1 \qquad \rho = \rho_0 &#43; \rho_1\]
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
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+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\vec j  = \vec j_0 &#43; \vec j_1 \qquad \vec B = \vec B_0 &#43; \vec B_1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
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+        
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+        
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+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>and for a static equilibrium
+<span>
+  \[\vec v = \vec v_1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>In our momentum equations of the perturbed quantities, we assume that the static equilibrium holds, so most of the equilibrium terms drop out. We can define a velocity displacement <span>
+  \( \vec \xi = \int_0 ^t \vec v_1 \dd t \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. As we integrate the field and pressure in time,</p>
+<p><span>
+  \[\pdv{B_1}{t} = \curl (\vec v_1 \cross \vec B_0)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
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+        "\\sen": '\\text{sen}\\,',
+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\int \pdv{B_1}{t} = \vec B_1 = \curl \int \vec v_1 \cross \vec B_0 \dd t \\
+ = \curl (\vec \xi \cross \vec B_0)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>If we do the same for the pressure equation, we get</p>
+<p><span>
+  \[p_1 = - \vec \xi \cdot \grad p_0 - \Gamma p_0 \div \vec \xi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where <span>
+  \( \Gamma \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the ratio of specific heats, to avoid confusion with typical perturbation growth rate <span>
+  \( \gamma \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>.</p>
+<p>If we combine all of these together, substituting into the momentum equation, we can express the perturbation entirely in terms of <span>
+  \( \vec \xi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and the equilibrium properties:</p>
+<span>
+  \[\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \grad (\vec \xi \cdot \grad p_0 &#43; \Gamma p_0 \div \vec \xi)\\
+ &#43; \frac{1}{\mu_0} \left[(\curl \vec B_0) \cross \curl (\vec \xi \cross \vec B_0) \right] \\
+ &#43; \frac{1}{\mu_0} \left[ \curl \curl ( \vec \xi \cross \vec B_0) \cross \vec B_0 \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
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+        "\\gr": "\\gamma",
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+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We define the right-hand-side as the linearized forcing function of our equilibrium
+<span>
+  \[\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \vec F(\vec \xi _i , p_0, \vec B_0)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>For a linear force function, we can also write it in terms of a spring constant tensor</p>
+<span>
+  \[\rho_0 \pdv{ ^2 \vec \xi }{t^2} = \vec F(\vec \xi) = - \overline \vec K \cdot \vec \xi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>We can determine the stability behavior of a configuration by specifying an initial condition</p>
+<p><span>
+  \[\vec \xi (t = 0) = 0 \qquad \text{and} \qquad \left. \pdv{\xi}{t} \right| _{t = 0} = f(\vec r)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+and boundary conditions. A boundary condition may be a rigid wall
+<span>
+  \[\vec \xi \cdot \vu n |_{wall} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>One way we can tell whether a given solution is unstable is to assume a variation of the form
+<span>
+  \[\vec \xi \propto e^{-i \omega t}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+If <span>
+  \( \omega^2 &gt; 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, the displacement will oscillate in time without growth, and if <span>
+  \( \omega^2 &lt; 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> then the displacement will grow. In other words, if <span>
+  \( \omega \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is real, then the mode is stable, and if <span>
+  \( \omega \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is imaginary then the mode is unstable. The eigenvalue equation to be solved is</p>
+<p><span>
+  \[- \omega ^2 \rho_0 \vec \xi = \vec F(\vec \xi)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+which we can write as a matrix equation
+<span>
+  \[\overline \vec A\, \overline X = \lambda \overline X\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\frac{1}{\rho_0} \vec F (\vec \xi) = - \omega ^2 \vec \xi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>For any arbitrary linear forcing function, we might get an infinite number of eigenvalues. How do we know which ones to look at? It turns out that the linearized force function <span>
+  \( \vec F(\vec \xi) \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> has the property of being self-adjoint, so
+<span>
+  \[\int \vec \eta \cdot \vec F(\vec \xi) \dd V = \int \xi \cdot \vec F( \vec \eta) \dd V\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where <span>
+  \( \vec \eta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( \vec \xi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> are arbitrary displacements that satisfy the same boundary conditions. If <span>
+  \( \vec F \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is self-adjoint, then the system is Hermitian, which guarantees that we get real eigenvalues (<span>
+  \( \omega^2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>) , orthogonal eigenfunctions, and most importantly we are guaranteed to have an ordered spectrum of eigenvalues. That is to say <span>
+  \( \omega_0 ^2 &lt; \omega _1 ^2 &lt; \omega _2 ^2 &lt; \ldots \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. This means that the eigenvalue of the lowest mode is guaranteed to be the most negative, and therefore dictates the stability of the system. If the lowest eigenvalue is negative, then the system is necessarily unstable, and if the lowest eigenvalue is positive, then we are guaranteed that all modes are stable.</p>
+<p>Because <span>
+  \( \vec F \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is self-adjoint, we can make use of the energy principle to write the variation in the sum of the kinetic and potential energy as:</p>
+<p><span>
+  \[0 = \dv{}{t} \left[ \frac{1}{2} \int \rho_0 \left( \pdv{\vec \xi}{t} \right) ^2 \dd V - \frac{1}{2} \int \vec \xi \cdot \vec F ( \vec \xi) \dd V \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+The kinetic energy term will always be positive, so we can formulate the stability based on the potential energy, often called a <span>
+  \( \delta W \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> approach
+<span>
+  \[\delta W = - \frac{1}{2} \int \vec \xi \cdot \vec F ( \vec \xi) \dd V\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+is the change in potential energy due to a displacement <span>
+  \( \xi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. If the potential energy decreases due to a displacement <span>
+  \( \xi \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, then the kinetic energy must necessarily increase, so <span>
+  \( \delta W &lt; 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> indicates instability.</p>
+<p>We can write the change in kinetic energy for our normal mode decomposition as
+<span>
+  \[\delta T = \frac{1}{2} \int \rho _0 \left( \pdv{\xi}{t} \right) ^2 \dd V = - \frac{1}{2} \omega ^2 \int \rho _0 \vec \xi ^ \star \cdot \vec \xi \dd V \\
+= - \delta W = \frac{1}{2} \int \vec \xi ^\star \cdot \vec F(\vec \xi) \dd V\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\omega^2 = \frac{- \int \xi ^\star \cdot \vec F \dd V}{\int \rho_0 \xi ^\star \cdot \xi \dd V} = \frac{\delta W}{\frac{1}{2} \int \rho_0 \xi ^\star \cdot \xi \dd V}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>The denominator is strictly positive, so the sign of <span>
+  \( \omega^2 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is determined by the sign of <span>
+  \( \delta W \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\delta W &lt; 0 \rightarrow \omega^2 &lt; 0 \rightarrow  \text{unstable} \\
+\delta W &gt; 0 \rightarrow \omega^2 &gt; 0 \rightarrow \text{stable}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Analyzing the form of <span>
+  \( \delta W \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> (within the plasma volume)</p>
+<span>
+  \[\delta W = \frac{1}{2} \int_{plasma} \dd V \Gamma p_0(\div \vec \xi) ^2 &#43; \vec \xi \cdot \grad p_0 (\div \vec \xi) \qquad \qquad  \\
+\qquad \qquad &#43; \frac{1}{\mu_0} \left[ \curl ( \vec \xi \cross \vec B_0) \right]^2 \\
+\qquad \qquad - \frac{1}{\mu_0} \left[\vec \xi \cross ( \curl \vec B_0) \right] \cdot \left[ \curl ( \vec \xi \cross \vec B_0) \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Generally speaking, the plasma volume does not extend to infinity, and we care very much about the boundary. The total <span>
+  \( \delta W \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the sum of that in the plasma volume <span>
+  \( \delta W_F \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, the surface <span>
+  \( \delta W_S \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, and the vacuum region <span>
+  \( \delta W_V \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The vacuum term looks like</p>
+<span>
+  \[\delta W_V = \frac{1}{2} \int _{vac} \dd V \frac{ (\curl ( \vec \xi \cross \vec B_0))^2}{\mu_0} = \int_{vac} \dd V \frac{\vec B_1 ^2}{\mu_0} &gt; 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>so the vacuum term is always positive, and has a stabilizing influence. The surface contribution offsets this</p>
+<span>
+  \[\delta W_S = \frac{1}{2} \oint \dd S ( \vu n \cdot \vec \xi) ^2 \left[ \left[ \grad \left( p_0 &#43; \frac{B_0^2}{2 \mu_0} \right) \right] \right] \cdot \vu n\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Instabilities can be characterized as:</p>
+<ul>
+<li>Internal/fixed boundary <span>
+  \( \delta W = \delta W_F \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></li>
+<li>External/free boundary <span>
+  \( \delta W = \delta W_F &#43; \delta W_S &#43; \delta W_V \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></li>
+</ul>
+<p>The plasma portion can be re-written slightly as</p>
+<span>
+  \[\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\
+ &#43; \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0}  - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\
+   &#43; \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\
+   &#43; \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1  \quad \leftarrow \text{Current-driven (kink)}  \\
+- 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where <span>
+  \( \vec \kappa \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the curvature vector <span>
+  \( \vu e_B \cdot \grad \vu e_B \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. If we look at each of these terms, the first three terms are all going to be stabilizing effects, which means that all instability is going to come from the last two terms, the current-driven instability term and the pressure-driven instability term.</p>
+<p>Going back to the screw pinch,
+<span>
+  \[\dv{p}{r} = j_\theta B_z - j_z B_\theta\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+we have current in the same direction as magnetic field (<span>
+  \( j_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> with <span>
+  \( B_\theta \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> and <span>
+  \( j_z \)
+</span>
+<script>
+  function loadKatex() {
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        "\\dd": "\\mathop{\\mathrm{d} #1}",
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+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> with <span>
+  \( B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>), so kink instabilities are possible. We also have a pressure gradient, so interchange instabilities are also possible.</p>
+<p>As a concrete example, look at the pressure driven instability term in a Z-pinch.
+<span>
+  \[\kappa = - \frac{ \vu r}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\vec \xi = \xi _r \vu r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<span>
+  \[\delta W_{F, pressure} = \int \dd V \xi _r \dv{p_0}{r} \frac{\xi_r}{r} \\
+ = \int \dd V \frac{2 \xi _r ^2}{r} \dv{p_0}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<p>In a Z-pinch, it is always the case that <span>
+  \( \dv{p_0}{r} &lt; 0 \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. As shown by Kadomtsev (1965) it turns out that these modes can be stabilized by adding <span>
+  \( B_z \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>, but this also introduces kink modes.</p>
+<p>Going back to our stabilizing quantities of wellness and shear, current-driven instabilities are generally managed through shear, and pressure-driven instabilities are stabilized by well.</p>
+</article>
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+    <a href="https://peppyhare.github.io/r/notes/working/crews2018/" class="">Crews (2018)</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
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+</h1>
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Kinetic Description&rdquo;</strong></p>
+<span>
+  \[\dv{\vec v}{t} = \frac{q_i}{m_i} (\vec E &#43; \vec v_i \cross \vec B) &#43; \sum_{j \neq i} \left[ \left. \dv{\vec v_{ij}}{t} \right|_{coll} \delta(\vec r_i - \vec r_j) \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\pdv{\vec B}{t} = - \curl \vec E\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\frac{1}{c^2} \pdv{\vec E}{t} = \curl \vec B - \mu_0 \sum_i q_i \vec v_i \delta (\vec r - \vec r_i)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\div \vec{B} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\div \vec E = \frac{1}{\epsilon_0} \sum_i q_i \delta (\vec r - \vec r_i)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<p>Klimontovich equation:</p>
+<span>
+  \[\dv{N}{t} = 0 = \pdv{N}{t} &#43; \pdv{}{q_i} \cdot (\dot{q_i} N) \\
+N \equiv \sum_i \delta (p - p_i) \delta(q - q_i)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Plasma Fluid Description&rdquo;</strong></p>
+<p>Boltzmann Equation</p>
+<span>
+  \[\pdv{f_\alpha}{t} &#43; \vec v \cdot \pdv{f_\alpha}{t} &#43; \frac{q_\alpha}{m_\alpha} (\vec E &#43; \vec v \cross \vec B) \cdot \pdv{f_\alpha}{\vec v} = \left. \pdv{f_\alpha}{t} \right|_{coll} = \sum_{\beta \neq \alpha} C_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<p>Maxwellian distribution:</p>
+<span>
+  \[f_\alpha (\vec v) = n_\alpha \left( \frac{m_\alpha}{2 \pi T} \right)^{3/2} e^{- \frac{m_\alpha(\vec v - \vec v_\alpha)^2}{2T}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Moments of fluid model (moments of distribution \( \rightarrow \) moments of Boltzmann equation:</p>
+<span>
+  \[\text{Continuity:} \qquad n_\alpha = \int f_\alpha \dd \vec v \\
+\quad \rightarrow \pdv{n_\alpha}{t} &#43; \div (n_\alpha \vec v_\alpha) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\text{Momentum:} \qquad n_\alpha \vec v_\alpha = \int \vec v f_\alpha \dd v \\
+\quad \rightarrow \quad \pdv{}{t} (n_\alpha \vec v_\alpha ) &#43; \div (n_\alpha \vec v_\alpha \vec v_\alpha) &#43; \frac{1}{m_\alpha} \div \vec P_\alpha - \frac{q_\alpha}{m_\alpha} n_\alpha ( \vec E &#43; \vec v _\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \int \vec w C_{\alpha \beta} \dd \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\rightarrow \rho_\alpha \left(\pdv{\vec v_\alpha}{t} &#43; \vec v_\alpha \cdot \grad \vec v_\alpha \right) &#43; \grad \vec P_\alpha &#43; \div \vec \Pi_\alpha - q_\alpha n_\alpha (\vec E &#43; \vec v_\alpha \cross \vec B) = \sum_{\beta \neq \alpha} \vec R_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\text{Energy:} \qquad \int \vec v \vec v \pdv{f_\alpha}{t} \dd \vec v = \pdv{}{t} \int \vec v \vec v f_\alpha \dd \vec v = \pdv{}{t} \vec E_\alpha / m_\alpha \rightarrow \pdv{}{t} \vec P_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\rightarrow \quad \frac{3}{2} n_\alpha \left( \pdv{T_\alpha}{t} &#43; \vec v_\alpha \cdot \grad T_\alpha \right) &#43; P_\alpha \div \vec v_\alpha &#43; \vec \Pi_\alpha \cdot \cdot \grad \vec v_\alpha &#43; \div \vec h_\alpha = \sum_{\beta \neq \alpha} Q_{\alpha \beta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Closure relations</p>
+<span>
+  \[\vec h_\alpha = - \kappa \grad T_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\overline \Pi_ \alpha = \nu \grad \vec v_\alpha\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<h2 id="ideal-mhd">
+  Ideal MHD
+  <a class="anchor" href="#ideal-mhd">#</a>
+</h2>
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Ideal MHD&rdquo;</strong></p>
+<p>Continuity:</p>
+<span>
+  \[\pdv{\rho}{t} &#43; \div (\rho \vec v) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Momentum:</p>
+<span>
+  \[\rho \left( \pdv{\vec v}{t} &#43; \vec v \cdot \grad \vec v \right) &#43; \grad p - \vec j \cross \vec B = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Generalized Ohm&rsquo;s Law</p>
+<span>
+  \[\vec E &#43; \vec v \cross \vec B = \frac{1}{Zen}\cancel{(\vec j \cross \vec B - \grad p_e)} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Energy</p>
+<span>
+  \[\dv{}{t} \left( \frac{p}{\rho^\gamma} \right) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<blockquote class="book-hint info">
+  <p>&ldquo;Lawson Criterion&rdquo;</p>
+<span>
+  \[n \tau_E &gt; 10^4 s/m^3\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Conservation Law Form of Ideal MHD&rdquo;</strong></p>
+<p>Continuity:</p>
+<span>
+  \[\pdv{\rho}{t} &#43; \div (\rho \vec v) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Momentum:</p>
+<span>
+  \[\pdv{(\rho \vec v)}{t} &#43; \div \left[ \rho \vec v \vec v - \frac{\vec B \vec B}{\mu_0} &#43; \left( p &#43; \frac{B^2}{2 \mu_0} \right) \overline{I} \right] = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Energy:</p>
+<span>
+  \[\pdv{\epsilon}{t} &#43; \div \left[ \left( \epsilon &#43; p &#43; \frac{B^2}{2 \mu_0} \right) \vec v - (\vec B \cdot \vec v) \frac{\vec B}{\mu_0} \right] = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\pdv{\vec B}{t} &#43; \div ( \vec v \vec B - \vec B \vec v) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+where</p>
+<span>
+  \[\epsilon = \frac{1}{\gamma - 1} p &#43; \frac{1}{2} \rho v^2 &#43; \frac{B^2}{2\mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<p>Static Equilibrium:</p>
+<span>
+  \[\vec j \cross \vec B = \grad p\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\frac{B^2}{\mu_0} \vec K = \grad_\perp (p &#43; \frac{B^2}{2 \mu_0})\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\vec K \equiv \frac{\vec B}{|B|} \cdot \grad \frac{ \vec B}{|B|}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<p>Conservation of flux:</p>
+<span>
+  \[\vec E &#43; \vec v \cross \vec B = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\pdv{\vec B}{t} = - \curl \vec E\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<h2 id="1d-equilibria">
+  1D Equilibria
+  <a class="anchor" href="#1d-equilibria">#</a>
+</h2>
+<blockquote class="book-hint info">
+  <p><strong>&quot;\( \theta \)-pinch&quot;</strong></p>
+<span>
+  \[B_\theta = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[j_\theta B_z = \dv{p}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[j_\theta = - \frac{1}{\mu_0} \dv{B_z}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\rightarrow p &#43; \frac{B_z ^2}{2 \mu_0} = \frac{B_0 ^2}{2 \mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\langle \beta \rangle = \frac{2}{a^2} \int_0 ^a \frac{r p}{B_0 ^2 / 2 \mu_0} \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[q = \infty\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[W = \frac{\mu_0 r}{B_z ^2} \dv{}{r} \left( p &#43; \frac{B_z ^2}{2 \mu_0} \right) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Z-pinch&rdquo;</strong></p>
+<span>
+  \[B_z =0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\grad p = \dv{p}{r} = - j_z B_\theta\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[- \dv{}{r} \left( p &#43; \frac{B_\theta ^2}{2 \mu_0} \right) = \frac{B_\theta ^2}{\mu_0 r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\langle \beta \rangle = \frac{2 \mu_0}{B_0 ^2 \pi a^2} \int _0 ^a 2 \pi r p \dd r = 1 \quad \text{ if } \quad p(a) = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[q = S = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[W = 1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Screw pinch&rdquo;</strong></p>
+<span>
+  \[\dv{}{r} \left( p &#43; \frac{B^2}{2 \mu_0} \right) = - \frac{B_\theta ^2}{\mu_0 r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\beta_t = \frac{2 \mu_0}{B_z (a) ^2} \left( \frac{1}{\pi a^2} \int_0 ^a 2 \pi r p \dd r \right)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\beta_p = \left( 1 - \frac{\alpha_t}{\beta _t} \right)^{-1} \qquad \alpha_t \equiv \frac{2}{a^2} \int_0 ^a \left(1 - \frac{B_z ^2}{B_0 ^2} \right) r \dd r\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[q = \frac{2 \pi r B_z}{L B_\theta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[q_a = \frac{4 \pi ^2 a^2 B_0}{\mu_0 I_a}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[S = \frac{r}{q} \dv{q}{r}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[W = - \frac{B_\theta ^2}{B_\theta ^2 &#43; B_z ^2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Stability&rdquo;</strong></p>
+<p>Shear:</p>
+<span>
+  \[S = 2 \frac{ dq / q}{dV / V} = 2 \frac{d \ln q}{d \ln V}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[q = \frac{\text{\# long windings}}{\text{\# short windings}} = \dv{\psi_t}{\psi_p}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Shear for toroid</p>
+<span>
+  \[q = \frac{r B_\phi}{R B_\theta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Shear for cylinder</p>
+<span>
+  \[q = 2 \pi \frac{r B_z}{L B_\theta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Well</p>
+<span>
+  \[W = \frac{ d \langle p &#43; B^2 / 2 \mu_0 \rangle / \langle B^2 / 2 \mu_0 \rangle}{dV / V}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+For stabilization, \( B^2 / 2 \mu_0 \) should increase faster than \( p \) decreases</p>
+
+</blockquote>
+
+<h2 id="2d-equilibria">
+  2D Equilibria
+  <a class="anchor" href="#2d-equilibria">#</a>
+</h2>
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Grad-Shafranov Equation: Static toroidal equilibrium&rdquo;</strong></p>
+<span>
+  \[\grad p = \vec j_\theta \cross \vec B_\phi &#43; \vec j_\phi \cross \vec B_\theta\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[A_\phi = \frac{\phi}{R} \vu \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\phi = \frac{\phi_p}{2 \pi}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\vec B_\theta = - \frac{\vu R}{R} \pdv{\psi}{z} &#43; \frac{ \vu z}{R} \pdv{\psi}{R} = \frac{ \grad \psi}{R} \cross \vu \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[F \equiv R B_\phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\Delta ^\star \equiv R \pdv{}{R} \frac{1}{R} \pdv{}{R} &#43; \pdv{^2}{z^2}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\Delta ^\star \psi = \pdv{^2 \psi}{z^2} &#43; \pdv{^2 \psi}{R^2} - \frac{1}{R} \pdv{\psi}{R}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\vec j_\phi = - \frac{1}{\mu_0 R} \Delta ^\star \psi \vu \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\vec j_\theta = \frac{1}{\mu_0 R} \grad (F) \cross \vu \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[R^2 \mu_0 \pdv{p}{\psi} = - \Delta ^\star \psi - F \pdv{F}{\psi}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[q(\psi) = \frac{F(\psi)}{2 \pi} \oint_{p} \frac{r \dd \theta}{R^2 B_\theta}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Limits:</p>
+<span>
+  \[\text{Force-free:} \qquad \vec j \parallel \vec B\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\rightarrow \Delta ^\star \psi &#43; F F&#39; = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\text{Connected $\theta$ pinch:} \qquad FF&#39; \gg \Delta ^\star \psi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\rightarrow \grad p \approx \vec j_\theta \cross \vec B_\phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\text{Connected Z-pinch:} \qquad FF&#39; \ll \Delta ^\star \phi\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\rightarrow \grad p \approx j_\phi \cross B_\theta\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+
+</blockquote>
+
+<h2 id="mhd-stability">
+  MHD Stability
+  <a class="anchor" href="#mhd-stability">#</a>
+</h2>
+<blockquote class="book-hint info">
+  <p><strong>&ldquo;Linear stability&rdquo;</strong></p>
+<span>
+  \[\pdv{\rho_1}{t} = - \vec v_1 \grad \rho_0 - \rho_0 \div \vec v_1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\pdv{\vec B_1}{t} = \curl ( \vec v_1 \cross \vec B_0)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\rho_0 \pdv{\vec v_1}{t} = - \grad p_1 &#43; \vec j_0 \cross \vec B_1 - \vec j_1 \cross \vec B_0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\pdv{p_1}{t} = - \vec v_1 \cdot \grad p_0 - \gamma p_0 \div \vec v_1\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+For linear perturbation \( \vec \xi = \int_0 ^t \vec v_1 \dd t \) the momentum equation becomes</p>
+<span>
+  \[\rho_0 \pdv{^2 \xi}{t^2} = \vec F(\xi)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+where</p>
+<span>
+  \[F(\xi) = \grad (\xi \cdot \grad p_0 &#43; \gamma p_0 \div \xi) &#43; \frac{1}{\mu_0} \left[ ( \curl \vec B_0) \cross \curl (\xi \cross \vec B_0) &#43; \curl \curl (\xi \cross \vec B_0) \cross \vec B_0 \right]\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
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+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<p>Eigenvalues of \( \frac{1}{\rho_0} \vec F (\xi) = \omega^2 \xi \) are real and ordered. Only need to check \( n=0 \) to determine stability/instability of configuration.</p>
+
+</blockquote>
+
+<blockquote class="book-hint info">
+  <p><strong>&quot;\( \delta W \) Approach&quot;</strong></p>
+<p>\( \delta W = \) change in potential energy due to a displacement \( \xi \)</p>
+<span>
+  \[\delta W &lt; 0  \rightarrow \text{instability}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script></p>
+<span>
+  \[\delta W = - \frac{1}{2} \int \xi \cdot F(\xi) \dd V = \delta W_F &#43; \delta W_S &#43; \delta W_V\]
+</span>
+<script>
+  function loadKatex() {
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+    
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+    &quot;\\cross&quot;: &quot;\\times&quot;,
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+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
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+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Surface term:</p>
+<span>
+  \[\delta W_s = \frac{1}{2} \oint \dd S (\vu n \cdot \xi) ^2 \left( \vu n \cdot \grad p_0 &#43; \left[ \vu n \cdot \grad \frac{B_0 ^2}{2 \mu_0} \right]_{jump} \right)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
+    &quot;\\dd&quot;: &quot;\\mathop{\\mathrm{d} #1}&quot;,
+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Vacuum term:</p>
+<span>
+  \[\delta W_V = \int_{vac} \dd V \frac{B_1 ^2}{\mu_0}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+<pre><code>    &quot;\\lcm&quot;: &quot;\\mathop{\\mathrm{lcm}}&quot;,
+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
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+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Plasma (free) term:</p>
+<span>
+  \[\delta W_F = \frac{1}{2} \int \dd V \frac{ |B_{1, \perp}|^2}{\mu_0} \quad \leftarrow \text{Shear Alfven} \\
+&#43; \mu_0 \left| \frac{B_{1, \parallel}}{\mu_0}  - \frac{B_0 \xi \cdot \grad p_0}{B_0} ^2 \right|^2 \quad \leftarrow \text{Fast magnetosonic} \\
+&#43; \Gamma p_0 |\div \xi|^2 \quad \leftarrow \text{Acoustic}\\
+&#43; \frac{\vec j_0 \cdot \vec B_0}{B_0 ^2} (\vec B_0 \cross \vec \xi) \cdot \vec B_1  \quad \leftarrow \text{Current-driven (kink)}  \\
+- 2 ( \vec \xi \cdot \grad p_0)(\vec \xi \cdot \vec \kappa) \quad \leftarrow \text{pressure-driven (interchange/balooning)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
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+    &quot;\\sen&quot;: '\\text{sen}\\,',
+    
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+    &quot;\\abs&quot;: &quot;\\lvert #1 \\rvert&quot;,
+    
+    &quot;\\dd&quot;: &quot;\\text{d}&quot;,
+    &quot;\\cross&quot;: &quot;\\times&quot;,
+    &quot;\\pdv&quot;: &quot;\\frac{\\partial #1}{\\partial #2}&quot;,
+    &quot;\\curl&quot;: &quot;\\nabla \\cross #1&quot;,
+    &quot;\\vu&quot;: &quot;\\mathbf{\\hat{#1}}&quot;,
+    &quot;\\vdot&quot;: &quot;\\dot&quot;,
+    &quot;\\div&quot;: &quot;\\nabla \\cdot #1&quot;,
+    &quot;\\grad&quot;: &quot;\\nabla&quot;,
+    &quot;\\dv&quot;: &quot;\\frac{d\\,#1}{d\\,#2}&quot;,
+    &quot;\\gr&quot;: &quot;\\gamma&quot;,
+    &quot;\\vec&quot;: &quot;\\mathbf{#1}&quot;,
+    &quot;\\tagl&quot;: &quot;\\tag{#1}&quot;,
+    &quot;\\eqref&quot;: &quot;(#1)&quot;,
+    &quot;\\laplacian&quot;: &quot;\\mathbf{\\nabla ^2}&quot;,
+    &quot;\\label&quot;: &quot;{}&quot;,
+    &quot;\\negadelta&quot;: &quot;\\nabla&quot;,
+  }
+});
+</code></pre>
+<p>};
+</script>
+Shear Alfven, fast magnetosonic, and acoustic modes are stabilizing. Current-driven and pressure-driven modes can lead to instability.</p>
+
+</blockquote>
+
+</article>
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+      
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+    <a href="https://peppyhare.github.io/r/notes/working/crews2018/" class="">Crews (2018)</a>
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+          
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+  
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/" class="">Plasma Models</a>
+  
+
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+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/" class="">Plasma Fluid Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
+  
+
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+    
+      
+        <li>
+          
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+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/" class="">Boundary Conditions</a>
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+        <li>
+          
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+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class="">Equilibrium for Fusion</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/" class="">1-D Equilibria</a>
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/" class="">Equilibrium of 3D Configurations</a>
+  
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+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/" class="">MHD Stability</a>
+  
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diff --git a/docs/notes/UWAA558/index.xml b/docs/notes/UWAA558/index.xml
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@@ -0,0 +1,129 @@
+<?xml version="1.0" encoding="utf-8" standalone="yes"?>
+<rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom">
+  <channel>
+    <title>MHD Theory on My Notes</title>
+    <link>https://peppyhare.github.io/r/notes/UWAA558/</link>
+    <description>Recent content in MHD Theory on My Notes</description>
+    <generator>Hugo -- gohugo.io</generator><atom:link href="https://peppyhare.github.io/r/notes/UWAA558/index.xml" rel="self" type="application/rss+xml" />
+    <item>
+      <title>Syllabus</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/</guid>
+      <description>Syllabus #  The course topics planned for this section are (in rough order):
+Particle Model, Boltzmann-Maxwell Model, Magnetohydrodynamic (MHD) Model, Region of Validity, Common Assumptions, Ideal MHD Model, General Properties (Equilibrium, Boundary Conditions, Conservation Laws, &amp;ldquo;Frozen-In&amp;rdquo; Flux)
+Ideal MHD Equilibrium, Virial Theorem, Magnetic Flux Surfaces
+One-Dimensional Equilibria, Theta-Pinch, Z-Pinch, Screw-Pinch, Safety Factor q
+Two-Dimensional Equilibria, Toroidal Geometry, Grad-Shafranov Equation, Closed Flux Surfaces, Safety Factor q, Magnetic Shear, Magnetic Well, Shafranov Shift, Spheromak, Reversed Field Pinch (RFP), Tokamaks, Stellarators (Elmo Bumpy Torus)</description>
+    </item>
+    
+    <item>
+      <title>Plasma Models</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/</guid>
+      <description>Plasma Models #  Working towards MHD #  Let&amp;rsquo;s start from a full-particle description with the goal of reaching a continuum description (kinetic model). Then, we&amp;rsquo;ll look at the forces on the separate species and form a multi-fluid model, finally simplifying to a single-fluid MHD model.
+The most important question to ask ourselves is &amp;ldquo;when is this model going to be useful?&amp;rdquo; The MHD model is the mathematical model for magnetized plasmas that are treated as a fluid.</description>
+    </item>
+    
+    <item>
+      <title>Plasma Fluid Model</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/</guid>
+      <description>Plasma Fluid Model #  We take velocity moments of each of the pieces of the kinetic model:
+ distribution function,   \( f_\alpha \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false }], macros: { &#34;\\lcm&#34;: &#34;</description>
+    </item>
+    
+    <item>
+      <title>Two-Fluid Plasma Model</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/</guid>
+      <description>Two-Fluid Plasma Model (ions-electrons) #  Restricting our multi-species fluid model to ions and electrons, what can we say about wave behavior in a magnetized 2-fluid plasma? Let&amp;rsquo;s start with a cold plasma approximation (   \( p = 0 \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;</description>
+    </item>
+    
+    <item>
+      <title>Magnetohydrodynamic (MHD) Model</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/</guid>
+      <description>Magnetohydrodynamic (MHD) Model #  Applying approximations to the two-fluid plasma model will allow us to arrive at a single-fluid (center-of-mass) description. The result is the ideal magnetohydrodynamic model (MHD).
+First, define the MHD variables:
+  \[\text{mass density:} \qquad \rho = n_i m_i &amp;#43; n_i m_e\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;</description>
+    </item>
+    
+    <item>
+      <title>Boundary Conditions</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/</guid>
+      <description>Boundary Conditions #  Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region.
+Perfectly Conducting Wall #  For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:</description>
+    </item>
+    
+    <item>
+      <title>Equilibrium for Fusion</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/</guid>
+      <description>Equilibrium for Fusion (   \( \beta \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false }], macros: { &#34;\\lcm&#34;: &#34;\\mathop{\\mathrm{lcm}}&#34;, &#34;\\sen&#34;: &#39;\\text{sen}\\,&#39;, &#34;\\dd&#34;: &#34;\\mathop{\\mathrm{d} #1}&#34;, &#34;\\abs&#34;: &#34;\\lvert #1 \\rvert&#34;, &#34;\\dd&#34;: &#34;\\text{d}&#34;, &#34;\\cross&#34;: &#34;\\times&#34;, &#34;\\pdv&#34;: &#34;\\frac{\\partial #1}{\\partial #2}&#34;</description>
+    </item>
+    
+    <item>
+      <title>1-D Equilibria</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/</guid>
+      <description>1-Dimensional Equilibria #  The   \( \theta \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false }], macros: { &#34;\\lcm&#34;: &#34;\\mathop{\\mathrm{lcm}}&#34;, &#34;\\sen&#34;: &#39;\\text{sen}\\,&#39;, &#34;\\dd&#34;: &#34;\\mathop{\\mathrm{d} #1}&#34;, &#34;\\abs&#34;: &#34;\\lvert #1 \\rvert&#34;, &#34;\\dd&#34;: &#34;\\text{d}&#34;, &#34;\\cross&#34;: &#34;\\times&#34;, &#34;\\pdv&#34;: &#34;\\frac{\\partial #1}{\\partial #2}&#34;</description>
+    </item>
+    
+    <item>
+      <title>2D Equilibria</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/</guid>
+      <description>2D Equilibria #  Let&amp;rsquo;s connect the ends of our 1D equilibria. Doing so is what gives us inherently toroidal configurations. From the 1-dimensional picture:
+  \[\vec j \cross \vec B = j_\theta B_z - j_z B_\theta \\ = \grad p = \dv{p}{r}\]   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;</description>
+    </item>
+    
+    <item>
+      <title>Equilibrium of 3D Configurations</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/</guid>
+      <description>Equilibrium of 3D Configurations #  In 3 dimensions, we lose the axisymmetry that allowed us to reach the Grad-Shefranov equation and we need to solve the full momentum equation in three dimensions. This is not something that we can actually do in this class, and the existing codes that do this are quite sophisticated.
+Some general features of 3D equilibria are:
+ No net toroidal current. This means that they tend to be steady-state configurations.</description>
+    </item>
+    
+    <item>
+      <title>MHD Stability</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/</guid>
+      <description>MHD Stability #  Equilibrium is simply a balance of forces that results in a steady state. Beyond equilibrium, stability is the tendency of a perturbation to return to equilibrium, rather than increasing. We are very interested in analyzing the stability of MHD equilibria, including the plasma dynamics, so we need to use the complete ideal MHD model. The MHD equations are non-linear, which means that any evolution/dynamics are also going to be non-linear.</description>
+    </item>
+    
+    <item>
+      <title>Formulary</title>
+      <link>https://peppyhare.github.io/r/notes/UWAA558/formulary/</link>
+      <pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate>
+      
+      <guid>https://peppyhare.github.io/r/notes/UWAA558/formulary/</guid>
+      <description>\( \)   function loadKatex() { renderMathInElement(document.body, { delimiters: [ { left: &#34;$$&#34;, right: &#34;$$&#34;, display: true }, { left: &#34;\\[&#34;, right: &#34;\\]&#34;, display: true }, { left: &#34;$&#34;, right: &#34;$&#34;, display: false }, { left: &#34;\\(&#34;, right: &#34;\\)&#34;, display: false }], macros: { &#34;\\lcm&#34;: &#34;\\mathop{\\mathrm{lcm}}&#34;, &#34;\\sen&#34;: &#39;\\text{sen}\\,&#39;, &#34;\\dd&#34;: &#34;\\mathop{\\mathrm{d} #1}&#34;, &#34;\\abs&#34;: &#34;\\lvert #1 \\rvert&#34;, &#34;\\dd&#34;: &#34;\\text{d}&#34;, &#34;\\cross&#34;: &#34;\\times&#34;, &#34;\\pdv&#34;: &#34;\\frac{\\partial #1}{\\partial #2}&#34;, &#34;\\curl&#34;: &#34;\\nabla \\cross #1&#34;</description>
+    </item>
+    
+  </channel>
+</rss>
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+  <ul>
+    
+      
+        <li class="book-section-flat" >
+          
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+    <span>Notes</span>
+  
+
+          
+  <ul>
+    
+      
+        <li class="book-section-flat" >
+          
+  
+  
+
+  
+    <span>Working</span>
+  
+
+          
+  <ul>
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/working/crews2018/" class="">Crews (2018)</a>
+  
+
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+      
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+      <a  class="">MHD Theory</a>
+      <span>▾</span>
+    </label>
+  
+
+          
+  <ul>
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/" class="">Syllabus</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/" class="">Plasma Models</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/" class="">Plasma Fluid Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/" class="">Boundary Conditions</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class="">Equilibrium for Fusion</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/" class="">1-D Equilibria</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/" class="">2D Equilibria</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/" class="">Equilibrium of 3D Configurations</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/" class="">MHD Stability</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/formulary/" class="">Formulary</a>
+  
+
+        </li>
+      
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+  <article class="markdown"><h1 id="development-of-a-collisionless-plasma-kinetic-solver-and-an-investigation-of-one-dimensional-plasma-waves-and-instabilities">
+  Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities
+  <a class="anchor" href="#development-of-a-collisionless-plasma-kinetic-solver-and-an-investigation-of-one-dimensional-plasma-waves-and-instabilities">#</a>
+</h1>
+<p>Shielded potential of a test electron:</p>
+
+<link rel="stylesheet" href="/r/katex/katex.min.css" />
+<script defer src="/r/katex/katex.min.js"></script>
+<script defer src="/r/katex/auto-render.min.js" onload="loadKatex();"></script><span>
+  \[\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>where the Debye length is <span>
+  \( \lambda_D = \sqrt{\frac{\epsilon_0 T_e}{ n_e e}} \)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The mean free path between large-angle collisions is estimated as</p>
+<p><span>
+  \[\lambda_{mfp} \sim \frac{\epsilon_0 T_e ^2}{\phi_e n_e \log ( \Lambda)}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+where <span>
+  \(\phi_e = e^2 / 4 \pi \epsilon_0\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> are the constants from the Coulomb force law.</p>
+<p>Smooth out the discreteness of particles via spatial average over small volumes:</p>
+<span>
+  \[\rho \rightarrow \langle \rho_c \rangle \qquad \vec E \rightarrow \langle \vec E \rangle &#43; \delta \vec E\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>The mean field <span>
+  \(\langle \vec E \rangle\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales <span>
+  \(\lambda_{mfp} / \lambda_D\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+<span>
+  \[\frac{\lambda_{mfp}}{\lambda_D} \sim \frac{T_e ^{3/2}}{n_e ^{1/2}}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
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+        { left: "\\(", right: "\\)", display: false }],
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+        
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+        
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+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified.</p>
+<h2 id="phase-space-mechanics">
+  Phase space mechanics
+  <a class="anchor" href="#phase-space-mechanics">#</a>
+</h2>
+<p>To arrive at a kinetic equation governing the collisionless mechanics, consider the one-dimensional motion of a single particle</p>
+<span>
+  \[\dot{x} = v \qquad \dot v = F(x)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>and define the phase space coordinates as <span>
+  \(\vec{\dot r} = \vec F \equiv [ v, F(x) ]\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>. The flux vector is similar to the velocity field of a fluid flow. The streamlines of <span>
+  \(\vec F\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> are the streamlines which a particle will follow if the flux is constant in time. Phase flow is always analogous to that of an incompressible fluid because the flux divergence is zero:</p>
+<span>
+  \[\div [v, F(x)] = \pdv{v}{x} &#43; \pdv{F(x)}{v}\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+<p>If the phase fluid density is given by a function <span>
+  \(f(x, v, t)\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> where <span>
+  \(t\)
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script> is the time parameter, because any instantiation of a particle can not leave the phase plane, the probability density will be conserved. We can write a conservation law:
+<span>
+  \[\pdv{f(x, v, t)}{t} = - \div ( f (x, v, t) \vec F)\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
+and due to the flow&rsquo;s incompressibility
+<span>
+  \[\div (f \vec F) = f ( \div \vec F) &#43; \vec F \cdot \grad f = \vec F \cdot \grad f \\
+\rightarrow \pdv{f}{t} = - \left[ v, F(x) \right] \cdot \left[ \pdv{f}{x}, \pdv{f}{v} \right] \\
+\rightarrow \pdv{f}{t} &#43; v \pdv{f}{x} &#43; F(x) \pdv{f}{v} = 0\]
+</span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script></p>
+</article>
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+
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+  
+  <ul>
+    
+      
+        <li class="book-section-flat" >
+          
+  
+  
+
+  
+    <span>Notes</span>
+  
+
+          
+  <ul>
+    
+      
+        <li class="book-section-flat" >
+          
+  
+  
+
+  
+    <span>Working</span>
+  
+
+          
+  <ul>
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/working/crews2018/" class="">Crews (2018)</a>
+  
+
+        </li>
+      
+    
+  </ul>
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <input type="checkbox" id="section-edf338bf880234e7aba93d04934ebde0" class="toggle"  />
+    <label for="section-edf338bf880234e7aba93d04934ebde0" class="flex justify-between">
+      <a  class="">MHD Theory</a>
+      <span>▾</span>
+    </label>
+  
+
+          
+  <ul>
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/01-syllabus/" class="">Syllabus</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/02-plasma-models/" class="">Plasma Models</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/" class="">Plasma Fluid Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/" class="">Boundary Conditions</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class="">Equilibrium for Fusion</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/" class="">1-D Equilibria</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/" class="">2D Equilibria</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/" class="">Equilibrium of 3D Configurations</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/" class="">MHD Stability</a>
+  
+
+        </li>
+      
+    
+      
+        <li>
+          
+  
+  
+
+  
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/formulary/" class="">Formulary</a>
+  
+
+        </li>
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+        </li>
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+
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+
+
+
+
+
+  
+<ul>
+  
+  <li>
+    <a href="https://github.com/Peppyhare/r" target="_blank" rel="noopener">
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+  </li>
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+    <a href="https://themes.gohugo.io/hugo-book/" target="_blank" rel="noopener">
+        Hugo Themes
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+
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+  <strong>Working</strong>
+
+  <label for="toc-control">
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+    <img src="/r/svg/toc.svg" class="book-icon" alt="Table of Contents" />
+    
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+
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+
+
+      <footer class="book-footer">
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+  <div class="flex flex-wrap justify-between">
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+  
+
+
+  
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+      English
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+      <a href="https://peppyhare.github.io/r/" class="flex align-center">
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+      <a href="https://peppyhare.github.io/r/zh/" class="flex align-center">
+        <img src="/r/svg/translate.svg" class="book-icon" alt="Languages" />
+        Chinese
+      </a>
+    </li>
+    
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+</div>
+
+
+
+
+
+
+</div>
+
+ 
+        
+      </footer>
+
+      
+  
+ 
+
+      <label for="menu-control" class="hidden book-menu-overlay"></label>
+    </div>
+
+    
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+      
+  <nav id="TableOfContents"></nav>
+
+ 
+    </aside>
+    
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+
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+
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diff --git a/docs/notes/working/index.xml b/docs/notes/working/index.xml
new file mode 100644
index 00000000..bdeee870
--- /dev/null
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+      
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+      <description>Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities #  Shielded potential of a test electron:
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index 6fc2ed89..b189b675 100644
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index 04fee355..01c00e66 100644
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+    <a href="https://peppyhare.github.io/r/notes/UWAA558/03-plasma-fluid-model/" class="">Plasma Fluid Model</a>
   
 
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@@ -245,7 +175,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/columns/" class="">Columns</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/04-two-fluid-plasma-model/" class="">Two-Fluid Plasma Model</a>
   
 
         </li>
@@ -258,7 +188,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/details/" class="">Details</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/05-mhd-model/" class="">Magnetohydrodynamic (MHD) Model</a>
   
 
         </li>
@@ -271,7 +201,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/expand/" class="">Expand</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/06-boundary-conditions/" class="">Boundary Conditions</a>
   
 
         </li>
@@ -284,7 +214,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/hints/" class="">Hints</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/07-equilibrium-for-fusion/" class="">Equilibrium for Fusion</a>
   
 
         </li>
@@ -297,7 +227,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/katex/" class="">Katex</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/08-1d-equilibria/" class="">1-D Equilibria</a>
   
 
         </li>
@@ -310,7 +240,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/mermaid/" class="">Mermaid</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/09-2d-equilibria/" class="">2D Equilibria</a>
   
 
         </li>
@@ -323,15 +253,11 @@ <h2 class="book-brand">
   
 
   
-    <input type="checkbox" id="section-d3fc1bf6d66cd84b896a0af9f40cb1d5" class="toggle"  />
-    <label for="section-d3fc1bf6d66cd84b896a0af9f40cb1d5" class="flex justify-between">
-      <a href="https://peppyhare.github.io/r/docs/shortcodes/section/" class="">Section</a>
-      <span>▾</span>
-    </label>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/10-equilibrium-of-3d-configurations/" class="">Equilibrium of 3D Configurations</a>
   
 
-          
-  <ul>
+        </li>
+      
     
       
         <li>
@@ -340,7 +266,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/section/page1/" class="">Page1</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/11-mhd-stability/" class="">MHD Stability</a>
   
 
         </li>
@@ -353,7 +279,7 @@ <h2 class="book-brand">
   
 
   
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/section/page2/" class="">Page2</a>
+    <a href="https://peppyhare.github.io/r/notes/UWAA558/formulary/" class="">Formulary</a>
   
 
         </li>
@@ -364,19 +290,6 @@ <h2 class="book-brand">
         </li>
       
     
-      
-        <li>
-          
-  
-  
-
-  
-    <a href="https://peppyhare.github.io/r/docs/shortcodes/tabs/" class="">Tabs</a>
-  
-
-        </li>
-      
-    
   </ul>
 
         </li>
@@ -490,6 +403,7 @@ <h2 class="book-brand">
     window.scrollTo(0,document.body.scrollHeight);
 };</script>
 
+
       <footer class="book-footer">
         
   <div class="flex flex-wrap justify-between">
diff --git a/docs/tags/index.xml b/docs/tags/index.xml
index ed42229c..283b05ed 100644
--- a/docs/tags/index.xml
+++ b/docs/tags/index.xml
@@ -4,10 +4,6 @@
     <title>Tags on My Notes</title>
     <link>https://peppyhare.github.io/r/tags/</link>
     <description>Recent content in Tags on My Notes</description>
-    <generator>Hugo -- gohugo.io</generator>
-    
-	<atom:link href="https://peppyhare.github.io/r/tags/index.xml" rel="self" type="application/rss+xml" />
-    
-    
+    <generator>Hugo -- gohugo.io</generator><atom:link href="https://peppyhare.github.io/r/tags/index.xml" rel="self" type="application/rss+xml" />
   </channel>
-</rss>
\ No newline at end of file
+</rss>
diff --git a/docs/zh/404.html b/docs/zh/404.html
index 0b08daf3..575a39c3 100644
--- a/docs/zh/404.html
+++ b/docs/zh/404.html
@@ -13,7 +13,7 @@
 <title>404 Page not found | My Notes</title>
 <link rel="manifest" href="/r/manifest.json">
 <link rel="icon" href="/r/favicon.png" type="image/x-icon">
-<link rel="stylesheet" href="/r/book.min.df9df75f294a40a9e31f3fa2e59bca6c22879beb4e792e932789fc04009208dd.css" integrity="sha256-3533XylKQKnjHz&#43;i5ZvKbCKHm&#43;tOeS6TJ4n8BACSCN0=">
+<link rel="stylesheet" href="/r/book.min.380d288a2e7eab258a84160c4cb9893132f45d4b08499d9a8114e97c3dc3733e.css" integrity="sha256-OA0oii5&#43;qyWKhBYMTLmJMTL0XUsISZ2agRTpfD3Dcz4=">
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diff --git a/docs/zh/categories/index.html b/docs/zh/categories/index.html
index 6435701a..d4397d53 100644
--- a/docs/zh/categories/index.html
+++ b/docs/zh/categories/index.html
@@ -17,7 +17,7 @@
 
 <link rel="alternate" hreflang="ru" href="https://peppyhare.github.io/r/ru/categories/" title="Categories">
 
-<link rel="stylesheet" href="/r/book.min.df9df75f294a40a9e31f3fa2e59bca6c22879beb4e792e932789fc04009208dd.css" integrity="sha256-3533XylKQKnjHz&#43;i5ZvKbCKHm&#43;tOeS6TJ4n8BACSCN0=">
+<link rel="stylesheet" href="/r/book.min.380d288a2e7eab258a84160c4cb9893132f45d4b08499d9a8114e97c3dc3733e.css" integrity="sha256-OA0oii5&#43;qyWKhBYMTLmJMTL0XUsISZ2agRTpfD3Dcz4=">
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 <script defer src="/r/sw.min.7dd1b982abb2287010a43b99e3233ff99ced25fbe0e72dc2e6ad364145177485.js" integrity="sha256-fdG5gquyKHAQpDuZ4yM/&#43;ZztJfvg5y3C5q02QUUXdIU="></script>
@@ -61,6 +61,14 @@ <h2 class="book-brand">
 
   
 
+  
+
+
+
+  
+  <ul>
+    
+  </ul>
 
 
 
@@ -168,6 +176,7 @@ <h2 class="book-brand">
     window.scrollTo(0,document.body.scrollHeight);
 };</script>
 
+
       <footer class="book-footer">
         
   <div class="flex flex-wrap justify-between">
diff --git a/docs/zh/categories/index.xml b/docs/zh/categories/index.xml
index 4536e7d3..335d64f6 100644
--- a/docs/zh/categories/index.xml
+++ b/docs/zh/categories/index.xml
@@ -4,10 +4,6 @@
     <title>Categories on My Notes</title>
     <link>https://peppyhare.github.io/r/zh/categories/</link>
     <description>Recent content in Categories on My Notes</description>
-    <generator>Hugo -- gohugo.io</generator>
-    
-	<atom:link href="https://peppyhare.github.io/r/zh/categories/index.xml" rel="self" type="application/rss+xml" />
-    
-    
+    <generator>Hugo -- gohugo.io</generator><atom:link href="https://peppyhare.github.io/r/zh/categories/index.xml" rel="self" type="application/rss+xml" />
   </channel>
-</rss>
\ No newline at end of file
+</rss>
diff --git a/docs/zh/index.html b/docs/zh/index.html
index d59897c5..c0d64c30 100644
--- a/docs/zh/index.html
+++ b/docs/zh/index.html
@@ -2,7 +2,7 @@
 <html lang="zh" dir=>
 
 <head>
-	<meta name="generator" content="Hugo 0.74.3" />
+	<meta name="generator" content="Hugo 0.75.1" />
   <meta charset="UTF-8">
 <meta name="viewport" content="width=device-width, initial-scale=1.0">
 <meta name="description" content="">
@@ -18,7 +18,7 @@
 
 <link rel="alternate" hreflang="ru" href="https://peppyhare.github.io/r/ru/" title="My Notes">
 
-<link rel="stylesheet" href="/r/book.min.df9df75f294a40a9e31f3fa2e59bca6c22879beb4e792e932789fc04009208dd.css" integrity="sha256-3533XylKQKnjHz&#43;i5ZvKbCKHm&#43;tOeS6TJ4n8BACSCN0=">
+<link rel="stylesheet" href="/r/book.min.380d288a2e7eab258a84160c4cb9893132f45d4b08499d9a8114e97c3dc3733e.css" integrity="sha256-OA0oii5&#43;qyWKhBYMTLmJMTL0XUsISZ2agRTpfD3Dcz4=">
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 <script defer src="/r/sw.min.7dd1b982abb2287010a43b99e3233ff99ced25fbe0e72dc2e6ad364145177485.js" integrity="sha256-fdG5gquyKHAQpDuZ4yM/&#43;ZztJfvg5y3C5q02QUUXdIU="></script>
@@ -62,6 +62,14 @@ <h2 class="book-brand">
 
   
 
+  
+
+
+
+  
+  <ul>
+    
+  </ul>
 
 
 
@@ -143,6 +151,7 @@ <h2 class="book-brand">
     window.scrollTo(0,document.body.scrollHeight);
 };</script>
 
+
       <footer class="book-footer">
         
   <div class="flex flex-wrap justify-between">
diff --git a/docs/zh/index.xml b/docs/zh/index.xml
index be46bc02..b0f6156f 100644
--- a/docs/zh/index.xml
+++ b/docs/zh/index.xml
@@ -4,10 +4,6 @@
     <title>My Notes</title>
     <link>https://peppyhare.github.io/r/zh/</link>
     <description>Recent content on My Notes</description>
-    <generator>Hugo -- gohugo.io</generator>
-    
-	<atom:link href="https://peppyhare.github.io/r/zh/index.xml" rel="self" type="application/rss+xml" />
-    
-    
+    <generator>Hugo -- gohugo.io</generator><atom:link href="https://peppyhare.github.io/r/zh/index.xml" rel="self" type="application/rss+xml" />
   </channel>
-</rss>
\ No newline at end of file
+</rss>
diff --git a/docs/zh/sitemap.xml b/docs/zh/sitemap.xml
index 49243de2..8eebd439 100644
--- a/docs/zh/sitemap.xml
+++ b/docs/zh/sitemap.xml
@@ -59,4 +59,4 @@
                 />
   </url>
   
-</urlset>
\ No newline at end of file
+</urlset>
diff --git a/docs/zh/tags/index.html b/docs/zh/tags/index.html
index dfc92f8f..2e7bdb9f 100644
--- a/docs/zh/tags/index.html
+++ b/docs/zh/tags/index.html
@@ -17,7 +17,7 @@
 
 <link rel="alternate" hreflang="ru" href="https://peppyhare.github.io/r/ru/tags/" title="Tags">
 
-<link rel="stylesheet" href="/r/book.min.df9df75f294a40a9e31f3fa2e59bca6c22879beb4e792e932789fc04009208dd.css" integrity="sha256-3533XylKQKnjHz&#43;i5ZvKbCKHm&#43;tOeS6TJ4n8BACSCN0=">
+<link rel="stylesheet" href="/r/book.min.380d288a2e7eab258a84160c4cb9893132f45d4b08499d9a8114e97c3dc3733e.css" integrity="sha256-OA0oii5&#43;qyWKhBYMTLmJMTL0XUsISZ2agRTpfD3Dcz4=">
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@@ -61,6 +61,14 @@ <h2 class="book-brand">
 
   
 
+  
+
+
+
+  
+  <ul>
+    
+  </ul>
 
 
 
@@ -168,6 +176,7 @@ <h2 class="book-brand">
     window.scrollTo(0,document.body.scrollHeight);
 };</script>
 
+
       <footer class="book-footer">
         
   <div class="flex flex-wrap justify-between">
diff --git a/docs/zh/tags/index.xml b/docs/zh/tags/index.xml
index 3dd525e3..bb1df088 100644
--- a/docs/zh/tags/index.xml
+++ b/docs/zh/tags/index.xml
@@ -4,10 +4,6 @@
     <title>Tags on My Notes</title>
     <link>https://peppyhare.github.io/r/zh/tags/</link>
     <description>Recent content in Tags on My Notes</description>
-    <generator>Hugo -- gohugo.io</generator>
-    
-	<atom:link href="https://peppyhare.github.io/r/zh/tags/index.xml" rel="self" type="application/rss+xml" />
-    
-    
+    <generator>Hugo -- gohugo.io</generator><atom:link href="https://peppyhare.github.io/r/zh/tags/index.xml" rel="self" type="application/rss+xml" />
   </channel>
-</rss>
\ No newline at end of file
+</rss>
diff --git a/katex.bk/README.md b/katex.bk/README.md
new file mode 100644
index 00000000..307fdbf9
--- /dev/null
+++ b/katex.bk/README.md
@@ -0,0 +1,91 @@
+# [<img src="https://katex.org/img/katex-logo-black.svg" width="130" alt="KaTeX">](https://katex.org/)
+[![npm](https://img.shields.io/npm/v/katex.svg)](https://www.npmjs.com/package/katex)
+[![CircleCI](https://circleci.com/gh/KaTeX/KaTeX.svg?style=shield)](https://circleci.com/gh/KaTeX/KaTeX)
+[![codecov](https://codecov.io/gh/KaTeX/KaTeX/branch/master/graph/badge.svg)](https://codecov.io/gh/KaTeX/KaTeX)
+[![Join the chat at https://gitter.im/KaTeX/KaTeX](https://badges.gitter.im/Join%20Chat.svg)](https://gitter.im/KaTeX/KaTeX?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge)
+[![Dependabot Status](https://api.dependabot.com/badges/status?host=github&repo=KaTeX/KaTeX)](https://dependabot.com)
+[![jsDelivr](https://data.jsdelivr.com/v1/package/npm/katex/badge?style=rounded)](https://www.jsdelivr.com/package/npm/katex)
+![](https://img.badgesize.io/KaTeX/KaTeX/v0.12.0/dist/katex.min.js?compression=gzip)
+
+KaTeX is a fast, easy-to-use JavaScript library for TeX math rendering on the web.
+
+ * **Fast:** KaTeX renders its math synchronously and doesn't need to reflow the page. See how it compares to a competitor in [this speed test](http://www.intmath.com/cg5/katex-mathjax-comparison.php).
+ * **Print quality:** KaTeX's layout is based on Donald Knuth's TeX, the gold standard for math typesetting.
+ * **Self contained:** KaTeX has no dependencies and can easily be bundled with your website resources.
+ * **Server side rendering:** KaTeX produces the same output regardless of browser or environment, so you can pre-render expressions using Node.js and send them as plain HTML.
+
+KaTeX is compatible with all major browsers, including Chrome, Safari, Firefox, Opera, Edge, and IE 11.
+
+KaTeX supports much (but not all) of LaTeX and many LaTeX packages. See the [list of supported functions](https://katex.org/docs/supported.html).
+
+Try out KaTeX [on the demo page](https://katex.org/#demo)!
+
+## Getting started
+
+### Starter template
+
+```html
+<!DOCTYPE html>
+<!-- KaTeX requires the use of the HTML5 doctype. Without it, KaTeX may not render properly -->
+<html>
+  <head>
+    <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
+
+    <!-- The loading of KaTeX is deferred to speed up page rendering -->
+    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.js" integrity="sha384-g7c+Jr9ZivxKLnZTDUhnkOnsh30B4H0rpLUpJ4jAIKs4fnJI+sEnkvrMWph2EDg4" crossorigin="anonymous"></script>
+
+    <!-- To automatically render math in text elements, include the auto-render extension: -->
+    <script defer src="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/contrib/auto-render.min.js" integrity="sha384-mll67QQFJfxn0IYznZYonOWZ644AWYC+Pt2cHqMaRhXVrursRwvLnLaebdGIlYNa" crossorigin="anonymous"
+        onload="renderMathInElement(document.body);"></script>
+  </head>
+  ...
+</html>
+```
+
+You can also [download KaTeX](https://github.com/KaTeX/KaTeX/releases) and host it yourself.
+
+For details on how to configure auto-render extension, refer to [the documentation](https://katex.org/docs/autorender.html).
+
+### API
+
+Call `katex.render` to render a TeX expression directly into a DOM element.
+For example:
+
+```js
+katex.render("c = \\pm\\sqrt{a^2 + b^2}", element, {
+    throwOnError: false
+});
+```
+
+Call `katex.renderToString` to generate an HTML string of the rendered math,
+e.g., for server-side rendering.  For example:
+
+```js
+var html = katex.renderToString("c = \\pm\\sqrt{a^2 + b^2}", {
+    throwOnError: false
+});
+// '<span class="katex">...</span>'
+```
+
+Make sure to include the CSS and font files in both cases.
+If you are doing all rendering on the server, there is no need to include the
+JavaScript on the client.
+
+The examples above use the `throwOnError: false` option, which renders invalid
+inputs as the TeX source code in red (by default), with the error message as
+hover text.  For other available options, see the
+[API documentation](https://katex.org/docs/api.html),
+[options documentation](https://katex.org/docs/options.html), and
+[handling errors documentation](https://katex.org/docs/error.html).
+
+## Demo and Documentation
+
+Learn more about using KaTeX [on the website](https://katex.org)!
+
+## Contributing
+
+See [CONTRIBUTING.md](CONTRIBUTING.md)
+
+## License
+
+KaTeX is licensed under the [MIT License](http://opensource.org/licenses/MIT).
diff --git a/katex.bk/contrib/auto-render.js b/katex.bk/contrib/auto-render.js
new file mode 100644
index 00000000..d5944348
--- /dev/null
+++ b/katex.bk/contrib/auto-render.js
@@ -0,0 +1,350 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory(require("katex"));
+	else if(typeof define === 'function' && define.amd)
+		define(["katex"], factory);
+	else if(typeof exports === 'object')
+		exports["renderMathInElement"] = factory(require("katex"));
+	else
+		root["renderMathInElement"] = factory(root["katex"]);
+})((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__0__) {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
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+/******/
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+/******/ 		return module.exports;
+/******/ 	}
+/******/
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+/******/ 	__webpack_require__.m = modules;
+/******/
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+/******/
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+/******/ 		if(typeof Symbol !== 'undefined' && Symbol.toStringTag) {
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+/******/ 		Object.defineProperty(exports, '__esModule', { value: true });
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+/******/
+/******/ 	// create a fake namespace object
+/******/ 	// mode & 1: value is a module id, require it
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+/******/ 	// mode & 4: return value when already ns object
+/******/ 	// mode & 8|1: behave like require
+/******/ 	__webpack_require__.t = function(value, mode) {
+/******/ 		if(mode & 1) value = __webpack_require__(value);
+/******/ 		if(mode & 8) return value;
+/******/ 		if((mode & 4) && typeof value === 'object' && value && value.__esModule) return value;
+/******/ 		var ns = Object.create(null);
+/******/ 		__webpack_require__.r(ns);
+/******/ 		Object.defineProperty(ns, 'default', { enumerable: true, value: value });
+/******/ 		if(mode & 2 && typeof value != 'string') for(var key in value) __webpack_require__.d(ns, key, function(key) { return value[key]; }.bind(null, key));
+/******/ 		return ns;
+/******/ 	};
+/******/
+/******/ 	// getDefaultExport function for compatibility with non-harmony modules
+/******/ 	__webpack_require__.n = function(module) {
+/******/ 		var getter = module && module.__esModule ?
+/******/ 			function getDefault() { return module['default']; } :
+/******/ 			function getModuleExports() { return module; };
+/******/ 		__webpack_require__.d(getter, 'a', getter);
+/******/ 		return getter;
+/******/ 	};
+/******/
+/******/ 	// Object.prototype.hasOwnProperty.call
+/******/ 	__webpack_require__.o = function(object, property) { return Object.prototype.hasOwnProperty.call(object, property); };
+/******/
+/******/ 	// __webpack_public_path__
+/******/ 	__webpack_require__.p = "";
+/******/
+/******/
+/******/ 	// Load entry module and return exports
+/******/ 	return __webpack_require__(__webpack_require__.s = 1);
+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports) {
+
+module.exports = __WEBPACK_EXTERNAL_MODULE__0__;
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+
+// EXTERNAL MODULE: external "katex"
+var external_katex_ = __webpack_require__(0);
+var external_katex_default = /*#__PURE__*/__webpack_require__.n(external_katex_);
+
+// CONCATENATED MODULE: ./contrib/auto-render/splitAtDelimiters.js
+/* eslint no-constant-condition:0 */
+var findEndOfMath = function findEndOfMath(delimiter, text, startIndex) {
+  // Adapted from
+  // https://github.com/Khan/perseus/blob/master/src/perseus-markdown.jsx
+  var index = startIndex;
+  var braceLevel = 0;
+  var delimLength = delimiter.length;
+
+  while (index < text.length) {
+    var character = text[index];
+
+    if (braceLevel <= 0 && text.slice(index, index + delimLength) === delimiter) {
+      return index;
+    } else if (character === "\\") {
+      index++;
+    } else if (character === "{") {
+      braceLevel++;
+    } else if (character === "}") {
+      braceLevel--;
+    }
+
+    index++;
+  }
+
+  return -1;
+};
+
+var splitAtDelimiters = function splitAtDelimiters(startData, leftDelim, rightDelim, display) {
+  var finalData = [];
+
+  for (var i = 0; i < startData.length; i++) {
+    if (startData[i].type === "text") {
+      var text = startData[i].data;
+      var lookingForLeft = true;
+      var currIndex = 0;
+      var nextIndex = void 0;
+      nextIndex = text.indexOf(leftDelim);
+
+      if (nextIndex !== -1) {
+        currIndex = nextIndex;
+        finalData.push({
+          type: "text",
+          data: text.slice(0, currIndex)
+        });
+        lookingForLeft = false;
+      }
+
+      while (true) {
+        if (lookingForLeft) {
+          nextIndex = text.indexOf(leftDelim, currIndex);
+
+          if (nextIndex === -1) {
+            break;
+          }
+
+          finalData.push({
+            type: "text",
+            data: text.slice(currIndex, nextIndex)
+          });
+          currIndex = nextIndex;
+        } else {
+          nextIndex = findEndOfMath(rightDelim, text, currIndex + leftDelim.length);
+
+          if (nextIndex === -1) {
+            break;
+          }
+
+          finalData.push({
+            type: "math",
+            data: text.slice(currIndex + leftDelim.length, nextIndex),
+            rawData: text.slice(currIndex, nextIndex + rightDelim.length),
+            display: display
+          });
+          currIndex = nextIndex + rightDelim.length;
+        }
+
+        lookingForLeft = !lookingForLeft;
+      }
+
+      finalData.push({
+        type: "text",
+        data: text.slice(currIndex)
+      });
+    } else {
+      finalData.push(startData[i]);
+    }
+  }
+
+  return finalData;
+};
+
+/* harmony default export */ var auto_render_splitAtDelimiters = (splitAtDelimiters);
+// CONCATENATED MODULE: ./contrib/auto-render/auto-render.js
+/* eslint no-console:0 */
+
+
+
+var auto_render_splitWithDelimiters = function splitWithDelimiters(text, delimiters) {
+  var data = [{
+    type: "text",
+    data: text
+  }];
+
+  for (var i = 0; i < delimiters.length; i++) {
+    var delimiter = delimiters[i];
+    data = auto_render_splitAtDelimiters(data, delimiter.left, delimiter.right, delimiter.display || false);
+  }
+
+  return data;
+};
+/* Note: optionsCopy is mutated by this method. If it is ever exposed in the
+ * API, we should copy it before mutating.
+ */
+
+
+var auto_render_renderMathInText = function renderMathInText(text, optionsCopy) {
+  var data = auto_render_splitWithDelimiters(text, optionsCopy.delimiters);
+
+  if (data.length === 1 && data[0].type === 'text') {
+    // There is no formula in the text.
+    // Let's return null which means there is no need to replace
+    // the current text node with a new one.
+    return null;
+  }
+
+  var fragment = document.createDocumentFragment();
+
+  for (var i = 0; i < data.length; i++) {
+    if (data[i].type === "text") {
+      fragment.appendChild(document.createTextNode(data[i].data));
+    } else {
+      var span = document.createElement("span");
+      var math = data[i].data; // Override any display mode defined in the settings with that
+      // defined by the text itself
+
+      optionsCopy.displayMode = data[i].display;
+
+      try {
+        if (optionsCopy.preProcess) {
+          math = optionsCopy.preProcess(math);
+        }
+
+        external_katex_default.a.render(math, span, optionsCopy);
+      } catch (e) {
+        if (!(e instanceof external_katex_default.a.ParseError)) {
+          throw e;
+        }
+
+        optionsCopy.errorCallback("KaTeX auto-render: Failed to parse `" + data[i].data + "` with ", e);
+        fragment.appendChild(document.createTextNode(data[i].rawData));
+        continue;
+      }
+
+      fragment.appendChild(span);
+    }
+  }
+
+  return fragment;
+};
+
+var renderElem = function renderElem(elem, optionsCopy) {
+  for (var i = 0; i < elem.childNodes.length; i++) {
+    var childNode = elem.childNodes[i];
+
+    if (childNode.nodeType === 3) {
+      // Text node
+      var frag = auto_render_renderMathInText(childNode.textContent, optionsCopy);
+
+      if (frag) {
+        i += frag.childNodes.length - 1;
+        elem.replaceChild(frag, childNode);
+      }
+    } else if (childNode.nodeType === 1) {
+      (function () {
+        // Element node
+        var className = ' ' + childNode.className + ' ';
+        var shouldRender = optionsCopy.ignoredTags.indexOf(childNode.nodeName.toLowerCase()) === -1 && optionsCopy.ignoredClasses.every(function (x) {
+          return className.indexOf(' ' + x + ' ') === -1;
+        });
+
+        if (shouldRender) {
+          renderElem(childNode, optionsCopy);
+        }
+      })();
+    } // Otherwise, it's something else, and ignore it.
+
+  }
+};
+
+var renderMathInElement = function renderMathInElement(elem, options) {
+  if (!elem) {
+    throw new Error("No element provided to render");
+  }
+
+  var optionsCopy = {}; // Object.assign(optionsCopy, option)
+
+  for (var option in options) {
+    if (options.hasOwnProperty(option)) {
+      optionsCopy[option] = options[option];
+    }
+  } // default options
+
+
+  optionsCopy.delimiters = optionsCopy.delimiters || [{
+    left: "$$",
+    right: "$$",
+    display: true
+  }, {
+    left: "\\(",
+    right: "\\)",
+    display: false
+  }, // LaTeX uses $…$, but it ruins the display of normal `$` in text:
+  // {left: "$", right: "$", display: false},
+  //  \[…\] must come last in this array. Otherwise, renderMathInElement
+  //  will search for \[ before it searches for $$ or  \(
+  // That makes it susceptible to finding a \\[0.3em] row delimiter and
+  // treating it as if it were the start of a KaTeX math zone.
+  {
+    left: "\\[",
+    right: "\\]",
+    display: true
+  }];
+  optionsCopy.ignoredTags = optionsCopy.ignoredTags || ["script", "noscript", "style", "textarea", "pre", "code", "option"];
+  optionsCopy.ignoredClasses = optionsCopy.ignoredClasses || [];
+  optionsCopy.errorCallback = optionsCopy.errorCallback || console.error; // Enable sharing of global macros defined via `\gdef` between different
+  // math elements within a single call to `renderMathInElement`.
+
+  optionsCopy.macros = optionsCopy.macros || {};
+  renderElem(elem, optionsCopy);
+};
+
+/* harmony default export */ var auto_render = __webpack_exports__["default"] = (renderMathInElement);
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/katex.bk/contrib/auto-render.min.js b/katex.bk/contrib/auto-render.min.js
new file mode 100644
index 00000000..840e74bf
--- /dev/null
+++ b/katex.bk/contrib/auto-render.min.js
@@ -0,0 +1 @@
+!function(e,t){"object"==typeof exports&&"object"==typeof module?module.exports=t(require("katex")):"function"==typeof define&&define.amd?define(["katex"],t):"object"==typeof exports?exports.renderMathInElement=t(require("katex")):e.renderMathInElement=t(e.katex)}("undefined"!=typeof self?self:this,function(e){return function(e){var t={};function r(n){if(t[n])return t[n].exports;var o=t[n]={i:n,l:!1,exports:{}};return e[n].call(o.exports,o,o.exports,r),o.l=!0,o.exports}return r.m=e,r.c=t,r.d=function(e,t,n){r.o(e,t)||Object.defineProperty(e,t,{enumerable:!0,get:n})},r.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},r.t=function(e,t){if(1&t&&(e=r(e)),8&t)return e;if(4&t&&"object"==typeof e&&e&&e.__esModule)return e;var n=Object.create(null);if(r.r(n),Object.defineProperty(n,"default",{enumerable:!0,value:e}),2&t&&"string"!=typeof e)for(var o in e)r.d(n,o,function(t){return e[t]}.bind(null,o));return n},r.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return r.d(t,"a",t),t},r.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)},r.p="",r(r.s=1)}([function(t,r){t.exports=e},function(e,t,r){"use strict";r.r(t);var n=r(0),o=r.n(n),a=function(e,t,r){for(var n=r,o=0,a=e.length;n<t.length;){var i=t[n];if(o<=0&&t.slice(n,n+a)===e)return n;"\\"===i?n++:"{"===i?o++:"}"===i&&o--,n++}return-1},i=function(e,t,r,n){for(var o=[],i=0;i<e.length;i++)if("text"===e[i].type){var l=e[i].data,d=!0,s=0,f=void 0;for(-1!==(f=l.indexOf(t))&&(s=f,o.push({type:"text",data:l.slice(0,s)}),d=!1);;){if(d){if(-1===(f=l.indexOf(t,s)))break;o.push({type:"text",data:l.slice(s,f)}),s=f}else{if(-1===(f=a(r,l,s+t.length)))break;o.push({type:"math",data:l.slice(s+t.length,f),rawData:l.slice(s,f+r.length),display:n}),s=f+r.length}d=!d}o.push({type:"text",data:l.slice(s)})}else o.push(e[i]);return o},l=function(e,t){var r=function(e,t){for(var r=[{type:"text",data:e}],n=0;n<t.length;n++){var o=t[n];r=i(r,o.left,o.right,o.display||!1)}return r}(e,t.delimiters);if(1===r.length&&"text"===r[0].type)return null;for(var n=document.createDocumentFragment(),a=0;a<r.length;a++)if("text"===r[a].type)n.appendChild(document.createTextNode(r[a].data));else{var l=document.createElement("span"),d=r[a].data;t.displayMode=r[a].display;try{t.preProcess&&(d=t.preProcess(d)),o.a.render(d,l,t)}catch(e){if(!(e instanceof o.a.ParseError))throw e;t.errorCallback("KaTeX auto-render: Failed to parse `"+r[a].data+"` with ",e),n.appendChild(document.createTextNode(r[a].rawData));continue}n.appendChild(l)}return n};t.default=function(e,t){if(!e)throw new Error("No element provided to render");var r={};for(var n in t)t.hasOwnProperty(n)&&(r[n]=t[n]);r.delimiters=r.delimiters||[{left:"$$",right:"$$",display:!0},{left:"\\(",right:"\\)",display:!1},{left:"\\[",right:"\\]",display:!0}],r.ignoredTags=r.ignoredTags||["script","noscript","style","textarea","pre","code","option"],r.ignoredClasses=r.ignoredClasses||[],r.errorCallback=r.errorCallback||console.error,r.macros=r.macros||{},function e(t,r){for(var n=0;n<t.childNodes.length;n++){var o=t.childNodes[n];if(3===o.nodeType){var a=l(o.textContent,r);a&&(n+=a.childNodes.length-1,t.replaceChild(a,o))}else 1===o.nodeType&&function(){var t=" "+o.className+" ";-1===r.ignoredTags.indexOf(o.nodeName.toLowerCase())&&r.ignoredClasses.every(function(e){return-1===t.indexOf(" "+e+" ")})&&e(o,r)}()}}(e,r)}}]).default});
\ No newline at end of file
diff --git a/katex.bk/contrib/auto-render.mjs b/katex.bk/contrib/auto-render.mjs
new file mode 100644
index 00000000..bc9a71f9
--- /dev/null
+++ b/katex.bk/contrib/auto-render.mjs
@@ -0,0 +1,226 @@
+import katex from '../katex.mjs';
+
+/* eslint no-constant-condition:0 */
+const findEndOfMath = function findEndOfMath(delimiter, text, startIndex) {
+  // Adapted from
+  // https://github.com/Khan/perseus/blob/master/src/perseus-markdown.jsx
+  let index = startIndex;
+  let braceLevel = 0;
+  const delimLength = delimiter.length;
+
+  while (index < text.length) {
+    const character = text[index];
+
+    if (braceLevel <= 0 && text.slice(index, index + delimLength) === delimiter) {
+      return index;
+    } else if (character === "\\") {
+      index++;
+    } else if (character === "{") {
+      braceLevel++;
+    } else if (character === "}") {
+      braceLevel--;
+    }
+
+    index++;
+  }
+
+  return -1;
+};
+
+const splitAtDelimiters = function splitAtDelimiters(startData, leftDelim, rightDelim, display) {
+  const finalData = [];
+
+  for (let i = 0; i < startData.length; i++) {
+    if (startData[i].type === "text") {
+      const text = startData[i].data;
+      let lookingForLeft = true;
+      let currIndex = 0;
+      let nextIndex;
+      nextIndex = text.indexOf(leftDelim);
+
+      if (nextIndex !== -1) {
+        currIndex = nextIndex;
+        finalData.push({
+          type: "text",
+          data: text.slice(0, currIndex)
+        });
+        lookingForLeft = false;
+      }
+
+      while (true) {
+        if (lookingForLeft) {
+          nextIndex = text.indexOf(leftDelim, currIndex);
+
+          if (nextIndex === -1) {
+            break;
+          }
+
+          finalData.push({
+            type: "text",
+            data: text.slice(currIndex, nextIndex)
+          });
+          currIndex = nextIndex;
+        } else {
+          nextIndex = findEndOfMath(rightDelim, text, currIndex + leftDelim.length);
+
+          if (nextIndex === -1) {
+            break;
+          }
+
+          finalData.push({
+            type: "math",
+            data: text.slice(currIndex + leftDelim.length, nextIndex),
+            rawData: text.slice(currIndex, nextIndex + rightDelim.length),
+            display: display
+          });
+          currIndex = nextIndex + rightDelim.length;
+        }
+
+        lookingForLeft = !lookingForLeft;
+      }
+
+      finalData.push({
+        type: "text",
+        data: text.slice(currIndex)
+      });
+    } else {
+      finalData.push(startData[i]);
+    }
+  }
+
+  return finalData;
+};
+
+/* eslint no-console:0 */
+
+const splitWithDelimiters = function splitWithDelimiters(text, delimiters) {
+  let data = [{
+    type: "text",
+    data: text
+  }];
+
+  for (let i = 0; i < delimiters.length; i++) {
+    const delimiter = delimiters[i];
+    data = splitAtDelimiters(data, delimiter.left, delimiter.right, delimiter.display || false);
+  }
+
+  return data;
+};
+/* Note: optionsCopy is mutated by this method. If it is ever exposed in the
+ * API, we should copy it before mutating.
+ */
+
+
+const renderMathInText = function renderMathInText(text, optionsCopy) {
+  const data = splitWithDelimiters(text, optionsCopy.delimiters);
+
+  if (data.length === 1 && data[0].type === 'text') {
+    // There is no formula in the text.
+    // Let's return null which means there is no need to replace
+    // the current text node with a new one.
+    return null;
+  }
+
+  const fragment = document.createDocumentFragment();
+
+  for (let i = 0; i < data.length; i++) {
+    if (data[i].type === "text") {
+      fragment.appendChild(document.createTextNode(data[i].data));
+    } else {
+      const span = document.createElement("span");
+      let math = data[i].data; // Override any display mode defined in the settings with that
+      // defined by the text itself
+
+      optionsCopy.displayMode = data[i].display;
+
+      try {
+        if (optionsCopy.preProcess) {
+          math = optionsCopy.preProcess(math);
+        }
+
+        katex.render(math, span, optionsCopy);
+      } catch (e) {
+        if (!(e instanceof katex.ParseError)) {
+          throw e;
+        }
+
+        optionsCopy.errorCallback("KaTeX auto-render: Failed to parse `" + data[i].data + "` with ", e);
+        fragment.appendChild(document.createTextNode(data[i].rawData));
+        continue;
+      }
+
+      fragment.appendChild(span);
+    }
+  }
+
+  return fragment;
+};
+
+const renderElem = function renderElem(elem, optionsCopy) {
+  for (let i = 0; i < elem.childNodes.length; i++) {
+    const childNode = elem.childNodes[i];
+
+    if (childNode.nodeType === 3) {
+      // Text node
+      const frag = renderMathInText(childNode.textContent, optionsCopy);
+
+      if (frag) {
+        i += frag.childNodes.length - 1;
+        elem.replaceChild(frag, childNode);
+      }
+    } else if (childNode.nodeType === 1) {
+      // Element node
+      const className = ' ' + childNode.className + ' ';
+      const shouldRender = optionsCopy.ignoredTags.indexOf(childNode.nodeName.toLowerCase()) === -1 && optionsCopy.ignoredClasses.every(x => className.indexOf(' ' + x + ' ') === -1);
+
+      if (shouldRender) {
+        renderElem(childNode, optionsCopy);
+      }
+    } // Otherwise, it's something else, and ignore it.
+
+  }
+};
+
+const renderMathInElement = function renderMathInElement(elem, options) {
+  if (!elem) {
+    throw new Error("No element provided to render");
+  }
+
+  const optionsCopy = {}; // Object.assign(optionsCopy, option)
+
+  for (const option in options) {
+    if (options.hasOwnProperty(option)) {
+      optionsCopy[option] = options[option];
+    }
+  } // default options
+
+
+  optionsCopy.delimiters = optionsCopy.delimiters || [{
+    left: "$$",
+    right: "$$",
+    display: true
+  }, {
+    left: "\\(",
+    right: "\\)",
+    display: false
+  }, // LaTeX uses $…$, but it ruins the display of normal `$` in text:
+  // {left: "$", right: "$", display: false},
+  //  \[…\] must come last in this array. Otherwise, renderMathInElement
+  //  will search for \[ before it searches for $$ or  \(
+  // That makes it susceptible to finding a \\[0.3em] row delimiter and
+  // treating it as if it were the start of a KaTeX math zone.
+  {
+    left: "\\[",
+    right: "\\]",
+    display: true
+  }];
+  optionsCopy.ignoredTags = optionsCopy.ignoredTags || ["script", "noscript", "style", "textarea", "pre", "code", "option"];
+  optionsCopy.ignoredClasses = optionsCopy.ignoredClasses || [];
+  optionsCopy.errorCallback = optionsCopy.errorCallback || console.error; // Enable sharing of global macros defined via `\gdef` between different
+  // math elements within a single call to `renderMathInElement`.
+
+  optionsCopy.macros = optionsCopy.macros || {};
+  renderElem(elem, optionsCopy);
+};
+
+export default renderMathInElement;
diff --git a/katex.bk/contrib/copy-tex.css b/katex.bk/contrib/copy-tex.css
new file mode 100644
index 00000000..90801d84
--- /dev/null
+++ b/katex.bk/contrib/copy-tex.css
@@ -0,0 +1,14 @@
+/* Force selection of entire .katex/.katex-display blocks, so that we can
+ * copy/paste the entire source code.  If you omit this CSS, partial
+ * selections of a formula will work, but will copy the ugly HTML
+ * representation instead of the LaTeX source code.  (Full selections will
+ * still produce the LaTeX source code.)
+ */
+.katex,
+.katex-display {
+    user-select: all;
+    -moz-user-select: all;
+    -webkit-user-select: all;
+    -ms-user-select: all;
+}
+
diff --git a/katex.bk/contrib/copy-tex.js b/katex.bk/contrib/copy-tex.js
new file mode 100644
index 00000000..4649f8f7
--- /dev/null
+++ b/katex.bk/contrib/copy-tex.js
@@ -0,0 +1,213 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory();
+	else if(typeof define === 'function' && define.amd)
+		define([], factory);
+	else {
+		var a = factory();
+		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
+	}
+})((typeof self !== 'undefined' ? self : this), function() {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
+/******/ 	function __webpack_require__(moduleId) {
+/******/
+/******/ 		// Check if module is in cache
+/******/ 		if(installedModules[moduleId]) {
+/******/ 			return installedModules[moduleId].exports;
+/******/ 		}
+/******/ 		// Create a new module (and put it into the cache)
+/******/ 		var module = installedModules[moduleId] = {
+/******/ 			i: moduleId,
+/******/ 			l: false,
+/******/ 			exports: {}
+/******/ 		};
+/******/
+/******/ 		// Execute the module function
+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
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+/******/ ([
+/* 0 */
+/***/ (function(module, exports, __webpack_require__) {
+
+// extracted by mini-css-extract-plugin
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+
+// EXTERNAL MODULE: ./contrib/copy-tex/copy-tex.css
+var copy_tex = __webpack_require__(0);
+
+// CONCATENATED MODULE: ./contrib/copy-tex/katex2tex.js
+// Set these to how you want inline and display math to be delimited.
+var defaultCopyDelimiters = {
+  inline: ['$', '$'],
+  // alternative: ['\(', '\)']
+  display: ['$$', '$$'] // alternative: ['\[', '\]']
+
+}; // Replace .katex elements with their TeX source (<annotation> element).
+// Modifies fragment in-place.  Useful for writing your own 'copy' handler,
+// as in copy-tex.js.
+
+var katexReplaceWithTex = function katexReplaceWithTex(fragment, copyDelimiters) {
+  if (copyDelimiters === void 0) {
+    copyDelimiters = defaultCopyDelimiters;
+  }
+
+  // Remove .katex-html blocks that are preceded by .katex-mathml blocks
+  // (which will get replaced below).
+  var katexHtml = fragment.querySelectorAll('.katex-mathml + .katex-html');
+
+  for (var i = 0; i < katexHtml.length; i++) {
+    var element = katexHtml[i];
+
+    if (element.remove) {
+      element.remove(null);
+    } else {
+      element.parentNode.removeChild(element);
+    }
+  } // Replace .katex-mathml elements with their annotation (TeX source)
+  // descendant, with inline delimiters.
+
+
+  var katexMathml = fragment.querySelectorAll('.katex-mathml');
+
+  for (var _i = 0; _i < katexMathml.length; _i++) {
+    var _element = katexMathml[_i];
+
+    var texSource = _element.querySelector('annotation');
+
+    if (texSource) {
+      if (_element.replaceWith) {
+        _element.replaceWith(texSource);
+      } else {
+        _element.parentNode.replaceChild(texSource, _element);
+      }
+
+      texSource.innerHTML = copyDelimiters.inline[0] + texSource.innerHTML + copyDelimiters.inline[1];
+    }
+  } // Switch display math to display delimiters.
+
+
+  var displays = fragment.querySelectorAll('.katex-display annotation');
+
+  for (var _i2 = 0; _i2 < displays.length; _i2++) {
+    var _element2 = displays[_i2];
+    _element2.innerHTML = copyDelimiters.display[0] + _element2.innerHTML.substr(copyDelimiters.inline[0].length, _element2.innerHTML.length - copyDelimiters.inline[0].length - copyDelimiters.inline[1].length) + copyDelimiters.display[1];
+  }
+
+  return fragment;
+};
+/* harmony default export */ var katex2tex = (katexReplaceWithTex);
+// CONCATENATED MODULE: ./contrib/copy-tex/copy-tex.js
+ // Global copy handler to modify behavior on .katex elements.
+
+document.addEventListener('copy', function (event) {
+  var selection = window.getSelection();
+
+  if (selection.isCollapsed) {
+    return; // default action OK if selection is empty
+  }
+
+  var fragment = selection.getRangeAt(0).cloneContents();
+
+  if (!fragment.querySelector('.katex-mathml')) {
+    return; // default action OK if no .katex-mathml elements
+  } // Preserve usual HTML copy/paste behavior.
+
+
+  var html = [];
+
+  for (var i = 0; i < fragment.childNodes.length; i++) {
+    html.push(fragment.childNodes[i].outerHTML);
+  }
+
+  event.clipboardData.setData('text/html', html.join('')); // Rewrite plain-text version.
+
+  event.clipboardData.setData('text/plain', katex2tex(fragment).textContent); // Prevent normal copy handling.
+
+  event.preventDefault();
+});
+// CONCATENATED MODULE: ./contrib/copy-tex/copy-tex.webpack.js
+/**
+ * This is the webpack entry point for KaTeX. As ECMAScript doesn't support
+ * CSS modules natively, a separate entry point is used.
+ */
+
+
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/katex.bk/contrib/copy-tex.min.css b/katex.bk/contrib/copy-tex.min.css
new file mode 100644
index 00000000..555ed115
--- /dev/null
+++ b/katex.bk/contrib/copy-tex.min.css
@@ -0,0 +1 @@
+.katex,.katex-display{user-select:all;-moz-user-select:all;-webkit-user-select:all;-ms-user-select:all}
diff --git a/katex.bk/contrib/copy-tex.min.js b/katex.bk/contrib/copy-tex.min.js
new file mode 100644
index 00000000..e0354d6b
--- /dev/null
+++ b/katex.bk/contrib/copy-tex.min.js
@@ -0,0 +1 @@
+!function(e,t){if("object"==typeof exports&&"object"==typeof module)module.exports=t();else if("function"==typeof define&&define.amd)define([],t);else{var n=t();for(var r in n)("object"==typeof exports?exports:e)[r]=n[r]}}("undefined"!=typeof self?self:this,function(){return function(e){var t={};function n(r){if(t[r])return t[r].exports;var o=t[r]={i:r,l:!1,exports:{}};return e[r].call(o.exports,o,o.exports,n),o.l=!0,o.exports}return n.m=e,n.c=t,n.d=function(e,t,r){n.o(e,t)||Object.defineProperty(e,t,{enumerable:!0,get:r})},n.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},n.t=function(e,t){if(1&t&&(e=n(e)),8&t)return e;if(4&t&&"object"==typeof e&&e&&e.__esModule)return e;var r=Object.create(null);if(n.r(r),Object.defineProperty(r,"default",{enumerable:!0,value:e}),2&t&&"string"!=typeof e)for(var o in e)n.d(r,o,function(t){return e[t]}.bind(null,o));return r},n.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return n.d(t,"a",t),t},n.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)},n.p="",n(n.s=1)}([function(e,t,n){},function(e,t,n){"use strict";n.r(t);n(0);var r={inline:["$","$"],display:["$$","$$"]},o=function(e,t){void 0===t&&(t=r);for(var n=e.querySelectorAll(".katex-mathml + .katex-html"),o=0;o<n.length;o++){var l=n[o];l.remove?l.remove(null):l.parentNode.removeChild(l)}for(var i=e.querySelectorAll(".katex-mathml"),a=0;a<i.length;a++){var u=i[a],f=u.querySelector("annotation");f&&(u.replaceWith?u.replaceWith(f):u.parentNode.replaceChild(f,u),f.innerHTML=t.inline[0]+f.innerHTML+t.inline[1])}for(var c=e.querySelectorAll(".katex-display annotation"),d=0;d<c.length;d++){var p=c[d];p.innerHTML=t.display[0]+p.innerHTML.substr(t.inline[0].length,p.innerHTML.length-t.inline[0].length-t.inline[1].length)+t.display[1]}return e};document.addEventListener("copy",function(e){var t=window.getSelection();if(!t.isCollapsed){var n=t.getRangeAt(0).cloneContents();if(n.querySelector(".katex-mathml")){for(var r=[],l=0;l<n.childNodes.length;l++)r.push(n.childNodes[l].outerHTML);e.clipboardData.setData("text/html",r.join("")),e.clipboardData.setData("text/plain",o(n).textContent),e.preventDefault()}}})}]).default});
\ No newline at end of file
diff --git a/katex.bk/contrib/copy-tex.mjs b/katex.bk/contrib/copy-tex.mjs
new file mode 100644
index 00000000..b6ed1eef
--- /dev/null
+++ b/katex.bk/contrib/copy-tex.mjs
@@ -0,0 +1,85 @@
+// Set these to how you want inline and display math to be delimited.
+const defaultCopyDelimiters = {
+  inline: ['$', '$'],
+  // alternative: ['\(', '\)']
+  display: ['$$', '$$'] // alternative: ['\[', '\]']
+
+}; // Replace .katex elements with their TeX source (<annotation> element).
+// Modifies fragment in-place.  Useful for writing your own 'copy' handler,
+// as in copy-tex.js.
+
+const katexReplaceWithTex = function katexReplaceWithTex(fragment, copyDelimiters) {
+  if (copyDelimiters === void 0) {
+    copyDelimiters = defaultCopyDelimiters;
+  }
+
+  // Remove .katex-html blocks that are preceded by .katex-mathml blocks
+  // (which will get replaced below).
+  const katexHtml = fragment.querySelectorAll('.katex-mathml + .katex-html');
+
+  for (let i = 0; i < katexHtml.length; i++) {
+    const element = katexHtml[i];
+
+    if (element.remove) {
+      element.remove(null);
+    } else {
+      element.parentNode.removeChild(element);
+    }
+  } // Replace .katex-mathml elements with their annotation (TeX source)
+  // descendant, with inline delimiters.
+
+
+  const katexMathml = fragment.querySelectorAll('.katex-mathml');
+
+  for (let i = 0; i < katexMathml.length; i++) {
+    const element = katexMathml[i];
+    const texSource = element.querySelector('annotation');
+
+    if (texSource) {
+      if (element.replaceWith) {
+        element.replaceWith(texSource);
+      } else {
+        element.parentNode.replaceChild(texSource, element);
+      }
+
+      texSource.innerHTML = copyDelimiters.inline[0] + texSource.innerHTML + copyDelimiters.inline[1];
+    }
+  } // Switch display math to display delimiters.
+
+
+  const displays = fragment.querySelectorAll('.katex-display annotation');
+
+  for (let i = 0; i < displays.length; i++) {
+    const element = displays[i];
+    element.innerHTML = copyDelimiters.display[0] + element.innerHTML.substr(copyDelimiters.inline[0].length, element.innerHTML.length - copyDelimiters.inline[0].length - copyDelimiters.inline[1].length) + copyDelimiters.display[1];
+  }
+
+  return fragment;
+};
+
+document.addEventListener('copy', function (event) {
+  const selection = window.getSelection();
+
+  if (selection.isCollapsed) {
+    return; // default action OK if selection is empty
+  }
+
+  const fragment = selection.getRangeAt(0).cloneContents();
+
+  if (!fragment.querySelector('.katex-mathml')) {
+    return; // default action OK if no .katex-mathml elements
+  } // Preserve usual HTML copy/paste behavior.
+
+
+  const html = [];
+
+  for (let i = 0; i < fragment.childNodes.length; i++) {
+    html.push(fragment.childNodes[i].outerHTML);
+  }
+
+  event.clipboardData.setData('text/html', html.join('')); // Rewrite plain-text version.
+
+  event.clipboardData.setData('text/plain', katexReplaceWithTex(fragment).textContent); // Prevent normal copy handling.
+
+  event.preventDefault();
+});
diff --git a/katex.bk/contrib/mathtex-script-type.js b/katex.bk/contrib/mathtex-script-type.js
new file mode 100644
index 00000000..daa01a63
--- /dev/null
+++ b/katex.bk/contrib/mathtex-script-type.js
@@ -0,0 +1,137 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory(require("katex"));
+	else if(typeof define === 'function' && define.amd)
+		define(["katex"], factory);
+	else {
+		var a = typeof exports === 'object' ? factory(require("katex")) : factory(root["katex"]);
+		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
+	}
+})((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__0__) {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
+/******/ 	function __webpack_require__(moduleId) {
+/******/
+/******/ 		// Check if module is in cache
+/******/ 		if(installedModules[moduleId]) {
+/******/ 			return installedModules[moduleId].exports;
+/******/ 		}
+/******/ 		// Create a new module (and put it into the cache)
+/******/ 		var module = installedModules[moduleId] = {
+/******/ 			i: moduleId,
+/******/ 			l: false,
+/******/ 			exports: {}
+/******/ 		};
+/******/
+/******/ 		// Execute the module function
+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
+/******/ 		// Flag the module as loaded
+/******/ 		module.l = true;
+/******/
+/******/ 		// Return the exports of the module
+/******/ 		return module.exports;
+/******/ 	}
+/******/
+/******/
+/******/ 	// expose the modules object (__webpack_modules__)
+/******/ 	__webpack_require__.m = modules;
+/******/
+/******/ 	// expose the module cache
+/******/ 	__webpack_require__.c = installedModules;
+/******/
+/******/ 	// define getter function for harmony exports
+/******/ 	__webpack_require__.d = function(exports, name, getter) {
+/******/ 		if(!__webpack_require__.o(exports, name)) {
+/******/ 			Object.defineProperty(exports, name, { enumerable: true, get: getter });
+/******/ 		}
+/******/ 	};
+/******/
+/******/ 	// define __esModule on exports
+/******/ 	__webpack_require__.r = function(exports) {
+/******/ 		if(typeof Symbol !== 'undefined' && Symbol.toStringTag) {
+/******/ 			Object.defineProperty(exports, Symbol.toStringTag, { value: 'Module' });
+/******/ 		}
+/******/ 		Object.defineProperty(exports, '__esModule', { value: true });
+/******/ 	};
+/******/
+/******/ 	// create a fake namespace object
+/******/ 	// mode & 1: value is a module id, require it
+/******/ 	// mode & 2: merge all properties of value into the ns
+/******/ 	// mode & 4: return value when already ns object
+/******/ 	// mode & 8|1: behave like require
+/******/ 	__webpack_require__.t = function(value, mode) {
+/******/ 		if(mode & 1) value = __webpack_require__(value);
+/******/ 		if(mode & 8) return value;
+/******/ 		if((mode & 4) && typeof value === 'object' && value && value.__esModule) return value;
+/******/ 		var ns = Object.create(null);
+/******/ 		__webpack_require__.r(ns);
+/******/ 		Object.defineProperty(ns, 'default', { enumerable: true, value: value });
+/******/ 		if(mode & 2 && typeof value != 'string') for(var key in value) __webpack_require__.d(ns, key, function(key) { return value[key]; }.bind(null, key));
+/******/ 		return ns;
+/******/ 	};
+/******/
+/******/ 	// getDefaultExport function for compatibility with non-harmony modules
+/******/ 	__webpack_require__.n = function(module) {
+/******/ 		var getter = module && module.__esModule ?
+/******/ 			function getDefault() { return module['default']; } :
+/******/ 			function getModuleExports() { return module; };
+/******/ 		__webpack_require__.d(getter, 'a', getter);
+/******/ 		return getter;
+/******/ 	};
+/******/
+/******/ 	// Object.prototype.hasOwnProperty.call
+/******/ 	__webpack_require__.o = function(object, property) { return Object.prototype.hasOwnProperty.call(object, property); };
+/******/
+/******/ 	// __webpack_public_path__
+/******/ 	__webpack_require__.p = "";
+/******/
+/******/
+/******/ 	// Load entry module and return exports
+/******/ 	return __webpack_require__(__webpack_require__.s = 1);
+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports) {
+
+module.exports = __WEBPACK_EXTERNAL_MODULE__0__;
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0__ = __webpack_require__(0);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0___default = /*#__PURE__*/__webpack_require__.n(katex__WEBPACK_IMPORTED_MODULE_0__);
+
+var scripts = document.body.getElementsByTagName("script");
+scripts = Array.prototype.slice.call(scripts);
+scripts.forEach(function (script) {
+  if (!script.type || !script.type.match(/math\/tex/i)) {
+    return -1;
+  }
+
+  var display = script.type.match(/mode\s*=\s*display(;|\s|\n|$)/) != null;
+  var katexElement = document.createElement(display ? "div" : "span");
+  katexElement.setAttribute("class", display ? "equation" : "inline-equation");
+
+  try {
+    katex__WEBPACK_IMPORTED_MODULE_0___default.a.render(script.text, katexElement, {
+      displayMode: display
+    });
+  } catch (err) {
+    //console.error(err); linter doesn't like this
+    katexElement.textContent = script.text;
+  }
+
+  script.parentNode.replaceChild(katexElement, script);
+});
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/katex.bk/contrib/mathtex-script-type.min.js b/katex.bk/contrib/mathtex-script-type.min.js
new file mode 100644
index 00000000..ae9f5286
--- /dev/null
+++ b/katex.bk/contrib/mathtex-script-type.min.js
@@ -0,0 +1 @@
+!function(e,t){if("object"==typeof exports&&"object"==typeof module)module.exports=t(require("katex"));else if("function"==typeof define&&define.amd)define(["katex"],t);else{var r="object"==typeof exports?t(require("katex")):t(e.katex);for(var n in r)("object"==typeof exports?exports:e)[n]=r[n]}}("undefined"!=typeof self?self:this,function(e){return function(e){var t={};function r(n){if(t[n])return t[n].exports;var o=t[n]={i:n,l:!1,exports:{}};return e[n].call(o.exports,o,o.exports,r),o.l=!0,o.exports}return r.m=e,r.c=t,r.d=function(e,t,n){r.o(e,t)||Object.defineProperty(e,t,{enumerable:!0,get:n})},r.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},r.t=function(e,t){if(1&t&&(e=r(e)),8&t)return e;if(4&t&&"object"==typeof e&&e&&e.__esModule)return e;var n=Object.create(null);if(r.r(n),Object.defineProperty(n,"default",{enumerable:!0,value:e}),2&t&&"string"!=typeof e)for(var o in e)r.d(n,o,function(t){return e[t]}.bind(null,o));return n},r.n=function(e){var t=e&&e.__esModule?function(){return e.default}:function(){return e};return r.d(t,"a",t),t},r.o=function(e,t){return Object.prototype.hasOwnProperty.call(e,t)},r.p="",r(r.s=1)}([function(t,r){t.exports=e},function(e,t,r){"use strict";r.r(t);var n=r(0),o=r.n(n),u=document.body.getElementsByTagName("script");(u=Array.prototype.slice.call(u)).forEach(function(e){if(!e.type||!e.type.match(/math\/tex/i))return-1;var t=null!=e.type.match(/mode\s*=\s*display(;|\s|\n|$)/),r=document.createElement(t?"div":"span");r.setAttribute("class",t?"equation":"inline-equation");try{o.a.render(e.text,r,{displayMode:t})}catch(t){r.textContent=e.text}e.parentNode.replaceChild(r,e)})}]).default});
\ No newline at end of file
diff --git a/katex.bk/contrib/mathtex-script-type.mjs b/katex.bk/contrib/mathtex-script-type.mjs
new file mode 100644
index 00000000..7cfb90e9
--- /dev/null
+++ b/katex.bk/contrib/mathtex-script-type.mjs
@@ -0,0 +1,24 @@
+import katex from '../katex.mjs';
+
+let scripts = document.body.getElementsByTagName("script");
+scripts = Array.prototype.slice.call(scripts);
+scripts.forEach(function (script) {
+  if (!script.type || !script.type.match(/math\/tex/i)) {
+    return -1;
+  }
+
+  const display = script.type.match(/mode\s*=\s*display(;|\s|\n|$)/) != null;
+  const katexElement = document.createElement(display ? "div" : "span");
+  katexElement.setAttribute("class", display ? "equation" : "inline-equation");
+
+  try {
+    katex.render(script.text, katexElement, {
+      displayMode: display
+    });
+  } catch (err) {
+    //console.error(err); linter doesn't like this
+    katexElement.textContent = script.text;
+  }
+
+  script.parentNode.replaceChild(katexElement, script);
+});
diff --git a/katex.bk/contrib/mhchem.js b/katex.bk/contrib/mhchem.js
new file mode 100644
index 00000000..f84566ef
--- /dev/null
+++ b/katex.bk/contrib/mhchem.js
@@ -0,0 +1,3241 @@
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+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory(require("katex"));
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+		define(["katex"], factory);
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+/* 0 */
+/***/ (function(module, exports) {
+
+module.exports = __WEBPACK_EXTERNAL_MODULE__0__;
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0__ = __webpack_require__(0);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0___default = /*#__PURE__*/__webpack_require__.n(katex__WEBPACK_IMPORTED_MODULE_0__);
+/* eslint-disable */
+
+/* -*- Mode: Javascript; indent-tabs-mode:nil; js-indent-level: 2 -*- */
+
+/* vim: set ts=2 et sw=2 tw=80: */
+
+/*************************************************************
+ *
+ *  KaTeX mhchem.js
+ *
+ *  This file implements a KaTeX version of mhchem version 3.3.0.
+ *  It is adapted from MathJax/extensions/TeX/mhchem.js
+ *  It differs from the MathJax version as follows:
+ *    1. The interface is changed so that it can be called from KaTeX, not MathJax.
+ *    2. \rlap and \llap are replaced with \mathrlap and \mathllap.
+ *    3. Four lines of code are edited in order to use \raisebox instead of \raise.
+ *    4. The reaction arrow code is simplified. All reaction arrows are rendered
+ *       using KaTeX extensible arrows instead of building non-extensible arrows.
+ *    5. \tripledash vertical alignment is slightly adjusted.
+ *
+ *    This code, as other KaTeX code, is released under the MIT license.
+ * 
+ * /*************************************************************
+ *
+ *  MathJax/extensions/TeX/mhchem.js
+ *
+ *  Implements the \ce command for handling chemical formulas
+ *  from the mhchem LaTeX package.
+ *
+ *  ---------------------------------------------------------------------
+ *
+ *  Copyright (c) 2011-2015 The MathJax Consortium
+ *  Copyright (c) 2015-2018 Martin Hensel
+ *
+ *  Licensed under the Apache License, Version 2.0 (the "License");
+ *  you may not use this file except in compliance with the License.
+ *  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ *  Unless required by applicable law or agreed to in writing, software
+ *  distributed under the License is distributed on an "AS IS" BASIS,
+ *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ *  See the License for the specific language governing permissions and
+ *  limitations under the License.
+ */
+//
+// Coding Style
+//   - use '' for identifiers that can by minified/uglified
+//   - use "" for strings that need to stay untouched
+// version: "3.3.0" for MathJax and KaTeX
+// Add \ce, \pu, and \tripledash to the KaTeX macros.
+katex__WEBPACK_IMPORTED_MODULE_0___default.a.__defineMacro("\\ce", function (context) {
+  return chemParse(context.consumeArgs(1)[0], "ce");
+});
+
+katex__WEBPACK_IMPORTED_MODULE_0___default.a.__defineMacro("\\pu", function (context) {
+  return chemParse(context.consumeArgs(1)[0], "pu");
+}); //  Needed for \bond for the ~ forms
+//  Raise by 2.56mu, not 2mu. We're raising a hyphen-minus, U+002D, not 
+//  a mathematical minus, U+2212. So we need that extra 0.56.
+
+
+katex__WEBPACK_IMPORTED_MODULE_0___default.a.__defineMacro("\\tripledash", "{\\vphantom{-}\\raisebox{2.56mu}{$\\mkern2mu" + "\\tiny\\text{-}\\mkern1mu\\text{-}\\mkern1mu\\text{-}\\mkern2mu$}}");
+
+ //
+//  This is the main function for handing the \ce and \pu commands.
+//  It takes the argument to \ce or \pu and returns the corresponding TeX string.
+//
+
+var chemParse = function chemParse(tokens, stateMachine) {
+  // Recreate the argument string from KaTeX's array of tokens.
+  var str = "";
+  var expectedLoc = tokens[tokens.length - 1].loc.start;
+
+  for (var i = tokens.length - 1; i >= 0; i--) {
+    if (tokens[i].loc.start > expectedLoc) {
+      // context.consumeArgs has eaten a space.
+      str += " ";
+      expectedLoc = tokens[i].loc.start;
+    }
+
+    str += tokens[i].text;
+    expectedLoc += tokens[i].text.length;
+  }
+
+  var tex = texify.go(mhchemParser.go(str, stateMachine));
+  return tex;
+}; //
+// Core parser for mhchem syntax  (recursive)
+//
+
+/** @type {MhchemParser} */
+
+
+var mhchemParser = {
+  //
+  // Parses mchem \ce syntax
+  //
+  // Call like
+  //   go("H2O");
+  //
+  go: function go(input, stateMachine) {
+    if (!input) {
+      return [];
+    }
+
+    if (stateMachine === undefined) {
+      stateMachine = 'ce';
+    }
+
+    var state = '0'; //
+    // String buffers for parsing:
+    //
+    // buffer.a == amount
+    // buffer.o == element
+    // buffer.b == left-side superscript
+    // buffer.p == left-side subscript
+    // buffer.q == right-side subscript
+    // buffer.d == right-side superscript
+    //
+    // buffer.r == arrow
+    // buffer.rdt == arrow, script above, type
+    // buffer.rd == arrow, script above, content
+    // buffer.rqt == arrow, script below, type
+    // buffer.rq == arrow, script below, content
+    //
+    // buffer.text_
+    // buffer.rm
+    // etc.
+    //
+    // buffer.parenthesisLevel == int, starting at 0
+    // buffer.sb == bool, space before
+    // buffer.beginsWithBond == bool
+    //
+    // These letters are also used as state names.
+    //
+    // Other states:
+    // 0 == begin of main part (arrow/operator unlikely)
+    // 1 == next entity
+    // 2 == next entity (arrow/operator unlikely)
+    // 3 == next atom
+    // c == macro
+    //
+
+    /** @type {Buffer} */
+
+    var buffer = {};
+    buffer['parenthesisLevel'] = 0;
+    input = input.replace(/\n/g, " ");
+    input = input.replace(/[\u2212\u2013\u2014\u2010]/g, "-");
+    input = input.replace(/[\u2026]/g, "..."); //
+    // Looks through mhchemParser.transitions, to execute a matching action
+    // (recursive)
+    //
+
+    var lastInput;
+    var watchdog = 10;
+    /** @type {ParserOutput[]} */
+
+    var output = [];
+
+    while (true) {
+      if (lastInput !== input) {
+        watchdog = 10;
+        lastInput = input;
+      } else {
+        watchdog--;
+      } //
+      // Find actions in transition table
+      //
+
+
+      var machine = mhchemParser.stateMachines[stateMachine];
+      var t = machine.transitions[state] || machine.transitions['*'];
+
+      iterateTransitions: for (var i = 0; i < t.length; i++) {
+        var matches = mhchemParser.patterns.match_(t[i].pattern, input);
+
+        if (matches) {
+          //
+          // Execute actions
+          //
+          var task = t[i].task;
+
+          for (var iA = 0; iA < task.action_.length; iA++) {
+            var o; //
+            // Find and execute action
+            //
+
+            if (machine.actions[task.action_[iA].type_]) {
+              o = machine.actions[task.action_[iA].type_](buffer, matches.match_, task.action_[iA].option);
+            } else if (mhchemParser.actions[task.action_[iA].type_]) {
+              o = mhchemParser.actions[task.action_[iA].type_](buffer, matches.match_, task.action_[iA].option);
+            } else {
+              throw ["MhchemBugA", "mhchem bug A. Please report. (" + task.action_[iA].type_ + ")"]; // Trying to use non-existing action
+            } //
+            // Add output
+            //
+
+
+            mhchemParser.concatArray(output, o);
+          } //
+          // Set next state,
+          // Shorten input,
+          // Continue with next character
+          //   (= apply only one transition per position)
+          //
+
+
+          state = task.nextState || state;
+
+          if (input.length > 0) {
+            if (!task.revisit) {
+              input = matches.remainder;
+            }
+
+            if (!task.toContinue) {
+              break iterateTransitions;
+            }
+          } else {
+            return output;
+          }
+        }
+      } //
+      // Prevent infinite loop
+      //
+
+
+      if (watchdog <= 0) {
+        throw ["MhchemBugU", "mhchem bug U. Please report."]; // Unexpected character
+      }
+    }
+  },
+  concatArray: function concatArray(a, b) {
+    if (b) {
+      if (Array.isArray(b)) {
+        for (var iB = 0; iB < b.length; iB++) {
+          a.push(b[iB]);
+        }
+      } else {
+        a.push(b);
+      }
+    }
+  },
+  patterns: {
+    //
+    // Matching patterns
+    // either regexps or function that return null or {match_:"a", remainder:"bc"}
+    //
+    patterns: {
+      // property names must not look like integers ("2") for correct property traversal order, later on
+      'empty': /^$/,
+      'else': /^./,
+      'else2': /^./,
+      'space': /^\s/,
+      'space A': /^\s(?=[A-Z\\$])/,
+      'space$': /^\s$/,
+      'a-z': /^[a-z]/,
+      'x': /^x/,
+      'x$': /^x$/,
+      'i$': /^i$/,
+      'letters': /^(?:[a-zA-Z\u03B1-\u03C9\u0391-\u03A9?@]|(?:\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega|Gamma|Delta|Theta|Lambda|Xi|Pi|Sigma|Upsilon|Phi|Psi|Omega)(?:\s+|\{\}|(?![a-zA-Z]))))+/,
+      '\\greek': /^\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega|Gamma|Delta|Theta|Lambda|Xi|Pi|Sigma|Upsilon|Phi|Psi|Omega)(?:\s+|\{\}|(?![a-zA-Z]))/,
+      'one lowercase latin letter $': /^(?:([a-z])(?:$|[^a-zA-Z]))$/,
+      '$one lowercase latin letter$ $': /^\$(?:([a-z])(?:$|[^a-zA-Z]))\$$/,
+      'one lowercase greek letter $': /^(?:\$?[\u03B1-\u03C9]\$?|\$?\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega)\s*\$?)(?:\s+|\{\}|(?![a-zA-Z]))$/,
+      'digits': /^[0-9]+/,
+      '-9.,9': /^[+\-]?(?:[0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))/,
+      '-9.,9 no missing 0': /^[+\-]?[0-9]+(?:[.,][0-9]+)?/,
+      '(-)(9.,9)(e)(99)': function e99(input) {
+        var m = input.match(/^(\+\-|\+\/\-|\+|\-|\\pm\s?)?([0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))?(\((?:[0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))\))?(?:([eE]|\s*(\*|x|\\times|\u00D7)\s*10\^)([+\-]?[0-9]+|\{[+\-]?[0-9]+\}))?/);
+
+        if (m && m[0]) {
+          return {
+            match_: m.splice(1),
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      '(-)(9)^(-9)': function _(input) {
+        var m = input.match(/^(\+\-|\+\/\-|\+|\-|\\pm\s?)?([0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+)?)\^([+\-]?[0-9]+|\{[+\-]?[0-9]+\})/);
+
+        if (m && m[0]) {
+          return {
+            match_: m.splice(1),
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      'state of aggregation $': function stateOfAggregation$(input) {
+        // ... or crystal system
+        var a = mhchemParser.patterns.findObserveGroups(input, "", /^\([a-z]{1,3}(?=[\),])/, ")", ""); // (aq), (aq,$\infty$), (aq, sat)
+
+        if (a && a.remainder.match(/^($|[\s,;\)\]\}])/)) {
+          return a;
+        } //  AND end of 'phrase'
+
+
+        var m = input.match(/^(?:\((?:\\ca\s?)?\$[amothc]\$\))/); // OR crystal system ($o$) (\ca$c$)
+
+        if (m) {
+          return {
+            match_: m[0],
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      '_{(state of aggregation)}$': /^_\{(\([a-z]{1,3}\))\}/,
+      '{[(': /^(?:\\\{|\[|\()/,
+      ')]}': /^(?:\)|\]|\\\})/,
+      ', ': /^[,;]\s*/,
+      ',': /^[,;]/,
+      '.': /^[.]/,
+      '. ': /^([.\u22C5\u00B7\u2022])\s*/,
+      '...': /^\.\.\.(?=$|[^.])/,
+      '* ': /^([*])\s*/,
+      '^{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^{", "", "", "}");
+      },
+      '^($...$)': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", "$", "$", "");
+      },
+      '^a': /^\^([0-9]+|[^\\_])/,
+      '^\\x{}{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '^\\x{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '^\\x': /^\^(\\[a-zA-Z]+)\s*/,
+      '^(-1)': /^\^(-?\d+)/,
+      '\'': /^'/,
+      '_{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_{", "", "", "}");
+      },
+      '_($...$)': function _$$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", "$", "$", "");
+      },
+      '_9': /^_([+\-]?[0-9]+|[^\\])/,
+      '_\\x{}{}': function _X(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '_\\x{}': function _X(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '_\\x': /^_(\\[a-zA-Z]+)\s*/,
+      '^_': /^(?:\^(?=_)|\_(?=\^)|[\^_]$)/,
+      '{}': /^\{\}/,
+      '{...}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", "{", "}", "");
+      },
+      '{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "{", "", "", "}");
+      },
+      '$...$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", "$", "$", "");
+      },
+      '${(...)}$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "${", "", "", "}$");
+      },
+      '$(...)$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "$", "", "", "$");
+      },
+      '=<>': /^[=<>]/,
+      '#': /^[#\u2261]/,
+      '+': /^\+/,
+      '-$': /^-(?=[\s_},;\]/]|$|\([a-z]+\))/,
+      // -space -, -; -] -/ -$ -state-of-aggregation
+      '-9': /^-(?=[0-9])/,
+      '- orbital overlap': /^-(?=(?:[spd]|sp)(?:$|[\s,;\)\]\}]))/,
+      '-': /^-/,
+      'pm-operator': /^(?:\\pm|\$\\pm\$|\+-|\+\/-)/,
+      'operator': /^(?:\+|(?:[\-=<>]|<<|>>|\\approx|\$\\approx\$)(?=\s|$|-?[0-9]))/,
+      'arrowUpDown': /^(?:v|\(v\)|\^|\(\^\))(?=$|[\s,;\)\]\}])/,
+      '\\bond{(...)}': function bond(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\bond{", "", "", "}");
+      },
+      '->': /^(?:<->|<-->|->|<-|<=>>|<<=>|<=>|[\u2192\u27F6\u21CC])/,
+      'CMT': /^[CMT](?=\[)/,
+      '[(...)]': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "[", "", "", "]");
+      },
+      '1st-level escape': /^(&|\\\\|\\hline)\s*/,
+      '\\,': /^(?:\\[,\ ;:])/,
+      // \\x - but output no space before
+      '\\x{}{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '\\x{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '\\ca': /^\\ca(?:\s+|(?![a-zA-Z]))/,
+      '\\x': /^(?:\\[a-zA-Z]+\s*|\\[_&{}%])/,
+      'orbital': /^(?:[0-9]{1,2}[spdfgh]|[0-9]{0,2}sp)(?=$|[^a-zA-Z])/,
+      // only those with numbers in front, because the others will be formatted correctly anyway
+      'others': /^[\/~|]/,
+      '\\frac{(...)}': function frac(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\frac{", "", "", "}", "{", "", "", "}");
+      },
+      '\\overset{(...)}': function overset(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\overset{", "", "", "}", "{", "", "", "}");
+      },
+      "\\underset{(...)}": function underset(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\underset{", "", "", "}", "{", "", "", "}");
+      },
+      "\\underbrace{(...)}": function underbrace(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\underbrace{", "", "", "}_", "{", "", "", "}");
+      },
+      '\\color{(...)}0': function color0(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color{", "", "", "}");
+      },
+      '\\color{(...)}{(...)}1': function color1(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color{", "", "", "}", "{", "", "", "}");
+      },
+      '\\color(...){(...)}2': function color2(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color", "\\", "", /^(?=\{)/, "{", "", "", "}");
+      },
+      '\\ce{(...)}': function ce(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\ce{", "", "", "}");
+      },
+      'oxidation$': /^(?:[+-][IVX]+|\\pm\s*0|\$\\pm\$\s*0)$/,
+      'd-oxidation$': /^(?:[+-]?\s?[IVX]+|\\pm\s*0|\$\\pm\$\s*0)$/,
+      // 0 could be oxidation or charge
+      'roman numeral': /^[IVX]+/,
+      '1/2$': /^[+\-]?(?:[0-9]+|\$[a-z]\$|[a-z])\/[0-9]+(?:\$[a-z]\$|[a-z])?$/,
+      'amount': function amount(input) {
+        var match; // e.g. 2, 0.5, 1/2, -2, n/2, +;  $a$ could be added later in parsing
+
+        match = input.match(/^(?:(?:(?:\([+\-]?[0-9]+\/[0-9]+\)|[+\-]?(?:[0-9]+|\$[a-z]\$|[a-z])\/[0-9]+|[+\-]?[0-9]+[.,][0-9]+|[+\-]?\.[0-9]+|[+\-]?[0-9]+)(?:[a-z](?=\s*[A-Z]))?)|[+\-]?[a-z](?=\s*[A-Z])|\+(?!\s))/);
+
+        if (match) {
+          return {
+            match_: match[0],
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        var a = mhchemParser.patterns.findObserveGroups(input, "", "$", "$", "");
+
+        if (a) {
+          // e.g. $2n-1$, $-$
+          match = a.match_.match(/^\$(?:\(?[+\-]?(?:[0-9]*[a-z]?[+\-])?[0-9]*[a-z](?:[+\-][0-9]*[a-z]?)?\)?|\+|-)\$$/);
+
+          if (match) {
+            return {
+              match_: match[0],
+              remainder: input.substr(match[0].length)
+            };
+          }
+        }
+
+        return null;
+      },
+      'amount2': function amount2(input) {
+        return this['amount'](input);
+      },
+      '(KV letters),': /^(?:[A-Z][a-z]{0,2}|i)(?=,)/,
+      'formula$': function formula$(input) {
+        if (input.match(/^\([a-z]+\)$/)) {
+          return null;
+        } // state of aggregation = no formula
+
+
+        var match = input.match(/^(?:[a-z]|(?:[0-9\ \+\-\,\.\(\)]+[a-z])+[0-9\ \+\-\,\.\(\)]*|(?:[a-z][0-9\ \+\-\,\.\(\)]+)+[a-z]?)$/);
+
+        if (match) {
+          return {
+            match_: match[0],
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        return null;
+      },
+      'uprightEntities': /^(?:pH|pOH|pC|pK|iPr|iBu)(?=$|[^a-zA-Z])/,
+      '/': /^\s*(\/)\s*/,
+      '//': /^\s*(\/\/)\s*/,
+      '*': /^\s*[*.]\s*/
+    },
+    findObserveGroups: function findObserveGroups(input, begExcl, begIncl, endIncl, endExcl, beg2Excl, beg2Incl, end2Incl, end2Excl, combine) {
+      /** @type {{(input: string, pattern: string | RegExp): string | string[] | null;}} */
+      var _match = function _match(input, pattern) {
+        if (typeof pattern === "string") {
+          if (input.indexOf(pattern) !== 0) {
+            return null;
+          }
+
+          return pattern;
+        } else {
+          var match = input.match(pattern);
+
+          if (!match) {
+            return null;
+          }
+
+          return match[0];
+        }
+      };
+      /** @type {{(input: string, i: number, endChars: string | RegExp): {endMatchBegin: number, endMatchEnd: number} | null;}} */
+
+
+      var _findObserveGroups = function _findObserveGroups(input, i, endChars) {
+        var braces = 0;
+
+        while (i < input.length) {
+          var a = input.charAt(i);
+
+          var match = _match(input.substr(i), endChars);
+
+          if (match !== null && braces === 0) {
+            return {
+              endMatchBegin: i,
+              endMatchEnd: i + match.length
+            };
+          } else if (a === "{") {
+            braces++;
+          } else if (a === "}") {
+            if (braces === 0) {
+              throw ["ExtraCloseMissingOpen", "Extra close brace or missing open brace"];
+            } else {
+              braces--;
+            }
+          }
+
+          i++;
+        }
+
+        if (braces > 0) {
+          return null;
+        }
+
+        return null;
+      };
+
+      var match = _match(input, begExcl);
+
+      if (match === null) {
+        return null;
+      }
+
+      input = input.substr(match.length);
+      match = _match(input, begIncl);
+
+      if (match === null) {
+        return null;
+      }
+
+      var e = _findObserveGroups(input, match.length, endIncl || endExcl);
+
+      if (e === null) {
+        return null;
+      }
+
+      var match1 = input.substring(0, endIncl ? e.endMatchEnd : e.endMatchBegin);
+
+      if (!(beg2Excl || beg2Incl)) {
+        return {
+          match_: match1,
+          remainder: input.substr(e.endMatchEnd)
+        };
+      } else {
+        var group2 = this.findObserveGroups(input.substr(e.endMatchEnd), beg2Excl, beg2Incl, end2Incl, end2Excl);
+
+        if (group2 === null) {
+          return null;
+        }
+        /** @type {string[]} */
+
+
+        var matchRet = [match1, group2.match_];
+        return {
+          match_: combine ? matchRet.join("") : matchRet,
+          remainder: group2.remainder
+        };
+      }
+    },
+    //
+    // Matching function
+    // e.g. match("a", input) will look for the regexp called "a" and see if it matches
+    // returns null or {match_:"a", remainder:"bc"}
+    //
+    match_: function match_(m, input) {
+      var pattern = mhchemParser.patterns.patterns[m];
+
+      if (pattern === undefined) {
+        throw ["MhchemBugP", "mhchem bug P. Please report. (" + m + ")"]; // Trying to use non-existing pattern
+      } else if (typeof pattern === "function") {
+        return mhchemParser.patterns.patterns[m](input); // cannot use cached var pattern here, because some pattern functions need this===mhchemParser
+      } else {
+        // RegExp
+        var match = input.match(pattern);
+
+        if (match) {
+          var mm;
+
+          if (match[2]) {
+            mm = [match[1], match[2]];
+          } else if (match[1]) {
+            mm = match[1];
+          } else {
+            mm = match[0];
+          }
+
+          return {
+            match_: mm,
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        return null;
+      }
+    }
+  },
+  //
+  // Generic state machine actions
+  //
+  actions: {
+    'a=': function a(buffer, m) {
+      buffer.a = (buffer.a || "") + m;
+    },
+    'b=': function b(buffer, m) {
+      buffer.b = (buffer.b || "") + m;
+    },
+    'p=': function p(buffer, m) {
+      buffer.p = (buffer.p || "") + m;
+    },
+    'o=': function o(buffer, m) {
+      buffer.o = (buffer.o || "") + m;
+    },
+    'q=': function q(buffer, m) {
+      buffer.q = (buffer.q || "") + m;
+    },
+    'd=': function d(buffer, m) {
+      buffer.d = (buffer.d || "") + m;
+    },
+    'rm=': function rm(buffer, m) {
+      buffer.rm = (buffer.rm || "") + m;
+    },
+    'text=': function text(buffer, m) {
+      buffer.text_ = (buffer.text_ || "") + m;
+    },
+    'insert': function insert(buffer, m, a) {
+      return {
+        type_: a
+      };
+    },
+    'insert+p1': function insertP1(buffer, m, a) {
+      return {
+        type_: a,
+        p1: m
+      };
+    },
+    'insert+p1+p2': function insertP1P2(buffer, m, a) {
+      return {
+        type_: a,
+        p1: m[0],
+        p2: m[1]
+      };
+    },
+    'copy': function copy(buffer, m) {
+      return m;
+    },
+    'rm': function rm(buffer, m) {
+      return {
+        type_: 'rm',
+        p1: m || ""
+      };
+    },
+    'text': function text(buffer, m) {
+      return mhchemParser.go(m, 'text');
+    },
+    '{text}': function text(buffer, m) {
+      var ret = ["{"];
+      mhchemParser.concatArray(ret, mhchemParser.go(m, 'text'));
+      ret.push("}");
+      return ret;
+    },
+    'tex-math': function texMath(buffer, m) {
+      return mhchemParser.go(m, 'tex-math');
+    },
+    'tex-math tight': function texMathTight(buffer, m) {
+      return mhchemParser.go(m, 'tex-math tight');
+    },
+    'bond': function bond(buffer, m, k) {
+      return {
+        type_: 'bond',
+        kind_: k || m
+      };
+    },
+    'color0-output': function color0Output(buffer, m) {
+      return {
+        type_: 'color0',
+        color: m[0]
+      };
+    },
+    'ce': function ce(buffer, m) {
+      return mhchemParser.go(m);
+    },
+    '1/2': function _(buffer, m) {
+      /** @type {ParserOutput[]} */
+      var ret = [];
+
+      if (m.match(/^[+\-]/)) {
+        ret.push(m.substr(0, 1));
+        m = m.substr(1);
+      }
+
+      var n = m.match(/^([0-9]+|\$[a-z]\$|[a-z])\/([0-9]+)(\$[a-z]\$|[a-z])?$/);
+      n[1] = n[1].replace(/\$/g, "");
+      ret.push({
+        type_: 'frac',
+        p1: n[1],
+        p2: n[2]
+      });
+
+      if (n[3]) {
+        n[3] = n[3].replace(/\$/g, "");
+        ret.push({
+          type_: 'tex-math',
+          p1: n[3]
+        });
+      }
+
+      return ret;
+    },
+    '9,9': function _(buffer, m) {
+      return mhchemParser.go(m, '9,9');
+    }
+  },
+  //
+  // createTransitions
+  // convert  { 'letter': { 'state': { action_: 'output' } } }  to  { 'state' => [ { pattern: 'letter', task: { action_: [{type_: 'output'}] } } ] }
+  // with expansion of 'a|b' to 'a' and 'b' (at 2 places)
+  //
+  createTransitions: function createTransitions(o) {
+    var pattern, state;
+    /** @type {string[]} */
+
+    var stateArray;
+    var i; //
+    // 1. Collect all states
+    //
+
+    /** @type {Transitions} */
+
+    var transitions = {};
+
+    for (pattern in o) {
+      for (state in o[pattern]) {
+        stateArray = state.split("|");
+        o[pattern][state].stateArray = stateArray;
+
+        for (i = 0; i < stateArray.length; i++) {
+          transitions[stateArray[i]] = [];
+        }
+      }
+    } //
+    // 2. Fill states
+    //
+
+
+    for (pattern in o) {
+      for (state in o[pattern]) {
+        stateArray = o[pattern][state].stateArray || [];
+
+        for (i = 0; i < stateArray.length; i++) {
+          //
+          // 2a. Normalize actions into array:  'text=' ==> [{type_:'text='}]
+          // (Note to myself: Resolving the function here would be problematic. It would need .bind (for *this*) and currying (for *option*).)
+          //
+
+          /** @type {any} */
+          var p = o[pattern][state];
+
+          if (p.action_) {
+            p.action_ = [].concat(p.action_);
+
+            for (var k = 0; k < p.action_.length; k++) {
+              if (typeof p.action_[k] === "string") {
+                p.action_[k] = {
+                  type_: p.action_[k]
+                };
+              }
+            }
+          } else {
+            p.action_ = [];
+          } //
+          // 2.b Multi-insert
+          //
+
+
+          var patternArray = pattern.split("|");
+
+          for (var j = 0; j < patternArray.length; j++) {
+            if (stateArray[i] === '*') {
+              // insert into all
+              for (var t in transitions) {
+                transitions[t].push({
+                  pattern: patternArray[j],
+                  task: p
+                });
+              }
+            } else {
+              transitions[stateArray[i]].push({
+                pattern: patternArray[j],
+                task: p
+              });
+            }
+          }
+        }
+      }
+    }
+
+    return transitions;
+  },
+  stateMachines: {}
+}; //
+// Definition of state machines
+//
+
+mhchemParser.stateMachines = {
+  //
+  // \ce state machines
+  //
+  //#region ce
+  'ce': {
+    // main parser
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      'else': {
+        '0|1|2': {
+          action_: 'beginsWithBond=false',
+          revisit: true,
+          toContinue: true
+        }
+      },
+      'oxidation$': {
+        '0': {
+          action_: 'oxidation-output'
+        }
+      },
+      'CMT': {
+        'r': {
+          action_: 'rdt=',
+          nextState: 'rt'
+        },
+        'rd': {
+          action_: 'rqt=',
+          nextState: 'rdt'
+        }
+      },
+      'arrowUpDown': {
+        '0|1|2|as': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '1'
+        }
+      },
+      'uprightEntities': {
+        '0|1|2': {
+          action_: ['o=', 'output'],
+          nextState: '1'
+        }
+      },
+      'orbital': {
+        '0|1|2|3': {
+          action_: 'o=',
+          nextState: 'o'
+        }
+      },
+      '->': {
+        '0|1|2|3': {
+          action_: 'r=',
+          nextState: 'r'
+        },
+        'a|as': {
+          action_: ['output', 'r='],
+          nextState: 'r'
+        },
+        '*': {
+          action_: ['output', 'r='],
+          nextState: 'r'
+        }
+      },
+      '+': {
+        'o': {
+          action_: 'd= kv',
+          nextState: 'd'
+        },
+        'd|D': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qd|qD': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'dq': {
+          action_: ['output', 'd='],
+          nextState: 'd'
+        },
+        '3': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '0'
+        }
+      },
+      'amount': {
+        '0|2': {
+          action_: 'a=',
+          nextState: 'a'
+        }
+      },
+      'pm-operator': {
+        '0|1|2|a|as': {
+          action_: ['sb=false', 'output', {
+            type_: 'operator',
+            option: '\\pm'
+          }],
+          nextState: '0'
+        }
+      },
+      'operator': {
+        '0|1|2|a|as': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '0'
+        }
+      },
+      '-$': {
+        'o|q': {
+          action_: ['charge or bond', 'output'],
+          nextState: 'qd'
+        },
+        'd': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'D': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qd': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qD|dq': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        }
+      },
+      '-9': {
+        '3|o': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '3'
+        }
+      },
+      '- orbital overlap': {
+        'o': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        },
+        'd': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        }
+      },
+      '-': {
+        '0|1|2': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'beginsWithBond=true', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        '3': {
+          action_: {
+            type_: 'bond',
+            option: "-"
+          }
+        },
+        'a': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        },
+        'as': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        'b': {
+          action_: 'b='
+        },
+        'o': {
+          action_: {
+            type_: '- after o/d',
+            option: false
+          },
+          nextState: '2'
+        },
+        'q': {
+          action_: {
+            type_: '- after o/d',
+            option: false
+          },
+          nextState: '2'
+        },
+        'd|qd|dq': {
+          action_: {
+            type_: '- after o/d',
+            option: true
+          },
+          nextState: '2'
+        },
+        'D|qD|p': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        }
+      },
+      'amount2': {
+        '1|3': {
+          action_: 'a=',
+          nextState: 'a'
+        }
+      },
+      'letters': {
+        '0|1|2|3|a|as|b|p|bp|o': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        'q|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        },
+        'd|D|qd|qD': {
+          action_: 'o after d',
+          nextState: 'o'
+        }
+      },
+      'digits': {
+        'o': {
+          action_: 'q=',
+          nextState: 'q'
+        },
+        'd|D': {
+          action_: 'q=',
+          nextState: 'dq'
+        },
+        'q': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        },
+        'a': {
+          action_: 'o=',
+          nextState: 'o'
+        }
+      },
+      'space A': {
+        'b|p|bp': {}
+      },
+      'space': {
+        'a': {
+          nextState: 'as'
+        },
+        '0': {
+          action_: 'sb=false'
+        },
+        '1|2': {
+          action_: 'sb=true'
+        },
+        'r|rt|rd|rdt|rdq': {
+          action_: 'output',
+          nextState: '0'
+        },
+        '*': {
+          action_: ['output', 'sb=true'],
+          nextState: '1'
+        }
+      },
+      '1st-level escape': {
+        '1|2': {
+          action_: ['output', {
+            type_: 'insert+p1',
+            option: '1st-level escape'
+          }]
+        },
+        '*': {
+          action_: ['output', {
+            type_: 'insert+p1',
+            option: '1st-level escape'
+          }],
+          nextState: '0'
+        }
+      },
+      '[(...)]': {
+        'r|rt': {
+          action_: 'rd=',
+          nextState: 'rd'
+        },
+        'rd|rdt': {
+          action_: 'rq=',
+          nextState: 'rdq'
+        }
+      },
+      '...': {
+        'o|d|D|dq|qd|qD': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "..."
+          }],
+          nextState: '3'
+        },
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, {
+            type_: 'insert',
+            option: 'ellipsis'
+          }],
+          nextState: '1'
+        }
+      },
+      '. |* ': {
+        '*': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'addition compound'
+          }],
+          nextState: '1'
+        }
+      },
+      'state of aggregation $': {
+        '*': {
+          action_: ['output', 'state of aggregation'],
+          nextState: '1'
+        }
+      },
+      '{[(': {
+        'a|as|o': {
+          action_: ['o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        },
+        '0|1|2|3': {
+          action_: ['o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        },
+        '*': {
+          action_: ['output', 'o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        }
+      },
+      ')]}': {
+        '0|1|2|3|b|p|bp|o': {
+          action_: ['o=', 'parenthesisLevel--'],
+          nextState: 'o'
+        },
+        'a|as|d|D|q|qd|qD|dq': {
+          action_: ['output', 'o=', 'parenthesisLevel--'],
+          nextState: 'o'
+        }
+      },
+      ', ': {
+        '*': {
+          action_: ['output', 'comma'],
+          nextState: '0'
+        }
+      },
+      '^_': {
+        // ^ and _ without a sensible argument
+        '*': {}
+      },
+      '^{(...)}|^($...$)': {
+        '0|1|2|as': {
+          action_: 'b=',
+          nextState: 'b'
+        },
+        'p': {
+          action_: 'b=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'd= kv',
+          nextState: 'D'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qD'
+        },
+        'd|D|qd|qD|dq': {
+          action_: ['output', 'd='],
+          nextState: 'D'
+        }
+      },
+      '^a|^\\x{}{}|^\\x{}|^\\x|\'': {
+        '0|1|2|as': {
+          action_: 'b=',
+          nextState: 'b'
+        },
+        'p': {
+          action_: 'b=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'd= kv',
+          nextState: 'd'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'd|qd|D|qD': {
+          action_: 'd='
+        },
+        'dq': {
+          action_: ['output', 'd='],
+          nextState: 'd'
+        }
+      },
+      '_{(state of aggregation)}$': {
+        'd|D|q|qd|qD|dq': {
+          action_: ['output', 'q='],
+          nextState: 'q'
+        }
+      },
+      '_{(...)}|_($...$)|_9|_\\x{}{}|_\\x{}|_\\x': {
+        '0|1|2|as': {
+          action_: 'p=',
+          nextState: 'p'
+        },
+        'b': {
+          action_: 'p=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'q=',
+          nextState: 'q'
+        },
+        'd|D': {
+          action_: 'q=',
+          nextState: 'dq'
+        },
+        'q|qd|qD|dq': {
+          action_: ['output', 'q='],
+          nextState: 'q'
+        }
+      },
+      '=<>': {
+        '0|1|2|3|a|as|o|q|d|D|qd|qD|dq': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'bond'],
+          nextState: '3'
+        }
+      },
+      '#': {
+        '0|1|2|3|a|as|o': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, {
+            type_: 'bond',
+            option: "#"
+          }],
+          nextState: '3'
+        }
+      },
+      '{}': {
+        '*': {
+          action_: {
+            type_: 'output',
+            option: 1
+          },
+          nextState: '1'
+        }
+      },
+      '{...}': {
+        '0|1|2|3|a|as|b|p|bp': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        'o|d|D|q|qd|qD|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        }
+      },
+      '$...$': {
+        'a': {
+          action_: 'a='
+        },
+        // 2$n$
+        '0|1|2|3|as|b|p|bp|o': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        // not 'amount'
+        'as|o': {
+          action_: 'o='
+        },
+        'q|d|D|qd|qD|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        }
+      },
+      '\\bond{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'bond'],
+          nextState: "3"
+        }
+      },
+      '\\frac{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'frac-output'],
+          nextState: '3'
+        }
+      },
+      '\\overset{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'overset-output'],
+          nextState: '3'
+        }
+      },
+      "\\underset{(...)}": {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'underset-output'],
+          nextState: '3'
+        }
+      },
+      "\\underbrace{(...)}": {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'underbrace-output'],
+          nextState: '3'
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'color-output'],
+          nextState: '3'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'color0-output']
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'ce'],
+          nextState: '3'
+        }
+      },
+      '\\,': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'copy'],
+          nextState: '1'
+        }
+      },
+      '\\x{}{}|\\x{}|\\x': {
+        '0|1|2|3|a|as|b|p|bp|o|c0': {
+          action_: ['o=', 'output'],
+          nextState: '3'
+        },
+        '*': {
+          action_: ['output', 'o=', 'output'],
+          nextState: '3'
+        }
+      },
+      'others': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'copy'],
+          nextState: '3'
+        }
+      },
+      'else2': {
+        'a': {
+          action_: 'a to o',
+          nextState: 'o',
+          revisit: true
+        },
+        'as': {
+          action_: ['output', 'sb=true'],
+          nextState: '1',
+          revisit: true
+        },
+        'r|rt|rd|rdt|rdq': {
+          action_: ['output'],
+          nextState: '0',
+          revisit: true
+        },
+        '*': {
+          action_: ['output', 'copy'],
+          nextState: '3'
+        }
+      }
+    }),
+    actions: {
+      'o after d': function oAfterD(buffer, m) {
+        var ret;
+
+        if ((buffer.d || "").match(/^[0-9]+$/)) {
+          var tmp = buffer.d;
+          buffer.d = undefined;
+          ret = this['output'](buffer);
+          buffer.b = tmp;
+        } else {
+          ret = this['output'](buffer);
+        }
+
+        mhchemParser.actions['o='](buffer, m);
+        return ret;
+      },
+      'd= kv': function dKv(buffer, m) {
+        buffer.d = m;
+        buffer.dType = 'kv';
+      },
+      'charge or bond': function chargeOrBond(buffer, m) {
+        if (buffer['beginsWithBond']) {
+          /** @type {ParserOutput[]} */
+          var ret = [];
+          mhchemParser.concatArray(ret, this['output'](buffer));
+          mhchemParser.concatArray(ret, mhchemParser.actions['bond'](buffer, m, "-"));
+          return ret;
+        } else {
+          buffer.d = m;
+        }
+      },
+      '- after o/d': function afterOD(buffer, m, isAfterD) {
+        var c1 = mhchemParser.patterns.match_('orbital', buffer.o || "");
+        var c2 = mhchemParser.patterns.match_('one lowercase greek letter $', buffer.o || "");
+        var c3 = mhchemParser.patterns.match_('one lowercase latin letter $', buffer.o || "");
+        var c4 = mhchemParser.patterns.match_('$one lowercase latin letter$ $', buffer.o || "");
+        var hyphenFollows = m === "-" && (c1 && c1.remainder === "" || c2 || c3 || c4);
+
+        if (hyphenFollows && !buffer.a && !buffer.b && !buffer.p && !buffer.d && !buffer.q && !c1 && c3) {
+          buffer.o = '$' + buffer.o + '$';
+        }
+        /** @type {ParserOutput[]} */
+
+
+        var ret = [];
+
+        if (hyphenFollows) {
+          mhchemParser.concatArray(ret, this['output'](buffer));
+          ret.push({
+            type_: 'hyphen'
+          });
+        } else {
+          c1 = mhchemParser.patterns.match_('digits', buffer.d || "");
+
+          if (isAfterD && c1 && c1.remainder === '') {
+            mhchemParser.concatArray(ret, mhchemParser.actions['d='](buffer, m));
+            mhchemParser.concatArray(ret, this['output'](buffer));
+          } else {
+            mhchemParser.concatArray(ret, this['output'](buffer));
+            mhchemParser.concatArray(ret, mhchemParser.actions['bond'](buffer, m, "-"));
+          }
+        }
+
+        return ret;
+      },
+      'a to o': function aToO(buffer) {
+        buffer.o = buffer.a;
+        buffer.a = undefined;
+      },
+      'sb=true': function sbTrue(buffer) {
+        buffer.sb = true;
+      },
+      'sb=false': function sbFalse(buffer) {
+        buffer.sb = false;
+      },
+      'beginsWithBond=true': function beginsWithBondTrue(buffer) {
+        buffer['beginsWithBond'] = true;
+      },
+      'beginsWithBond=false': function beginsWithBondFalse(buffer) {
+        buffer['beginsWithBond'] = false;
+      },
+      'parenthesisLevel++': function parenthesisLevel(buffer) {
+        buffer['parenthesisLevel']++;
+      },
+      'parenthesisLevel--': function parenthesisLevel(buffer) {
+        buffer['parenthesisLevel']--;
+      },
+      'state of aggregation': function stateOfAggregation(buffer, m) {
+        return {
+          type_: 'state of aggregation',
+          p1: mhchemParser.go(m, 'o')
+        };
+      },
+      'comma': function comma(buffer, m) {
+        var a = m.replace(/\s*$/, '');
+        var withSpace = a !== m;
+
+        if (withSpace && buffer['parenthesisLevel'] === 0) {
+          return {
+            type_: 'comma enumeration L',
+            p1: a
+          };
+        } else {
+          return {
+            type_: 'comma enumeration M',
+            p1: a
+          };
+        }
+      },
+      'output': function output(buffer, m, entityFollows) {
+        // entityFollows:
+        //   undefined = if we have nothing else to output, also ignore the just read space (buffer.sb)
+        //   1 = an entity follows, never omit the space if there was one just read before (can only apply to state 1)
+        //   2 = 1 + the entity can have an amount, so output a\, instead of converting it to o (can only apply to states a|as)
+
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret;
+
+        if (!buffer.r) {
+          ret = [];
+
+          if (!buffer.a && !buffer.b && !buffer.p && !buffer.o && !buffer.q && !buffer.d && !entityFollows) {//ret = [];
+          } else {
+            if (buffer.sb) {
+              ret.push({
+                type_: 'entitySkip'
+              });
+            }
+
+            if (!buffer.o && !buffer.q && !buffer.d && !buffer.b && !buffer.p && entityFollows !== 2) {
+              buffer.o = buffer.a;
+              buffer.a = undefined;
+            } else if (!buffer.o && !buffer.q && !buffer.d && (buffer.b || buffer.p)) {
+              buffer.o = buffer.a;
+              buffer.d = buffer.b;
+              buffer.q = buffer.p;
+              buffer.a = buffer.b = buffer.p = undefined;
+            } else {
+              if (buffer.o && buffer.dType === 'kv' && mhchemParser.patterns.match_('d-oxidation$', buffer.d || "")) {
+                buffer.dType = 'oxidation';
+              } else if (buffer.o && buffer.dType === 'kv' && !buffer.q) {
+                buffer.dType = undefined;
+              }
+            }
+
+            ret.push({
+              type_: 'chemfive',
+              a: mhchemParser.go(buffer.a, 'a'),
+              b: mhchemParser.go(buffer.b, 'bd'),
+              p: mhchemParser.go(buffer.p, 'pq'),
+              o: mhchemParser.go(buffer.o, 'o'),
+              q: mhchemParser.go(buffer.q, 'pq'),
+              d: mhchemParser.go(buffer.d, buffer.dType === 'oxidation' ? 'oxidation' : 'bd'),
+              dType: buffer.dType
+            });
+          }
+        } else {
+          // r
+
+          /** @type {ParserOutput[]} */
+          var rd;
+
+          if (buffer.rdt === 'M') {
+            rd = mhchemParser.go(buffer.rd, 'tex-math');
+          } else if (buffer.rdt === 'T') {
+            rd = [{
+              type_: 'text',
+              p1: buffer.rd || ""
+            }];
+          } else {
+            rd = mhchemParser.go(buffer.rd);
+          }
+          /** @type {ParserOutput[]} */
+
+
+          var rq;
+
+          if (buffer.rqt === 'M') {
+            rq = mhchemParser.go(buffer.rq, 'tex-math');
+          } else if (buffer.rqt === 'T') {
+            rq = [{
+              type_: 'text',
+              p1: buffer.rq || ""
+            }];
+          } else {
+            rq = mhchemParser.go(buffer.rq);
+          }
+
+          ret = {
+            type_: 'arrow',
+            r: buffer.r,
+            rd: rd,
+            rq: rq
+          };
+        }
+
+        for (var p in buffer) {
+          if (p !== 'parenthesisLevel' && p !== 'beginsWithBond') {
+            delete buffer[p];
+          }
+        }
+
+        return ret;
+      },
+      'oxidation-output': function oxidationOutput(buffer, m) {
+        var ret = ["{"];
+        mhchemParser.concatArray(ret, mhchemParser.go(m, 'oxidation'));
+        ret.push("}");
+        return ret;
+      },
+      'frac-output': function fracOutput(buffer, m) {
+        return {
+          type_: 'frac-ce',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'overset-output': function oversetOutput(buffer, m) {
+        return {
+          type_: 'overset',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'underset-output': function undersetOutput(buffer, m) {
+        return {
+          type_: 'underset',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'underbrace-output': function underbraceOutput(buffer, m) {
+        return {
+          type_: 'underbrace',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1])
+        };
+      },
+      'r=': function r(buffer, m) {
+        buffer.r = m;
+      },
+      'rdt=': function rdt(buffer, m) {
+        buffer.rdt = m;
+      },
+      'rd=': function rd(buffer, m) {
+        buffer.rd = m;
+      },
+      'rqt=': function rqt(buffer, m) {
+        buffer.rqt = m;
+      },
+      'rq=': function rq(buffer, m) {
+        buffer.rq = m;
+      },
+      'operator': function operator(buffer, m, p1) {
+        return {
+          type_: 'operator',
+          kind_: p1 || m
+        };
+      }
+    }
+  },
+  'a': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '1',
+          revisit: true
+        }
+      },
+      '$(...)$': {
+        '*': {
+          action_: 'tex-math tight',
+          nextState: '1'
+        }
+      },
+      ',': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'commaDecimal'
+          }
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {}
+  },
+  'o': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '1',
+          revisit: true
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '\\ca': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'circa'
+          }
+        }
+      },
+      '\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: '{text}'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {}
+  },
+  'text': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '{...}': {
+        '*': {
+          action_: 'text='
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '\\greek': {
+        '*': {
+          action_: ['output', 'rm']
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: ['output', 'copy']
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'text='
+        }
+      }
+    }),
+    actions: {
+      'output': function output(buffer) {
+        if (buffer.text_) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'text',
+            p1: buffer.text_
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  'pq': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'state of aggregation $': {
+        '*': {
+          action_: 'state of aggregation'
+        }
+      },
+      'i$': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '(KV letters),': {
+        '0': {
+          action_: 'rm',
+          nextState: '0'
+        }
+      },
+      'formula$': {
+        '0': {
+          nextState: 'f',
+          revisit: true
+        }
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: 'text'
+        }
+      },
+      'a-z': {
+        'f': {
+          action_: 'tex-math'
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '-9.,9': {
+        '*': {
+          action_: '9,9'
+        }
+      },
+      ',': {
+        '*': {
+          action_: {
+            type_: 'insert+p1',
+            option: 'comma enumeration S'
+          }
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: 'color-output'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: 'color0-output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: 'ce'
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'state of aggregation': function stateOfAggregation(buffer, m) {
+        return {
+          type_: 'state of aggregation subscript',
+          p1: mhchemParser.go(m, 'o')
+        };
+      },
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1], 'pq')
+        };
+      }
+    }
+  },
+  'bd': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'x$': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      'formula$': {
+        '0': {
+          nextState: 'f',
+          revisit: true
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '-9.,9 no missing 0': {
+        '*': {
+          action_: '9,9'
+        }
+      },
+      '.': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'electron dot'
+          }
+        }
+      },
+      'a-z': {
+        'f': {
+          action_: 'tex-math'
+        }
+      },
+      'x': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'KV x'
+          }
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '\'': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'prime'
+          }
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: 'text'
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: 'color-output'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: 'color0-output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: 'ce'
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1], 'bd')
+        };
+      }
+    }
+  },
+  'oxidation': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'roman numeral': {
+        '*': {
+          action_: 'roman-numeral'
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'roman-numeral': function romanNumeral(buffer, m) {
+        return {
+          type_: 'roman numeral',
+          p1: m || ""
+        };
+      }
+    }
+  },
+  'tex-math': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: ['output', 'ce']
+        }
+      },
+      '{...}|\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'o='
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'o='
+        }
+      }
+    }),
+    actions: {
+      'output': function output(buffer) {
+        if (buffer.o) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'tex-math',
+            p1: buffer.o
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  'tex-math tight': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: ['output', 'ce']
+        }
+      },
+      '{...}|\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'o='
+        }
+      },
+      '-|+': {
+        '*': {
+          action_: 'tight operator'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'o='
+        }
+      }
+    }),
+    actions: {
+      'tight operator': function tightOperator(buffer, m) {
+        buffer.o = (buffer.o || "") + "{" + m + "}";
+      },
+      'output': function output(buffer) {
+        if (buffer.o) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'tex-math',
+            p1: buffer.o
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  '9,9': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      ',': {
+        '*': {
+          action_: 'comma'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'comma': function comma() {
+        return {
+          type_: 'commaDecimal'
+        };
+      }
+    }
+  },
+  //#endregion
+  //
+  // \pu state machines
+  //
+  //#region pu
+  'pu': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      'space$': {
+        '*': {
+          action_: ['output', 'space']
+        }
+      },
+      '{[(|)]}': {
+        '0|a': {
+          action_: 'copy'
+        }
+      },
+      '(-)(9)^(-9)': {
+        '0': {
+          action_: 'number^',
+          nextState: 'a'
+        }
+      },
+      '(-)(9.,9)(e)(99)': {
+        '0': {
+          action_: 'enumber',
+          nextState: 'a'
+        }
+      },
+      'space': {
+        '0|a': {}
+      },
+      'pm-operator': {
+        '0|a': {
+          action_: {
+            type_: 'operator',
+            option: '\\pm'
+          },
+          nextState: '0'
+        }
+      },
+      'operator': {
+        '0|a': {
+          action_: 'copy',
+          nextState: '0'
+        }
+      },
+      '//': {
+        'd': {
+          action_: 'o=',
+          nextState: '/'
+        }
+      },
+      '/': {
+        'd': {
+          action_: 'o=',
+          nextState: '/'
+        }
+      },
+      '{...}|else': {
+        '0|d': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'a': {
+          action_: ['space', 'd='],
+          nextState: 'd'
+        },
+        '/|q': {
+          action_: 'q=',
+          nextState: 'q'
+        }
+      }
+    }),
+    actions: {
+      'enumber': function enumber(buffer, m) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+
+        if (m[0] === "+-" || m[0] === "+/-") {
+          ret.push("\\pm ");
+        } else if (m[0]) {
+          ret.push(m[0]);
+        }
+
+        if (m[1]) {
+          mhchemParser.concatArray(ret, mhchemParser.go(m[1], 'pu-9,9'));
+
+          if (m[2]) {
+            if (m[2].match(/[,.]/)) {
+              mhchemParser.concatArray(ret, mhchemParser.go(m[2], 'pu-9,9'));
+            } else {
+              ret.push(m[2]);
+            }
+          }
+
+          m[3] = m[4] || m[3];
+
+          if (m[3]) {
+            m[3] = m[3].trim();
+
+            if (m[3] === "e" || m[3].substr(0, 1) === "*") {
+              ret.push({
+                type_: 'cdot'
+              });
+            } else {
+              ret.push({
+                type_: 'times'
+              });
+            }
+          }
+        }
+
+        if (m[3]) {
+          ret.push("10^{" + m[5] + "}");
+        }
+
+        return ret;
+      },
+      'number^': function number(buffer, m) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+
+        if (m[0] === "+-" || m[0] === "+/-") {
+          ret.push("\\pm ");
+        } else if (m[0]) {
+          ret.push(m[0]);
+        }
+
+        mhchemParser.concatArray(ret, mhchemParser.go(m[1], 'pu-9,9'));
+        ret.push("^{" + m[2] + "}");
+        return ret;
+      },
+      'operator': function operator(buffer, m, p1) {
+        return {
+          type_: 'operator',
+          kind_: p1 || m
+        };
+      },
+      'space': function space() {
+        return {
+          type_: 'pu-space-1'
+        };
+      },
+      'output': function output(buffer) {
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret;
+        var md = mhchemParser.patterns.match_('{(...)}', buffer.d || "");
+
+        if (md && md.remainder === '') {
+          buffer.d = md.match_;
+        }
+
+        var mq = mhchemParser.patterns.match_('{(...)}', buffer.q || "");
+
+        if (mq && mq.remainder === '') {
+          buffer.q = mq.match_;
+        }
+
+        if (buffer.d) {
+          buffer.d = buffer.d.replace(/\u00B0C|\^oC|\^{o}C/g, "{}^{\\circ}C");
+          buffer.d = buffer.d.replace(/\u00B0F|\^oF|\^{o}F/g, "{}^{\\circ}F");
+        }
+
+        if (buffer.q) {
+          // fraction
+          buffer.q = buffer.q.replace(/\u00B0C|\^oC|\^{o}C/g, "{}^{\\circ}C");
+          buffer.q = buffer.q.replace(/\u00B0F|\^oF|\^{o}F/g, "{}^{\\circ}F");
+          var b5 = {
+            d: mhchemParser.go(buffer.d, 'pu'),
+            q: mhchemParser.go(buffer.q, 'pu')
+          };
+
+          if (buffer.o === '//') {
+            ret = {
+              type_: 'pu-frac',
+              p1: b5.d,
+              p2: b5.q
+            };
+          } else {
+            ret = b5.d;
+
+            if (b5.d.length > 1 || b5.q.length > 1) {
+              ret.push({
+                type_: ' / '
+              });
+            } else {
+              ret.push({
+                type_: '/'
+              });
+            }
+
+            mhchemParser.concatArray(ret, b5.q);
+          }
+        } else {
+          // no fraction
+          ret = mhchemParser.go(buffer.d, 'pu-2');
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    }
+  },
+  'pu-2': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '*': {
+        '*': {
+          action_: ['output', 'cdot'],
+          nextState: '0'
+        }
+      },
+      '\\x': {
+        '*': {
+          action_: 'rm='
+        }
+      },
+      'space': {
+        '*': {
+          action_: ['output', 'space'],
+          nextState: '0'
+        }
+      },
+      '^{(...)}|^(-1)': {
+        '1': {
+          action_: '^(-1)'
+        }
+      },
+      '-9.,9': {
+        '0': {
+          action_: 'rm=',
+          nextState: '0'
+        },
+        '1': {
+          action_: '^(-1)',
+          nextState: '0'
+        }
+      },
+      '{...}|else': {
+        '*': {
+          action_: 'rm=',
+          nextState: '1'
+        }
+      }
+    }),
+    actions: {
+      'cdot': function cdot() {
+        return {
+          type_: 'tight cdot'
+        };
+      },
+      '^(-1)': function _(buffer, m) {
+        buffer.rm += "^{" + m + "}";
+      },
+      'space': function space() {
+        return {
+          type_: 'pu-space-2'
+        };
+      },
+      'output': function output(buffer) {
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret = [];
+
+        if (buffer.rm) {
+          var mrm = mhchemParser.patterns.match_('{(...)}', buffer.rm || "");
+
+          if (mrm && mrm.remainder === '') {
+            ret = mhchemParser.go(mrm.match_, 'pu');
+          } else {
+            ret = {
+              type_: 'rm',
+              p1: buffer.rm
+            };
+          }
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    }
+  },
+  'pu-9,9': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '0': {
+          action_: 'output-0'
+        },
+        'o': {
+          action_: 'output-o'
+        }
+      },
+      ',': {
+        '0': {
+          action_: ['output-0', 'comma'],
+          nextState: 'o'
+        }
+      },
+      '.': {
+        '0': {
+          action_: ['output-0', 'copy'],
+          nextState: 'o'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'text='
+        }
+      }
+    }),
+    actions: {
+      'comma': function comma() {
+        return {
+          type_: 'commaDecimal'
+        };
+      },
+      'output-0': function output0(buffer) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+        buffer.text_ = buffer.text_ || "";
+
+        if (buffer.text_.length > 4) {
+          var a = buffer.text_.length % 3;
+
+          if (a === 0) {
+            a = 3;
+          }
+
+          for (var i = buffer.text_.length - 3; i > 0; i -= 3) {
+            ret.push(buffer.text_.substr(i, 3));
+            ret.push({
+              type_: '1000 separator'
+            });
+          }
+
+          ret.push(buffer.text_.substr(0, a));
+          ret.reverse();
+        } else {
+          ret.push(buffer.text_);
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      },
+      'output-o': function outputO(buffer) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+        buffer.text_ = buffer.text_ || "";
+
+        if (buffer.text_.length > 4) {
+          var a = buffer.text_.length - 3;
+
+          for (var i = 0; i < a; i += 3) {
+            ret.push(buffer.text_.substr(i, 3));
+            ret.push({
+              type_: '1000 separator'
+            });
+          }
+
+          ret.push(buffer.text_.substr(i));
+        } else {
+          ret.push(buffer.text_);
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    } //#endregion
+
+  }
+}; //
+// texify: Take MhchemParser output and convert it to TeX
+//
+
+/** @type {Texify} */
+
+var texify = {
+  go: function go(input, isInner) {
+    // (recursive, max 4 levels)
+    if (!input) {
+      return "";
+    }
+
+    var res = "";
+    var cee = false;
+
+    for (var i = 0; i < input.length; i++) {
+      var inputi = input[i];
+
+      if (typeof inputi === "string") {
+        res += inputi;
+      } else {
+        res += texify._go2(inputi);
+
+        if (inputi.type_ === '1st-level escape') {
+          cee = true;
+        }
+      }
+    }
+
+    if (!isInner && !cee && res) {
+      res = "{" + res + "}";
+    }
+
+    return res;
+  },
+  _goInner: function _goInner(input) {
+    if (!input) {
+      return input;
+    }
+
+    return texify.go(input, true);
+  },
+  _go2: function _go2(buf) {
+    /** @type {undefined | string} */
+    var res;
+
+    switch (buf.type_) {
+      case 'chemfive':
+        res = "";
+        var b5 = {
+          a: texify._goInner(buf.a),
+          b: texify._goInner(buf.b),
+          p: texify._goInner(buf.p),
+          o: texify._goInner(buf.o),
+          q: texify._goInner(buf.q),
+          d: texify._goInner(buf.d)
+        }; //
+        // a
+        //
+
+        if (b5.a) {
+          if (b5.a.match(/^[+\-]/)) {
+            b5.a = "{" + b5.a + "}";
+          }
+
+          res += b5.a + "\\,";
+        } //
+        // b and p
+        //
+
+
+        if (b5.b || b5.p) {
+          res += "{\\vphantom{X}}";
+          res += "^{\\hphantom{" + (b5.b || "") + "}}_{\\hphantom{" + (b5.p || "") + "}}";
+          res += "{\\vphantom{X}}";
+          res += "^{\\smash[t]{\\vphantom{2}}\\mathllap{" + (b5.b || "") + "}}";
+          res += "_{\\vphantom{2}\\mathllap{\\smash[t]{" + (b5.p || "") + "}}}";
+        } //
+        // o
+        //
+
+
+        if (b5.o) {
+          if (b5.o.match(/^[+\-]/)) {
+            b5.o = "{" + b5.o + "}";
+          }
+
+          res += b5.o;
+        } //
+        // q and d
+        //
+
+
+        if (buf.dType === 'kv') {
+          if (b5.d || b5.q) {
+            res += "{\\vphantom{X}}";
+          }
+
+          if (b5.d) {
+            res += "^{" + b5.d + "}";
+          }
+
+          if (b5.q) {
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+        } else if (buf.dType === 'oxidation') {
+          if (b5.d) {
+            res += "{\\vphantom{X}}";
+            res += "^{" + b5.d + "}";
+          }
+
+          if (b5.q) {
+            res += "{\\vphantom{X}}";
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+        } else {
+          if (b5.q) {
+            res += "{\\vphantom{X}}";
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+
+          if (b5.d) {
+            res += "{\\vphantom{X}}";
+            res += "^{" + b5.d + "}";
+          }
+        }
+
+        break;
+
+      case 'rm':
+        res = "\\mathrm{" + buf.p1 + "}";
+        break;
+
+      case 'text':
+        if (buf.p1.match(/[\^_]/)) {
+          buf.p1 = buf.p1.replace(" ", "~").replace("-", "\\text{-}");
+          res = "\\mathrm{" + buf.p1 + "}";
+        } else {
+          res = "\\text{" + buf.p1 + "}";
+        }
+
+        break;
+
+      case 'roman numeral':
+        res = "\\mathrm{" + buf.p1 + "}";
+        break;
+
+      case 'state of aggregation':
+        res = "\\mskip2mu " + texify._goInner(buf.p1);
+        break;
+
+      case 'state of aggregation subscript':
+        res = "\\mskip1mu " + texify._goInner(buf.p1);
+        break;
+
+      case 'bond':
+        res = texify._getBond(buf.kind_);
+
+        if (!res) {
+          throw ["MhchemErrorBond", "mhchem Error. Unknown bond type (" + buf.kind_ + ")"];
+        }
+
+        break;
+
+      case 'frac':
+        var c = "\\frac{" + buf.p1 + "}{" + buf.p2 + "}";
+        res = "\\mathchoice{\\textstyle" + c + "}{" + c + "}{" + c + "}{" + c + "}";
+        break;
+
+      case 'pu-frac':
+        var d = "\\frac{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        res = "\\mathchoice{\\textstyle" + d + "}{" + d + "}{" + d + "}{" + d + "}";
+        break;
+
+      case 'tex-math':
+        res = buf.p1 + " ";
+        break;
+
+      case 'frac-ce':
+        res = "\\frac{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'overset':
+        res = "\\overset{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'underset':
+        res = "\\underset{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'underbrace':
+        res = "\\underbrace{" + texify._goInner(buf.p1) + "}_{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'color':
+        res = "{\\color{" + buf.color1 + "}{" + texify._goInner(buf.color2) + "}}";
+        break;
+
+      case 'color0':
+        res = "\\color{" + buf.color + "}";
+        break;
+
+      case 'arrow':
+        var b6 = {
+          rd: texify._goInner(buf.rd),
+          rq: texify._goInner(buf.rq)
+        };
+
+        var arrow = "\\x" + texify._getArrow(buf.r);
+
+        if (b6.rq) {
+          arrow += "[{" + b6.rq + "}]";
+        }
+
+        if (b6.rd) {
+          arrow += "{" + b6.rd + "}";
+        } else {
+          arrow += "{}";
+        }
+
+        res = arrow;
+        break;
+
+      case 'operator':
+        res = texify._getOperator(buf.kind_);
+        break;
+
+      case '1st-level escape':
+        res = buf.p1 + " "; // &, \\\\, \\hlin
+
+        break;
+
+      case 'space':
+        res = " ";
+        break;
+
+      case 'entitySkip':
+        res = "~";
+        break;
+
+      case 'pu-space-1':
+        res = "~";
+        break;
+
+      case 'pu-space-2':
+        res = "\\mkern3mu ";
+        break;
+
+      case '1000 separator':
+        res = "\\mkern2mu ";
+        break;
+
+      case 'commaDecimal':
+        res = "{,}";
+        break;
+
+      case 'comma enumeration L':
+        res = "{" + buf.p1 + "}\\mkern6mu ";
+        break;
+
+      case 'comma enumeration M':
+        res = "{" + buf.p1 + "}\\mkern3mu ";
+        break;
+
+      case 'comma enumeration S':
+        res = "{" + buf.p1 + "}\\mkern1mu ";
+        break;
+
+      case 'hyphen':
+        res = "\\text{-}";
+        break;
+
+      case 'addition compound':
+        res = "\\,{\\cdot}\\,";
+        break;
+
+      case 'electron dot':
+        res = "\\mkern1mu \\bullet\\mkern1mu ";
+        break;
+
+      case 'KV x':
+        res = "{\\times}";
+        break;
+
+      case 'prime':
+        res = "\\prime ";
+        break;
+
+      case 'cdot':
+        res = "\\cdot ";
+        break;
+
+      case 'tight cdot':
+        res = "\\mkern1mu{\\cdot}\\mkern1mu ";
+        break;
+
+      case 'times':
+        res = "\\times ";
+        break;
+
+      case 'circa':
+        res = "{\\sim}";
+        break;
+
+      case '^':
+        res = "uparrow";
+        break;
+
+      case 'v':
+        res = "downarrow";
+        break;
+
+      case 'ellipsis':
+        res = "\\ldots ";
+        break;
+
+      case '/':
+        res = "/";
+        break;
+
+      case ' / ':
+        res = "\\,/\\,";
+        break;
+
+      default:
+        assertNever(buf);
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+      // Missing texify rule or unknown MhchemParser output
+    }
+
+    assertString(res);
+    return res;
+  },
+  _getArrow: function _getArrow(a) {
+    switch (a) {
+      case "->":
+        return "rightarrow";
+
+      case "\u2192":
+        return "rightarrow";
+
+      case "\u27F6":
+        return "rightarrow";
+
+      case "<-":
+        return "leftarrow";
+
+      case "<->":
+        return "leftrightarrow";
+
+      case "<-->":
+        return "rightleftarrows";
+
+      case "<=>":
+        return "rightleftharpoons";
+
+      case "\u21CC":
+        return "rightleftharpoons";
+
+      case "<=>>":
+        return "rightequilibrium";
+
+      case "<<=>":
+        return "leftequilibrium";
+
+      default:
+        assertNever(a);
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  },
+  _getBond: function _getBond(a) {
+    switch (a) {
+      case "-":
+        return "{-}";
+
+      case "1":
+        return "{-}";
+
+      case "=":
+        return "{=}";
+
+      case "2":
+        return "{=}";
+
+      case "#":
+        return "{\\equiv}";
+
+      case "3":
+        return "{\\equiv}";
+
+      case "~":
+        return "{\\tripledash}";
+
+      case "~-":
+        return "{\\mathrlap{\\raisebox{-.1em}{$-$}}\\raisebox{.1em}{$\\tripledash$}}";
+
+      case "~=":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";
+
+      case "~--":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";
+
+      case "-~-":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$-$}}\\tripledash}";
+
+      case "...":
+        return "{{\\cdot}{\\cdot}{\\cdot}}";
+
+      case "....":
+        return "{{\\cdot}{\\cdot}{\\cdot}{\\cdot}}";
+
+      case "->":
+        return "{\\rightarrow}";
+
+      case "<-":
+        return "{\\leftarrow}";
+
+      case "<":
+        return "{<}";
+
+      case ">":
+        return "{>}";
+
+      default:
+        assertNever(a);
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  },
+  _getOperator: function _getOperator(a) {
+    switch (a) {
+      case "+":
+        return " {}+{} ";
+
+      case "-":
+        return " {}-{} ";
+
+      case "=":
+        return " {}={} ";
+
+      case "<":
+        return " {}<{} ";
+
+      case ">":
+        return " {}>{} ";
+
+      case "<<":
+        return " {}\\ll{} ";
+
+      case ">>":
+        return " {}\\gg{} ";
+
+      case "\\pm":
+        return " {}\\pm{} ";
+
+      case "\\approx":
+        return " {}\\approx{} ";
+
+      case "$\\approx$":
+        return " {}\\approx{} ";
+
+      case "v":
+        return " \\downarrow{} ";
+
+      case "(v)":
+        return " \\downarrow{} ";
+
+      case "^":
+        return " \\uparrow{} ";
+
+      case "(^)":
+        return " \\uparrow{} ";
+
+      default:
+        assertNever(a);
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  }
+}; //
+// Helpers for code anaylsis
+// Will show type error at calling position
+//
+
+/** @param {number} a */
+
+function assertNever(a) {}
+/** @param {string} a */
+
+
+function assertString(a) {}
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/katex.bk/contrib/mhchem.min.js b/katex.bk/contrib/mhchem.min.js
new file mode 100644
index 00000000..432a76af
--- /dev/null
+++ b/katex.bk/contrib/mhchem.min.js
@@ -0,0 +1 @@
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n={a:c._goInner(t.a),b:c._goInner(t.b),p:c._goInner(t.p),o:c._goInner(t.o),q:c._goInner(t.q),d:c._goInner(t.d)};n.a&&(n.a.match(/^[+\-]/)&&(n.a="{"+n.a+"}"),e+=n.a+"\\,"),(n.b||n.p)&&(e+="{\\vphantom{X}}",e+="^{\\hphantom{"+(n.b||"")+"}}_{\\hphantom{"+(n.p||"")+"}}",e+="{\\vphantom{X}}",e+="^{\\smash[t]{\\vphantom{2}}\\mathllap{"+(n.b||"")+"}}",e+="_{\\vphantom{2}\\mathllap{\\smash[t]{"+(n.p||"")+"}}}"),n.o&&(n.o.match(/^[+\-]/)&&(n.o="{"+n.o+"}"),e+=n.o),"kv"===t.dType?((n.d||n.q)&&(e+="{\\vphantom{X}}"),n.d&&(e+="^{"+n.d+"}"),n.q&&(e+="_{\\smash[t]{"+n.q+"}}")):"oxidation"===t.dType?(n.d&&(e+="{\\vphantom{X}}",e+="^{"+n.d+"}"),n.q&&(e+="{\\vphantom{X}}",e+="_{\\smash[t]{"+n.q+"}}")):(n.q&&(e+="{\\vphantom{X}}",e+="_{\\smash[t]{"+n.q+"}}"),n.d&&(e+="{\\vphantom{X}}",e+="^{"+n.d+"}"));break;case"rm":e="\\mathrm{"+t.p1+"}";break;case"text":t.p1.match(/[\^_]/)?(t.p1=t.p1.replace(" ","~").replace("-","\\text{-}"),e="\\mathrm{"+t.p1+"}"):e="\\text{"+t.p1+"}";break;case"roman numeral":e="\\mathrm{"+t.p1+"}";break;case"state of aggregation":e="\\mskip2mu "+c._goInner(t.p1);break;case"state of aggregation subscript":e="\\mskip1mu "+c._goInner(t.p1);break;case"bond":if(!(e=c._getBond(t.kind_)))throw["MhchemErrorBond","mhchem Error. Unknown bond type ("+t.kind_+")"];break;case"frac":var o="\\frac{"+t.p1+"}{"+t.p2+"}";e="\\mathchoice{\\textstyle"+o+"}{"+o+"}{"+o+"}{"+o+"}";break;case"pu-frac":var a="\\frac{"+c._goInner(t.p1)+"}{"+c._goInner(t.p2)+"}";e="\\mathchoice{\\textstyle"+a+"}{"+a+"}{"+a+"}{"+a+"}";break;case"tex-math":e=t.p1+" ";break;case"frac-ce":e="\\frac{"+c._goInner(t.p1)+"}{"+c._goInner(t.p2)+"}";break;case"overset":e="\\overset{"+c._goInner(t.p1)+"}{"+c._goInner(t.p2)+"}";break;case"underset":e="\\underset{"+c._goInner(t.p1)+"}{"+c._goInner(t.p2)+"}";break;case"underbrace":e="\\underbrace{"+c._goInner(t.p1)+"}_{"+c._goInner(t.p2)+"}";break;case"color":e="{\\color{"+t.color1+"}{"+c._goInner(t.color2)+"}}";break;case"color0":e="\\color{"+t.color+"}";break;case"arrow":var r={rd:c._goInner(t.rd),rq:c._goInner(t.rq)},i="\\x"+c._getArrow(t.r);r.rq&&(i+="[{"+r.rq+"}]"),e=i+=r.rd?"{"+r.rd+"}":"{}";break;case"operator":e=c._getOperator(t.kind_);break;case"1st-level escape":e=t.p1+" ";break;case"space":e=" ";break;case"entitySkip":case"pu-space-1":e="~";break;case"pu-space-2":e="\\mkern3mu ";break;case"1000 separator":e="\\mkern2mu ";break;case"commaDecimal":e="{,}";break;case"comma enumeration L":e="{"+t.p1+"}\\mkern6mu ";break;case"comma enumeration M":e="{"+t.p1+"}\\mkern3mu ";break;case"comma enumeration S":e="{"+t.p1+"}\\mkern1mu ";break;case"hyphen":e="\\text{-}";break;case"addition compound":e="\\,{\\cdot}\\,";break;case"electron dot":e="\\mkern1mu \\bullet\\mkern1mu ";break;case"KV x":e="{\\times}";break;case"prime":e="\\prime ";break;case"cdot":e="\\cdot ";break;case"tight cdot":e="\\mkern1mu{\\cdot}\\mkern1mu ";break;case"times":e="\\times ";break;case"circa":e="{\\sim}";break;case"^":e="uparrow";break;case"v":e="downarrow";break;case"ellipsis":e="\\ldots ";break;case"/":e="/";break;case" / ":e="\\,/\\,";break;default:throw["MhchemBugT","mhchem bug T. Please report."]}return e},_getArrow:function(t){switch(t){case"->":case"\u2192":case"\u27f6":return"rightarrow";case"<-":return"leftarrow";case"<->":return"leftrightarrow";case"<--\x3e":return"rightleftarrows";case"<=>":case"\u21cc":return"rightleftharpoons";case"<=>>":return"rightequilibrium";case"<<=>":return"leftequilibrium";default:throw["MhchemBugT","mhchem bug T. Please report."]}},_getBond:function(t){switch(t){case"-":case"1":return"{-}";case"=":case"2":return"{=}";case"#":case"3":return"{\\equiv}";case"~":return"{\\tripledash}";case"~-":return"{\\mathrlap{\\raisebox{-.1em}{$-$}}\\raisebox{.1em}{$\\tripledash$}}";case"~=":case"~--":return"{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";case"-~-":return"{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$-$}}\\tripledash}";case"...":return"{{\\cdot}{\\cdot}{\\cdot}}";case"....":return"{{\\cdot}{\\cdot}{\\cdot}{\\cdot}}";case"->":return"{\\rightarrow}";case"<-":return"{\\leftarrow}";case"<":return"{<}";case">":return"{>}";default:throw["MhchemBugT","mhchem bug T. Please report."]}},_getOperator:function(t){switch(t){case"+":return" {}+{} ";case"-":return" {}-{} ";case"=":return" {}={} ";case"<":return" {}<{} ";case">":return" {}>{} ";case"<<":return" {}\\ll{} ";case">>":return" {}\\gg{} ";case"\\pm":return" {}\\pm{} ";case"\\approx":case"$\\approx$":return" {}\\approx{} ";case"v":case"(v)":return" \\downarrow{} ";case"^":case"(^)":return" \\uparrow{} ";default:throw["MhchemBugT","mhchem bug T. Please report."]}}}}]).default});
\ No newline at end of file
diff --git a/katex.bk/contrib/mhchem.mjs b/katex.bk/contrib/mhchem.mjs
new file mode 100644
index 00000000..46241b5f
--- /dev/null
+++ b/katex.bk/contrib/mhchem.mjs
@@ -0,0 +1,3109 @@
+import katex from '../katex.mjs';
+
+/* eslint-disable */
+
+/* -*- Mode: Javascript; indent-tabs-mode:nil; js-indent-level: 2 -*- */
+
+/* vim: set ts=2 et sw=2 tw=80: */
+
+/*************************************************************
+ *
+ *  KaTeX mhchem.js
+ *
+ *  This file implements a KaTeX version of mhchem version 3.3.0.
+ *  It is adapted from MathJax/extensions/TeX/mhchem.js
+ *  It differs from the MathJax version as follows:
+ *    1. The interface is changed so that it can be called from KaTeX, not MathJax.
+ *    2. \rlap and \llap are replaced with \mathrlap and \mathllap.
+ *    3. Four lines of code are edited in order to use \raisebox instead of \raise.
+ *    4. The reaction arrow code is simplified. All reaction arrows are rendered
+ *       using KaTeX extensible arrows instead of building non-extensible arrows.
+ *    5. \tripledash vertical alignment is slightly adjusted.
+ *
+ *    This code, as other KaTeX code, is released under the MIT license.
+ * 
+ * /*************************************************************
+ *
+ *  MathJax/extensions/TeX/mhchem.js
+ *
+ *  Implements the \ce command for handling chemical formulas
+ *  from the mhchem LaTeX package.
+ *
+ *  ---------------------------------------------------------------------
+ *
+ *  Copyright (c) 2011-2015 The MathJax Consortium
+ *  Copyright (c) 2015-2018 Martin Hensel
+ *
+ *  Licensed under the Apache License, Version 2.0 (the "License");
+ *  you may not use this file except in compliance with the License.
+ *  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ *  Unless required by applicable law or agreed to in writing, software
+ *  distributed under the License is distributed on an "AS IS" BASIS,
+ *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ *  See the License for the specific language governing permissions and
+ *  limitations under the License.
+ */
+//
+// Coding Style
+//   - use '' for identifiers that can by minified/uglified
+//   - use "" for strings that need to stay untouched
+// version: "3.3.0" for MathJax and KaTeX
+// Add \ce, \pu, and \tripledash to the KaTeX macros.
+katex.__defineMacro("\\ce", function (context) {
+  return chemParse(context.consumeArgs(1)[0], "ce");
+});
+
+katex.__defineMacro("\\pu", function (context) {
+  return chemParse(context.consumeArgs(1)[0], "pu");
+}); //  Needed for \bond for the ~ forms
+//  Raise by 2.56mu, not 2mu. We're raising a hyphen-minus, U+002D, not 
+//  a mathematical minus, U+2212. So we need that extra 0.56.
+
+
+katex.__defineMacro("\\tripledash", "{\\vphantom{-}\\raisebox{2.56mu}{$\\mkern2mu" + "\\tiny\\text{-}\\mkern1mu\\text{-}\\mkern1mu\\text{-}\\mkern2mu$}}");
+//  This is the main function for handing the \ce and \pu commands.
+//  It takes the argument to \ce or \pu and returns the corresponding TeX string.
+//
+
+var chemParse = function chemParse(tokens, stateMachine) {
+  // Recreate the argument string from KaTeX's array of tokens.
+  var str = "";
+  var expectedLoc = tokens[tokens.length - 1].loc.start;
+
+  for (var i = tokens.length - 1; i >= 0; i--) {
+    if (tokens[i].loc.start > expectedLoc) {
+      // context.consumeArgs has eaten a space.
+      str += " ";
+      expectedLoc = tokens[i].loc.start;
+    }
+
+    str += tokens[i].text;
+    expectedLoc += tokens[i].text.length;
+  }
+
+  var tex = texify.go(mhchemParser.go(str, stateMachine));
+  return tex;
+}; //
+// Core parser for mhchem syntax  (recursive)
+//
+
+/** @type {MhchemParser} */
+
+
+var mhchemParser = {
+  //
+  // Parses mchem \ce syntax
+  //
+  // Call like
+  //   go("H2O");
+  //
+  go: function go(input, stateMachine) {
+    if (!input) {
+      return [];
+    }
+
+    if (stateMachine === undefined) {
+      stateMachine = 'ce';
+    }
+
+    var state = '0'; //
+    // String buffers for parsing:
+    //
+    // buffer.a == amount
+    // buffer.o == element
+    // buffer.b == left-side superscript
+    // buffer.p == left-side subscript
+    // buffer.q == right-side subscript
+    // buffer.d == right-side superscript
+    //
+    // buffer.r == arrow
+    // buffer.rdt == arrow, script above, type
+    // buffer.rd == arrow, script above, content
+    // buffer.rqt == arrow, script below, type
+    // buffer.rq == arrow, script below, content
+    //
+    // buffer.text_
+    // buffer.rm
+    // etc.
+    //
+    // buffer.parenthesisLevel == int, starting at 0
+    // buffer.sb == bool, space before
+    // buffer.beginsWithBond == bool
+    //
+    // These letters are also used as state names.
+    //
+    // Other states:
+    // 0 == begin of main part (arrow/operator unlikely)
+    // 1 == next entity
+    // 2 == next entity (arrow/operator unlikely)
+    // 3 == next atom
+    // c == macro
+    //
+
+    /** @type {Buffer} */
+
+    var buffer = {};
+    buffer['parenthesisLevel'] = 0;
+    input = input.replace(/\n/g, " ");
+    input = input.replace(/[\u2212\u2013\u2014\u2010]/g, "-");
+    input = input.replace(/[\u2026]/g, "..."); //
+    // Looks through mhchemParser.transitions, to execute a matching action
+    // (recursive)
+    //
+
+    var lastInput;
+    var watchdog = 10;
+    /** @type {ParserOutput[]} */
+
+    var output = [];
+
+    while (true) {
+      if (lastInput !== input) {
+        watchdog = 10;
+        lastInput = input;
+      } else {
+        watchdog--;
+      } //
+      // Find actions in transition table
+      //
+
+
+      var machine = mhchemParser.stateMachines[stateMachine];
+      var t = machine.transitions[state] || machine.transitions['*'];
+
+      iterateTransitions: for (var i = 0; i < t.length; i++) {
+        var matches = mhchemParser.patterns.match_(t[i].pattern, input);
+
+        if (matches) {
+          //
+          // Execute actions
+          //
+          var task = t[i].task;
+
+          for (var iA = 0; iA < task.action_.length; iA++) {
+            var o; //
+            // Find and execute action
+            //
+
+            if (machine.actions[task.action_[iA].type_]) {
+              o = machine.actions[task.action_[iA].type_](buffer, matches.match_, task.action_[iA].option);
+            } else if (mhchemParser.actions[task.action_[iA].type_]) {
+              o = mhchemParser.actions[task.action_[iA].type_](buffer, matches.match_, task.action_[iA].option);
+            } else {
+              throw ["MhchemBugA", "mhchem bug A. Please report. (" + task.action_[iA].type_ + ")"]; // Trying to use non-existing action
+            } //
+            // Add output
+            //
+
+
+            mhchemParser.concatArray(output, o);
+          } //
+          // Set next state,
+          // Shorten input,
+          // Continue with next character
+          //   (= apply only one transition per position)
+          //
+
+
+          state = task.nextState || state;
+
+          if (input.length > 0) {
+            if (!task.revisit) {
+              input = matches.remainder;
+            }
+
+            if (!task.toContinue) {
+              break iterateTransitions;
+            }
+          } else {
+            return output;
+          }
+        }
+      } //
+      // Prevent infinite loop
+      //
+
+
+      if (watchdog <= 0) {
+        throw ["MhchemBugU", "mhchem bug U. Please report."]; // Unexpected character
+      }
+    }
+  },
+  concatArray: function concatArray(a, b) {
+    if (b) {
+      if (Array.isArray(b)) {
+        for (var iB = 0; iB < b.length; iB++) {
+          a.push(b[iB]);
+        }
+      } else {
+        a.push(b);
+      }
+    }
+  },
+  patterns: {
+    //
+    // Matching patterns
+    // either regexps or function that return null or {match_:"a", remainder:"bc"}
+    //
+    patterns: {
+      // property names must not look like integers ("2") for correct property traversal order, later on
+      'empty': /^$/,
+      'else': /^./,
+      'else2': /^./,
+      'space': /^\s/,
+      'space A': /^\s(?=[A-Z\\$])/,
+      'space$': /^\s$/,
+      'a-z': /^[a-z]/,
+      'x': /^x/,
+      'x$': /^x$/,
+      'i$': /^i$/,
+      'letters': /^(?:[a-zA-Z\u03B1-\u03C9\u0391-\u03A9?@]|(?:\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega|Gamma|Delta|Theta|Lambda|Xi|Pi|Sigma|Upsilon|Phi|Psi|Omega)(?:\s+|\{\}|(?![a-zA-Z]))))+/,
+      '\\greek': /^\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega|Gamma|Delta|Theta|Lambda|Xi|Pi|Sigma|Upsilon|Phi|Psi|Omega)(?:\s+|\{\}|(?![a-zA-Z]))/,
+      'one lowercase latin letter $': /^(?:([a-z])(?:$|[^a-zA-Z]))$/,
+      '$one lowercase latin letter$ $': /^\$(?:([a-z])(?:$|[^a-zA-Z]))\$$/,
+      'one lowercase greek letter $': /^(?:\$?[\u03B1-\u03C9]\$?|\$?\\(?:alpha|beta|gamma|delta|epsilon|zeta|eta|theta|iota|kappa|lambda|mu|nu|xi|omicron|pi|rho|sigma|tau|upsilon|phi|chi|psi|omega)\s*\$?)(?:\s+|\{\}|(?![a-zA-Z]))$/,
+      'digits': /^[0-9]+/,
+      '-9.,9': /^[+\-]?(?:[0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))/,
+      '-9.,9 no missing 0': /^[+\-]?[0-9]+(?:[.,][0-9]+)?/,
+      '(-)(9.,9)(e)(99)': function e99(input) {
+        var m = input.match(/^(\+\-|\+\/\-|\+|\-|\\pm\s?)?([0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))?(\((?:[0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+))\))?(?:([eE]|\s*(\*|x|\\times|\u00D7)\s*10\^)([+\-]?[0-9]+|\{[+\-]?[0-9]+\}))?/);
+
+        if (m && m[0]) {
+          return {
+            match_: m.splice(1),
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      '(-)(9)^(-9)': function _(input) {
+        var m = input.match(/^(\+\-|\+\/\-|\+|\-|\\pm\s?)?([0-9]+(?:[,.][0-9]+)?|[0-9]*(?:\.[0-9]+)?)\^([+\-]?[0-9]+|\{[+\-]?[0-9]+\})/);
+
+        if (m && m[0]) {
+          return {
+            match_: m.splice(1),
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      'state of aggregation $': function stateOfAggregation$(input) {
+        // ... or crystal system
+        var a = mhchemParser.patterns.findObserveGroups(input, "", /^\([a-z]{1,3}(?=[\),])/, ")", ""); // (aq), (aq,$\infty$), (aq, sat)
+
+        if (a && a.remainder.match(/^($|[\s,;\)\]\}])/)) {
+          return a;
+        } //  AND end of 'phrase'
+
+
+        var m = input.match(/^(?:\((?:\\ca\s?)?\$[amothc]\$\))/); // OR crystal system ($o$) (\ca$c$)
+
+        if (m) {
+          return {
+            match_: m[0],
+            remainder: input.substr(m[0].length)
+          };
+        }
+
+        return null;
+      },
+      '_{(state of aggregation)}$': /^_\{(\([a-z]{1,3}\))\}/,
+      '{[(': /^(?:\\\{|\[|\()/,
+      ')]}': /^(?:\)|\]|\\\})/,
+      ', ': /^[,;]\s*/,
+      ',': /^[,;]/,
+      '.': /^[.]/,
+      '. ': /^([.\u22C5\u00B7\u2022])\s*/,
+      '...': /^\.\.\.(?=$|[^.])/,
+      '* ': /^([*])\s*/,
+      '^{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^{", "", "", "}");
+      },
+      '^($...$)': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", "$", "$", "");
+      },
+      '^a': /^\^([0-9]+|[^\\_])/,
+      '^\\x{}{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '^\\x{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "^", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '^\\x': /^\^(\\[a-zA-Z]+)\s*/,
+      '^(-1)': /^\^(-?\d+)/,
+      '\'': /^'/,
+      '_{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_{", "", "", "}");
+      },
+      '_($...$)': function _$$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", "$", "$", "");
+      },
+      '_9': /^_([+\-]?[0-9]+|[^\\])/,
+      '_\\x{}{}': function _X(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '_\\x{}': function _X(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "_", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '_\\x': /^_(\\[a-zA-Z]+)\s*/,
+      '^_': /^(?:\^(?=_)|\_(?=\^)|[\^_]$)/,
+      '{}': /^\{\}/,
+      '{...}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", "{", "}", "");
+      },
+      '{(...)}': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "{", "", "", "}");
+      },
+      '$...$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", "$", "$", "");
+      },
+      '${(...)}$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "${", "", "", "}$");
+      },
+      '$(...)$': function $$(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "$", "", "", "$");
+      },
+      '=<>': /^[=<>]/,
+      '#': /^[#\u2261]/,
+      '+': /^\+/,
+      '-$': /^-(?=[\s_},;\]/]|$|\([a-z]+\))/,
+      // -space -, -; -] -/ -$ -state-of-aggregation
+      '-9': /^-(?=[0-9])/,
+      '- orbital overlap': /^-(?=(?:[spd]|sp)(?:$|[\s,;\)\]\}]))/,
+      '-': /^-/,
+      'pm-operator': /^(?:\\pm|\$\\pm\$|\+-|\+\/-)/,
+      'operator': /^(?:\+|(?:[\-=<>]|<<|>>|\\approx|\$\\approx\$)(?=\s|$|-?[0-9]))/,
+      'arrowUpDown': /^(?:v|\(v\)|\^|\(\^\))(?=$|[\s,;\)\]\}])/,
+      '\\bond{(...)}': function bond(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\bond{", "", "", "}");
+      },
+      '->': /^(?:<->|<-->|->|<-|<=>>|<<=>|<=>|[\u2192\u27F6\u21CC])/,
+      'CMT': /^[CMT](?=\[)/,
+      '[(...)]': function _(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "[", "", "", "]");
+      },
+      '1st-level escape': /^(&|\\\\|\\hline)\s*/,
+      '\\,': /^(?:\\[,\ ;:])/,
+      // \\x - but output no space before
+      '\\x{}{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", /^\\[a-zA-Z]+\{/, "}", "", "", "{", "}", "", true);
+      },
+      '\\x{}': function x(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "", /^\\[a-zA-Z]+\{/, "}", "");
+      },
+      '\\ca': /^\\ca(?:\s+|(?![a-zA-Z]))/,
+      '\\x': /^(?:\\[a-zA-Z]+\s*|\\[_&{}%])/,
+      'orbital': /^(?:[0-9]{1,2}[spdfgh]|[0-9]{0,2}sp)(?=$|[^a-zA-Z])/,
+      // only those with numbers in front, because the others will be formatted correctly anyway
+      'others': /^[\/~|]/,
+      '\\frac{(...)}': function frac(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\frac{", "", "", "}", "{", "", "", "}");
+      },
+      '\\overset{(...)}': function overset(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\overset{", "", "", "}", "{", "", "", "}");
+      },
+      '\\underset{(...)}': function underset(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\underset{", "", "", "}", "{", "", "", "}");
+      },
+      '\\underbrace{(...)}': function underbrace(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\underbrace{", "", "", "}_", "{", "", "", "}");
+      },
+      '\\color{(...)}0': function color0(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color{", "", "", "}");
+      },
+      '\\color{(...)}{(...)}1': function color1(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color{", "", "", "}", "{", "", "", "}");
+      },
+      '\\color(...){(...)}2': function color2(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\color", "\\", "", /^(?=\{)/, "{", "", "", "}");
+      },
+      '\\ce{(...)}': function ce(input) {
+        return mhchemParser.patterns.findObserveGroups(input, "\\ce{", "", "", "}");
+      },
+      'oxidation$': /^(?:[+-][IVX]+|\\pm\s*0|\$\\pm\$\s*0)$/,
+      'd-oxidation$': /^(?:[+-]?\s?[IVX]+|\\pm\s*0|\$\\pm\$\s*0)$/,
+      // 0 could be oxidation or charge
+      'roman numeral': /^[IVX]+/,
+      '1/2$': /^[+\-]?(?:[0-9]+|\$[a-z]\$|[a-z])\/[0-9]+(?:\$[a-z]\$|[a-z])?$/,
+      'amount': function amount(input) {
+        var match; // e.g. 2, 0.5, 1/2, -2, n/2, +;  $a$ could be added later in parsing
+
+        match = input.match(/^(?:(?:(?:\([+\-]?[0-9]+\/[0-9]+\)|[+\-]?(?:[0-9]+|\$[a-z]\$|[a-z])\/[0-9]+|[+\-]?[0-9]+[.,][0-9]+|[+\-]?\.[0-9]+|[+\-]?[0-9]+)(?:[a-z](?=\s*[A-Z]))?)|[+\-]?[a-z](?=\s*[A-Z])|\+(?!\s))/);
+
+        if (match) {
+          return {
+            match_: match[0],
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        var a = mhchemParser.patterns.findObserveGroups(input, "", "$", "$", "");
+
+        if (a) {
+          // e.g. $2n-1$, $-$
+          match = a.match_.match(/^\$(?:\(?[+\-]?(?:[0-9]*[a-z]?[+\-])?[0-9]*[a-z](?:[+\-][0-9]*[a-z]?)?\)?|\+|-)\$$/);
+
+          if (match) {
+            return {
+              match_: match[0],
+              remainder: input.substr(match[0].length)
+            };
+          }
+        }
+
+        return null;
+      },
+      'amount2': function amount2(input) {
+        return this['amount'](input);
+      },
+      '(KV letters),': /^(?:[A-Z][a-z]{0,2}|i)(?=,)/,
+      'formula$': function formula$(input) {
+        if (input.match(/^\([a-z]+\)$/)) {
+          return null;
+        } // state of aggregation = no formula
+
+
+        var match = input.match(/^(?:[a-z]|(?:[0-9\ \+\-\,\.\(\)]+[a-z])+[0-9\ \+\-\,\.\(\)]*|(?:[a-z][0-9\ \+\-\,\.\(\)]+)+[a-z]?)$/);
+
+        if (match) {
+          return {
+            match_: match[0],
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        return null;
+      },
+      'uprightEntities': /^(?:pH|pOH|pC|pK|iPr|iBu)(?=$|[^a-zA-Z])/,
+      '/': /^\s*(\/)\s*/,
+      '//': /^\s*(\/\/)\s*/,
+      '*': /^\s*[*.]\s*/
+    },
+    findObserveGroups: function findObserveGroups(input, begExcl, begIncl, endIncl, endExcl, beg2Excl, beg2Incl, end2Incl, end2Excl, combine) {
+      /** @type {{(input: string, pattern: string | RegExp): string | string[] | null;}} */
+      var _match = function _match(input, pattern) {
+        if (typeof pattern === "string") {
+          if (input.indexOf(pattern) !== 0) {
+            return null;
+          }
+
+          return pattern;
+        } else {
+          var match = input.match(pattern);
+
+          if (!match) {
+            return null;
+          }
+
+          return match[0];
+        }
+      };
+      /** @type {{(input: string, i: number, endChars: string | RegExp): {endMatchBegin: number, endMatchEnd: number} | null;}} */
+
+
+      var _findObserveGroups = function _findObserveGroups(input, i, endChars) {
+        var braces = 0;
+
+        while (i < input.length) {
+          var a = input.charAt(i);
+
+          var match = _match(input.substr(i), endChars);
+
+          if (match !== null && braces === 0) {
+            return {
+              endMatchBegin: i,
+              endMatchEnd: i + match.length
+            };
+          } else if (a === "{") {
+            braces++;
+          } else if (a === "}") {
+            if (braces === 0) {
+              throw ["ExtraCloseMissingOpen", "Extra close brace or missing open brace"];
+            } else {
+              braces--;
+            }
+          }
+
+          i++;
+        }
+
+        if (braces > 0) {
+          return null;
+        }
+
+        return null;
+      };
+
+      var match = _match(input, begExcl);
+
+      if (match === null) {
+        return null;
+      }
+
+      input = input.substr(match.length);
+      match = _match(input, begIncl);
+
+      if (match === null) {
+        return null;
+      }
+
+      var e = _findObserveGroups(input, match.length, endIncl || endExcl);
+
+      if (e === null) {
+        return null;
+      }
+
+      var match1 = input.substring(0, endIncl ? e.endMatchEnd : e.endMatchBegin);
+
+      if (!(beg2Excl || beg2Incl)) {
+        return {
+          match_: match1,
+          remainder: input.substr(e.endMatchEnd)
+        };
+      } else {
+        var group2 = this.findObserveGroups(input.substr(e.endMatchEnd), beg2Excl, beg2Incl, end2Incl, end2Excl);
+
+        if (group2 === null) {
+          return null;
+        }
+        /** @type {string[]} */
+
+
+        var matchRet = [match1, group2.match_];
+        return {
+          match_: combine ? matchRet.join("") : matchRet,
+          remainder: group2.remainder
+        };
+      }
+    },
+    //
+    // Matching function
+    // e.g. match("a", input) will look for the regexp called "a" and see if it matches
+    // returns null or {match_:"a", remainder:"bc"}
+    //
+    match_: function match_(m, input) {
+      var pattern = mhchemParser.patterns.patterns[m];
+
+      if (pattern === undefined) {
+        throw ["MhchemBugP", "mhchem bug P. Please report. (" + m + ")"]; // Trying to use non-existing pattern
+      } else if (typeof pattern === "function") {
+        return mhchemParser.patterns.patterns[m](input); // cannot use cached var pattern here, because some pattern functions need this===mhchemParser
+      } else {
+        // RegExp
+        var match = input.match(pattern);
+
+        if (match) {
+          var mm;
+
+          if (match[2]) {
+            mm = [match[1], match[2]];
+          } else if (match[1]) {
+            mm = match[1];
+          } else {
+            mm = match[0];
+          }
+
+          return {
+            match_: mm,
+            remainder: input.substr(match[0].length)
+          };
+        }
+
+        return null;
+      }
+    }
+  },
+  //
+  // Generic state machine actions
+  //
+  actions: {
+    'a=': function a(buffer, m) {
+      buffer.a = (buffer.a || "") + m;
+    },
+    'b=': function b(buffer, m) {
+      buffer.b = (buffer.b || "") + m;
+    },
+    'p=': function p(buffer, m) {
+      buffer.p = (buffer.p || "") + m;
+    },
+    'o=': function o(buffer, m) {
+      buffer.o = (buffer.o || "") + m;
+    },
+    'q=': function q(buffer, m) {
+      buffer.q = (buffer.q || "") + m;
+    },
+    'd=': function d(buffer, m) {
+      buffer.d = (buffer.d || "") + m;
+    },
+    'rm=': function rm(buffer, m) {
+      buffer.rm = (buffer.rm || "") + m;
+    },
+    'text=': function text(buffer, m) {
+      buffer.text_ = (buffer.text_ || "") + m;
+    },
+    'insert': function insert(buffer, m, a) {
+      return {
+        type_: a
+      };
+    },
+    'insert+p1': function insertP1(buffer, m, a) {
+      return {
+        type_: a,
+        p1: m
+      };
+    },
+    'insert+p1+p2': function insertP1P2(buffer, m, a) {
+      return {
+        type_: a,
+        p1: m[0],
+        p2: m[1]
+      };
+    },
+    'copy': function copy(buffer, m) {
+      return m;
+    },
+    'rm': function rm(buffer, m) {
+      return {
+        type_: 'rm',
+        p1: m || ""
+      };
+    },
+    'text': function text(buffer, m) {
+      return mhchemParser.go(m, 'text');
+    },
+    '{text}': function text(buffer, m) {
+      var ret = ["{"];
+      mhchemParser.concatArray(ret, mhchemParser.go(m, 'text'));
+      ret.push("}");
+      return ret;
+    },
+    'tex-math': function texMath(buffer, m) {
+      return mhchemParser.go(m, 'tex-math');
+    },
+    'tex-math tight': function texMathTight(buffer, m) {
+      return mhchemParser.go(m, 'tex-math tight');
+    },
+    'bond': function bond(buffer, m, k) {
+      return {
+        type_: 'bond',
+        kind_: k || m
+      };
+    },
+    'color0-output': function color0Output(buffer, m) {
+      return {
+        type_: 'color0',
+        color: m[0]
+      };
+    },
+    'ce': function ce(buffer, m) {
+      return mhchemParser.go(m);
+    },
+    '1/2': function _(buffer, m) {
+      /** @type {ParserOutput[]} */
+      var ret = [];
+
+      if (m.match(/^[+\-]/)) {
+        ret.push(m.substr(0, 1));
+        m = m.substr(1);
+      }
+
+      var n = m.match(/^([0-9]+|\$[a-z]\$|[a-z])\/([0-9]+)(\$[a-z]\$|[a-z])?$/);
+      n[1] = n[1].replace(/\$/g, "");
+      ret.push({
+        type_: 'frac',
+        p1: n[1],
+        p2: n[2]
+      });
+
+      if (n[3]) {
+        n[3] = n[3].replace(/\$/g, "");
+        ret.push({
+          type_: 'tex-math',
+          p1: n[3]
+        });
+      }
+
+      return ret;
+    },
+    '9,9': function _(buffer, m) {
+      return mhchemParser.go(m, '9,9');
+    }
+  },
+  //
+  // createTransitions
+  // convert  { 'letter': { 'state': { action_: 'output' } } }  to  { 'state' => [ { pattern: 'letter', task: { action_: [{type_: 'output'}] } } ] }
+  // with expansion of 'a|b' to 'a' and 'b' (at 2 places)
+  //
+  createTransitions: function createTransitions(o) {
+    var pattern, state;
+    /** @type {string[]} */
+
+    var stateArray;
+    var i; //
+    // 1. Collect all states
+    //
+
+    /** @type {Transitions} */
+
+    var transitions = {};
+
+    for (pattern in o) {
+      for (state in o[pattern]) {
+        stateArray = state.split("|");
+        o[pattern][state].stateArray = stateArray;
+
+        for (i = 0; i < stateArray.length; i++) {
+          transitions[stateArray[i]] = [];
+        }
+      }
+    } //
+    // 2. Fill states
+    //
+
+
+    for (pattern in o) {
+      for (state in o[pattern]) {
+        stateArray = o[pattern][state].stateArray || [];
+
+        for (i = 0; i < stateArray.length; i++) {
+          //
+          // 2a. Normalize actions into array:  'text=' ==> [{type_:'text='}]
+          // (Note to myself: Resolving the function here would be problematic. It would need .bind (for *this*) and currying (for *option*).)
+          //
+
+          /** @type {any} */
+          var p = o[pattern][state];
+
+          if (p.action_) {
+            p.action_ = [].concat(p.action_);
+
+            for (var k = 0; k < p.action_.length; k++) {
+              if (typeof p.action_[k] === "string") {
+                p.action_[k] = {
+                  type_: p.action_[k]
+                };
+              }
+            }
+          } else {
+            p.action_ = [];
+          } //
+          // 2.b Multi-insert
+          //
+
+
+          var patternArray = pattern.split("|");
+
+          for (var j = 0; j < patternArray.length; j++) {
+            if (stateArray[i] === '*') {
+              // insert into all
+              for (var t in transitions) {
+                transitions[t].push({
+                  pattern: patternArray[j],
+                  task: p
+                });
+              }
+            } else {
+              transitions[stateArray[i]].push({
+                pattern: patternArray[j],
+                task: p
+              });
+            }
+          }
+        }
+      }
+    }
+
+    return transitions;
+  },
+  stateMachines: {}
+}; //
+// Definition of state machines
+//
+
+mhchemParser.stateMachines = {
+  //
+  // \ce state machines
+  //
+  //#region ce
+  'ce': {
+    // main parser
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      'else': {
+        '0|1|2': {
+          action_: 'beginsWithBond=false',
+          revisit: true,
+          toContinue: true
+        }
+      },
+      'oxidation$': {
+        '0': {
+          action_: 'oxidation-output'
+        }
+      },
+      'CMT': {
+        'r': {
+          action_: 'rdt=',
+          nextState: 'rt'
+        },
+        'rd': {
+          action_: 'rqt=',
+          nextState: 'rdt'
+        }
+      },
+      'arrowUpDown': {
+        '0|1|2|as': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '1'
+        }
+      },
+      'uprightEntities': {
+        '0|1|2': {
+          action_: ['o=', 'output'],
+          nextState: '1'
+        }
+      },
+      'orbital': {
+        '0|1|2|3': {
+          action_: 'o=',
+          nextState: 'o'
+        }
+      },
+      '->': {
+        '0|1|2|3': {
+          action_: 'r=',
+          nextState: 'r'
+        },
+        'a|as': {
+          action_: ['output', 'r='],
+          nextState: 'r'
+        },
+        '*': {
+          action_: ['output', 'r='],
+          nextState: 'r'
+        }
+      },
+      '+': {
+        'o': {
+          action_: 'd= kv',
+          nextState: 'd'
+        },
+        'd|D': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qd|qD': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'dq': {
+          action_: ['output', 'd='],
+          nextState: 'd'
+        },
+        '3': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '0'
+        }
+      },
+      'amount': {
+        '0|2': {
+          action_: 'a=',
+          nextState: 'a'
+        }
+      },
+      'pm-operator': {
+        '0|1|2|a|as': {
+          action_: ['sb=false', 'output', {
+            type_: 'operator',
+            option: '\\pm'
+          }],
+          nextState: '0'
+        }
+      },
+      'operator': {
+        '0|1|2|a|as': {
+          action_: ['sb=false', 'output', 'operator'],
+          nextState: '0'
+        }
+      },
+      '-$': {
+        'o|q': {
+          action_: ['charge or bond', 'output'],
+          nextState: 'qd'
+        },
+        'd': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'D': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qd': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'qD|dq': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        }
+      },
+      '-9': {
+        '3|o': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '3'
+        }
+      },
+      '- orbital overlap': {
+        'o': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        },
+        'd': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        }
+      },
+      '-': {
+        '0|1|2': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'beginsWithBond=true', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        '3': {
+          action_: {
+            type_: 'bond',
+            option: "-"
+          }
+        },
+        'a': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'hyphen'
+          }],
+          nextState: '2'
+        },
+        'as': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        },
+        'b': {
+          action_: 'b='
+        },
+        'o': {
+          action_: {
+            type_: '- after o/d',
+            option: false
+          },
+          nextState: '2'
+        },
+        'q': {
+          action_: {
+            type_: '- after o/d',
+            option: false
+          },
+          nextState: '2'
+        },
+        'd|qd|dq': {
+          action_: {
+            type_: '- after o/d',
+            option: true
+          },
+          nextState: '2'
+        },
+        'D|qD|p': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "-"
+          }],
+          nextState: '3'
+        }
+      },
+      'amount2': {
+        '1|3': {
+          action_: 'a=',
+          nextState: 'a'
+        }
+      },
+      'letters': {
+        '0|1|2|3|a|as|b|p|bp|o': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        'q|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        },
+        'd|D|qd|qD': {
+          action_: 'o after d',
+          nextState: 'o'
+        }
+      },
+      'digits': {
+        'o': {
+          action_: 'q=',
+          nextState: 'q'
+        },
+        'd|D': {
+          action_: 'q=',
+          nextState: 'dq'
+        },
+        'q': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        },
+        'a': {
+          action_: 'o=',
+          nextState: 'o'
+        }
+      },
+      'space A': {
+        'b|p|bp': {}
+      },
+      'space': {
+        'a': {
+          nextState: 'as'
+        },
+        '0': {
+          action_: 'sb=false'
+        },
+        '1|2': {
+          action_: 'sb=true'
+        },
+        'r|rt|rd|rdt|rdq': {
+          action_: 'output',
+          nextState: '0'
+        },
+        '*': {
+          action_: ['output', 'sb=true'],
+          nextState: '1'
+        }
+      },
+      '1st-level escape': {
+        '1|2': {
+          action_: ['output', {
+            type_: 'insert+p1',
+            option: '1st-level escape'
+          }]
+        },
+        '*': {
+          action_: ['output', {
+            type_: 'insert+p1',
+            option: '1st-level escape'
+          }],
+          nextState: '0'
+        }
+      },
+      '[(...)]': {
+        'r|rt': {
+          action_: 'rd=',
+          nextState: 'rd'
+        },
+        'rd|rdt': {
+          action_: 'rq=',
+          nextState: 'rdq'
+        }
+      },
+      '...': {
+        'o|d|D|dq|qd|qD': {
+          action_: ['output', {
+            type_: 'bond',
+            option: "..."
+          }],
+          nextState: '3'
+        },
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, {
+            type_: 'insert',
+            option: 'ellipsis'
+          }],
+          nextState: '1'
+        }
+      },
+      '. |* ': {
+        '*': {
+          action_: ['output', {
+            type_: 'insert',
+            option: 'addition compound'
+          }],
+          nextState: '1'
+        }
+      },
+      'state of aggregation $': {
+        '*': {
+          action_: ['output', 'state of aggregation'],
+          nextState: '1'
+        }
+      },
+      '{[(': {
+        'a|as|o': {
+          action_: ['o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        },
+        '0|1|2|3': {
+          action_: ['o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        },
+        '*': {
+          action_: ['output', 'o=', 'output', 'parenthesisLevel++'],
+          nextState: '2'
+        }
+      },
+      ')]}': {
+        '0|1|2|3|b|p|bp|o': {
+          action_: ['o=', 'parenthesisLevel--'],
+          nextState: 'o'
+        },
+        'a|as|d|D|q|qd|qD|dq': {
+          action_: ['output', 'o=', 'parenthesisLevel--'],
+          nextState: 'o'
+        }
+      },
+      ', ': {
+        '*': {
+          action_: ['output', 'comma'],
+          nextState: '0'
+        }
+      },
+      '^_': {
+        // ^ and _ without a sensible argument
+        '*': {}
+      },
+      '^{(...)}|^($...$)': {
+        '0|1|2|as': {
+          action_: 'b=',
+          nextState: 'b'
+        },
+        'p': {
+          action_: 'b=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'd= kv',
+          nextState: 'D'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qD'
+        },
+        'd|D|qd|qD|dq': {
+          action_: ['output', 'd='],
+          nextState: 'D'
+        }
+      },
+      '^a|^\\x{}{}|^\\x{}|^\\x|\'': {
+        '0|1|2|as': {
+          action_: 'b=',
+          nextState: 'b'
+        },
+        'p': {
+          action_: 'b=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'd= kv',
+          nextState: 'd'
+        },
+        'q': {
+          action_: 'd=',
+          nextState: 'qd'
+        },
+        'd|qd|D|qD': {
+          action_: 'd='
+        },
+        'dq': {
+          action_: ['output', 'd='],
+          nextState: 'd'
+        }
+      },
+      '_{(state of aggregation)}$': {
+        'd|D|q|qd|qD|dq': {
+          action_: ['output', 'q='],
+          nextState: 'q'
+        }
+      },
+      '_{(...)}|_($...$)|_9|_\\x{}{}|_\\x{}|_\\x': {
+        '0|1|2|as': {
+          action_: 'p=',
+          nextState: 'p'
+        },
+        'b': {
+          action_: 'p=',
+          nextState: 'bp'
+        },
+        '3|o': {
+          action_: 'q=',
+          nextState: 'q'
+        },
+        'd|D': {
+          action_: 'q=',
+          nextState: 'dq'
+        },
+        'q|qd|qD|dq': {
+          action_: ['output', 'q='],
+          nextState: 'q'
+        }
+      },
+      '=<>': {
+        '0|1|2|3|a|as|o|q|d|D|qd|qD|dq': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'bond'],
+          nextState: '3'
+        }
+      },
+      '#': {
+        '0|1|2|3|a|as|o': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, {
+            type_: 'bond',
+            option: "#"
+          }],
+          nextState: '3'
+        }
+      },
+      '{}': {
+        '*': {
+          action_: {
+            type_: 'output',
+            option: 1
+          },
+          nextState: '1'
+        }
+      },
+      '{...}': {
+        '0|1|2|3|a|as|b|p|bp': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        'o|d|D|q|qd|qD|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        }
+      },
+      '$...$': {
+        'a': {
+          action_: 'a='
+        },
+        // 2$n$
+        '0|1|2|3|as|b|p|bp|o': {
+          action_: 'o=',
+          nextState: 'o'
+        },
+        // not 'amount'
+        'as|o': {
+          action_: 'o='
+        },
+        'q|d|D|qd|qD|dq': {
+          action_: ['output', 'o='],
+          nextState: 'o'
+        }
+      },
+      '\\bond{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'bond'],
+          nextState: "3"
+        }
+      },
+      '\\frac{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'frac-output'],
+          nextState: '3'
+        }
+      },
+      '\\overset{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'overset-output'],
+          nextState: '3'
+        }
+      },
+      '\\underset{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'underset-output'],
+          nextState: '3'
+        }
+      },
+      '\\underbrace{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'underbrace-output'],
+          nextState: '3'
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'color-output'],
+          nextState: '3'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'color0-output']
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 2
+          }, 'ce'],
+          nextState: '3'
+        }
+      },
+      '\\,': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'copy'],
+          nextState: '1'
+        }
+      },
+      '\\x{}{}|\\x{}|\\x': {
+        '0|1|2|3|a|as|b|p|bp|o|c0': {
+          action_: ['o=', 'output'],
+          nextState: '3'
+        },
+        '*': {
+          action_: ['output', 'o=', 'output'],
+          nextState: '3'
+        }
+      },
+      'others': {
+        '*': {
+          action_: [{
+            type_: 'output',
+            option: 1
+          }, 'copy'],
+          nextState: '3'
+        }
+      },
+      'else2': {
+        'a': {
+          action_: 'a to o',
+          nextState: 'o',
+          revisit: true
+        },
+        'as': {
+          action_: ['output', 'sb=true'],
+          nextState: '1',
+          revisit: true
+        },
+        'r|rt|rd|rdt|rdq': {
+          action_: ['output'],
+          nextState: '0',
+          revisit: true
+        },
+        '*': {
+          action_: ['output', 'copy'],
+          nextState: '3'
+        }
+      }
+    }),
+    actions: {
+      'o after d': function oAfterD(buffer, m) {
+        var ret;
+
+        if ((buffer.d || "").match(/^[0-9]+$/)) {
+          var tmp = buffer.d;
+          buffer.d = undefined;
+          ret = this['output'](buffer);
+          buffer.b = tmp;
+        } else {
+          ret = this['output'](buffer);
+        }
+
+        mhchemParser.actions['o='](buffer, m);
+        return ret;
+      },
+      'd= kv': function dKv(buffer, m) {
+        buffer.d = m;
+        buffer.dType = 'kv';
+      },
+      'charge or bond': function chargeOrBond(buffer, m) {
+        if (buffer['beginsWithBond']) {
+          /** @type {ParserOutput[]} */
+          var ret = [];
+          mhchemParser.concatArray(ret, this['output'](buffer));
+          mhchemParser.concatArray(ret, mhchemParser.actions['bond'](buffer, m, "-"));
+          return ret;
+        } else {
+          buffer.d = m;
+        }
+      },
+      '- after o/d': function afterOD(buffer, m, isAfterD) {
+        var c1 = mhchemParser.patterns.match_('orbital', buffer.o || "");
+        var c2 = mhchemParser.patterns.match_('one lowercase greek letter $', buffer.o || "");
+        var c3 = mhchemParser.patterns.match_('one lowercase latin letter $', buffer.o || "");
+        var c4 = mhchemParser.patterns.match_('$one lowercase latin letter$ $', buffer.o || "");
+        var hyphenFollows = m === "-" && (c1 && c1.remainder === "" || c2 || c3 || c4);
+
+        if (hyphenFollows && !buffer.a && !buffer.b && !buffer.p && !buffer.d && !buffer.q && !c1 && c3) {
+          buffer.o = '$' + buffer.o + '$';
+        }
+        /** @type {ParserOutput[]} */
+
+
+        var ret = [];
+
+        if (hyphenFollows) {
+          mhchemParser.concatArray(ret, this['output'](buffer));
+          ret.push({
+            type_: 'hyphen'
+          });
+        } else {
+          c1 = mhchemParser.patterns.match_('digits', buffer.d || "");
+
+          if (isAfterD && c1 && c1.remainder === '') {
+            mhchemParser.concatArray(ret, mhchemParser.actions['d='](buffer, m));
+            mhchemParser.concatArray(ret, this['output'](buffer));
+          } else {
+            mhchemParser.concatArray(ret, this['output'](buffer));
+            mhchemParser.concatArray(ret, mhchemParser.actions['bond'](buffer, m, "-"));
+          }
+        }
+
+        return ret;
+      },
+      'a to o': function aToO(buffer) {
+        buffer.o = buffer.a;
+        buffer.a = undefined;
+      },
+      'sb=true': function sbTrue(buffer) {
+        buffer.sb = true;
+      },
+      'sb=false': function sbFalse(buffer) {
+        buffer.sb = false;
+      },
+      'beginsWithBond=true': function beginsWithBondTrue(buffer) {
+        buffer['beginsWithBond'] = true;
+      },
+      'beginsWithBond=false': function beginsWithBondFalse(buffer) {
+        buffer['beginsWithBond'] = false;
+      },
+      'parenthesisLevel++': function parenthesisLevel(buffer) {
+        buffer['parenthesisLevel']++;
+      },
+      'parenthesisLevel--': function parenthesisLevel(buffer) {
+        buffer['parenthesisLevel']--;
+      },
+      'state of aggregation': function stateOfAggregation(buffer, m) {
+        return {
+          type_: 'state of aggregation',
+          p1: mhchemParser.go(m, 'o')
+        };
+      },
+      'comma': function comma(buffer, m) {
+        var a = m.replace(/\s*$/, '');
+        var withSpace = a !== m;
+
+        if (withSpace && buffer['parenthesisLevel'] === 0) {
+          return {
+            type_: 'comma enumeration L',
+            p1: a
+          };
+        } else {
+          return {
+            type_: 'comma enumeration M',
+            p1: a
+          };
+        }
+      },
+      'output': function output(buffer, m, entityFollows) {
+        // entityFollows:
+        //   undefined = if we have nothing else to output, also ignore the just read space (buffer.sb)
+        //   1 = an entity follows, never omit the space if there was one just read before (can only apply to state 1)
+        //   2 = 1 + the entity can have an amount, so output a\, instead of converting it to o (can only apply to states a|as)
+
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret;
+
+        if (!buffer.r) {
+          ret = [];
+
+          if (!buffer.a && !buffer.b && !buffer.p && !buffer.o && !buffer.q && !buffer.d && !entityFollows) ; else {
+            if (buffer.sb) {
+              ret.push({
+                type_: 'entitySkip'
+              });
+            }
+
+            if (!buffer.o && !buffer.q && !buffer.d && !buffer.b && !buffer.p && entityFollows !== 2) {
+              buffer.o = buffer.a;
+              buffer.a = undefined;
+            } else if (!buffer.o && !buffer.q && !buffer.d && (buffer.b || buffer.p)) {
+              buffer.o = buffer.a;
+              buffer.d = buffer.b;
+              buffer.q = buffer.p;
+              buffer.a = buffer.b = buffer.p = undefined;
+            } else {
+              if (buffer.o && buffer.dType === 'kv' && mhchemParser.patterns.match_('d-oxidation$', buffer.d || "")) {
+                buffer.dType = 'oxidation';
+              } else if (buffer.o && buffer.dType === 'kv' && !buffer.q) {
+                buffer.dType = undefined;
+              }
+            }
+
+            ret.push({
+              type_: 'chemfive',
+              a: mhchemParser.go(buffer.a, 'a'),
+              b: mhchemParser.go(buffer.b, 'bd'),
+              p: mhchemParser.go(buffer.p, 'pq'),
+              o: mhchemParser.go(buffer.o, 'o'),
+              q: mhchemParser.go(buffer.q, 'pq'),
+              d: mhchemParser.go(buffer.d, buffer.dType === 'oxidation' ? 'oxidation' : 'bd'),
+              dType: buffer.dType
+            });
+          }
+        } else {
+          // r
+
+          /** @type {ParserOutput[]} */
+          var rd;
+
+          if (buffer.rdt === 'M') {
+            rd = mhchemParser.go(buffer.rd, 'tex-math');
+          } else if (buffer.rdt === 'T') {
+            rd = [{
+              type_: 'text',
+              p1: buffer.rd || ""
+            }];
+          } else {
+            rd = mhchemParser.go(buffer.rd);
+          }
+          /** @type {ParserOutput[]} */
+
+
+          var rq;
+
+          if (buffer.rqt === 'M') {
+            rq = mhchemParser.go(buffer.rq, 'tex-math');
+          } else if (buffer.rqt === 'T') {
+            rq = [{
+              type_: 'text',
+              p1: buffer.rq || ""
+            }];
+          } else {
+            rq = mhchemParser.go(buffer.rq);
+          }
+
+          ret = {
+            type_: 'arrow',
+            r: buffer.r,
+            rd: rd,
+            rq: rq
+          };
+        }
+
+        for (var p in buffer) {
+          if (p !== 'parenthesisLevel' && p !== 'beginsWithBond') {
+            delete buffer[p];
+          }
+        }
+
+        return ret;
+      },
+      'oxidation-output': function oxidationOutput(buffer, m) {
+        var ret = ["{"];
+        mhchemParser.concatArray(ret, mhchemParser.go(m, 'oxidation'));
+        ret.push("}");
+        return ret;
+      },
+      'frac-output': function fracOutput(buffer, m) {
+        return {
+          type_: 'frac-ce',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'overset-output': function oversetOutput(buffer, m) {
+        return {
+          type_: 'overset',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'underset-output': function undersetOutput(buffer, m) {
+        return {
+          type_: 'underset',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'underbrace-output': function underbraceOutput(buffer, m) {
+        return {
+          type_: 'underbrace',
+          p1: mhchemParser.go(m[0]),
+          p2: mhchemParser.go(m[1])
+        };
+      },
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1])
+        };
+      },
+      'r=': function r(buffer, m) {
+        buffer.r = m;
+      },
+      'rdt=': function rdt(buffer, m) {
+        buffer.rdt = m;
+      },
+      'rd=': function rd(buffer, m) {
+        buffer.rd = m;
+      },
+      'rqt=': function rqt(buffer, m) {
+        buffer.rqt = m;
+      },
+      'rq=': function rq(buffer, m) {
+        buffer.rq = m;
+      },
+      'operator': function operator(buffer, m, p1) {
+        return {
+          type_: 'operator',
+          kind_: p1 || m
+        };
+      }
+    }
+  },
+  'a': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '1',
+          revisit: true
+        }
+      },
+      '$(...)$': {
+        '*': {
+          action_: 'tex-math tight',
+          nextState: '1'
+        }
+      },
+      ',': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'commaDecimal'
+          }
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {}
+  },
+  'o': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '1',
+          revisit: true
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '\\ca': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'circa'
+          }
+        }
+      },
+      '\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: '{text}'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {}
+  },
+  'text': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '{...}': {
+        '*': {
+          action_: 'text='
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '\\greek': {
+        '*': {
+          action_: ['output', 'rm']
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: ['output', 'copy']
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'text='
+        }
+      }
+    }),
+    actions: {
+      'output': function output(buffer) {
+        if (buffer.text_) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'text',
+            p1: buffer.text_
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  'pq': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'state of aggregation $': {
+        '*': {
+          action_: 'state of aggregation'
+        }
+      },
+      'i$': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '(KV letters),': {
+        '0': {
+          action_: 'rm',
+          nextState: '0'
+        }
+      },
+      'formula$': {
+        '0': {
+          nextState: 'f',
+          revisit: true
+        }
+      },
+      '1/2$': {
+        '0': {
+          action_: '1/2'
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: 'text'
+        }
+      },
+      'a-z': {
+        'f': {
+          action_: 'tex-math'
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '-9.,9': {
+        '*': {
+          action_: '9,9'
+        }
+      },
+      ',': {
+        '*': {
+          action_: {
+            type_: 'insert+p1',
+            option: 'comma enumeration S'
+          }
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: 'color-output'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: 'color0-output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: 'ce'
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'state of aggregation': function stateOfAggregation(buffer, m) {
+        return {
+          type_: 'state of aggregation subscript',
+          p1: mhchemParser.go(m, 'o')
+        };
+      },
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1], 'pq')
+        };
+      }
+    }
+  },
+  'bd': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'x$': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      'formula$': {
+        '0': {
+          nextState: 'f',
+          revisit: true
+        }
+      },
+      'else': {
+        '0': {
+          nextState: '!f',
+          revisit: true
+        }
+      },
+      '-9.,9 no missing 0': {
+        '*': {
+          action_: '9,9'
+        }
+      },
+      '.': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'electron dot'
+          }
+        }
+      },
+      'a-z': {
+        'f': {
+          action_: 'tex-math'
+        }
+      },
+      'x': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'KV x'
+          }
+        }
+      },
+      'letters': {
+        '*': {
+          action_: 'rm'
+        }
+      },
+      '\'': {
+        '*': {
+          action_: {
+            type_: 'insert',
+            option: 'prime'
+          }
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      '{(...)}': {
+        '*': {
+          action_: 'text'
+        }
+      },
+      '\\color{(...)}{(...)}1|\\color(...){(...)}2': {
+        '*': {
+          action_: 'color-output'
+        }
+      },
+      '\\color{(...)}0': {
+        '*': {
+          action_: 'color0-output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: 'ce'
+        }
+      },
+      '\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'copy'
+        }
+      },
+      'else2': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'color-output': function colorOutput(buffer, m) {
+        return {
+          type_: 'color',
+          color1: m[0],
+          color2: mhchemParser.go(m[1], 'bd')
+        };
+      }
+    }
+  },
+  'oxidation': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      'roman numeral': {
+        '*': {
+          action_: 'roman-numeral'
+        }
+      },
+      '${(...)}$|$(...)$': {
+        '*': {
+          action_: 'tex-math'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'roman-numeral': function romanNumeral(buffer, m) {
+        return {
+          type_: 'roman numeral',
+          p1: m || ""
+        };
+      }
+    }
+  },
+  'tex-math': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: ['output', 'ce']
+        }
+      },
+      '{...}|\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'o='
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'o='
+        }
+      }
+    }),
+    actions: {
+      'output': function output(buffer) {
+        if (buffer.o) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'tex-math',
+            p1: buffer.o
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  'tex-math tight': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '\\ce{(...)}': {
+        '*': {
+          action_: ['output', 'ce']
+        }
+      },
+      '{...}|\\,|\\x{}{}|\\x{}|\\x': {
+        '*': {
+          action_: 'o='
+        }
+      },
+      '-|+': {
+        '*': {
+          action_: 'tight operator'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'o='
+        }
+      }
+    }),
+    actions: {
+      'tight operator': function tightOperator(buffer, m) {
+        buffer.o = (buffer.o || "") + "{" + m + "}";
+      },
+      'output': function output(buffer) {
+        if (buffer.o) {
+          /** @type {ParserOutput} */
+          var ret = {
+            type_: 'tex-math',
+            p1: buffer.o
+          };
+
+          for (var p in buffer) {
+            delete buffer[p];
+          }
+
+          return ret;
+        }
+      }
+    }
+  },
+  '9,9': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {}
+      },
+      ',': {
+        '*': {
+          action_: 'comma'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'copy'
+        }
+      }
+    }),
+    actions: {
+      'comma': function comma() {
+        return {
+          type_: 'commaDecimal'
+        };
+      }
+    }
+  },
+  //#endregion
+  //
+  // \pu state machines
+  //
+  //#region pu
+  'pu': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      'space$': {
+        '*': {
+          action_: ['output', 'space']
+        }
+      },
+      '{[(|)]}': {
+        '0|a': {
+          action_: 'copy'
+        }
+      },
+      '(-)(9)^(-9)': {
+        '0': {
+          action_: 'number^',
+          nextState: 'a'
+        }
+      },
+      '(-)(9.,9)(e)(99)': {
+        '0': {
+          action_: 'enumber',
+          nextState: 'a'
+        }
+      },
+      'space': {
+        '0|a': {}
+      },
+      'pm-operator': {
+        '0|a': {
+          action_: {
+            type_: 'operator',
+            option: '\\pm'
+          },
+          nextState: '0'
+        }
+      },
+      'operator': {
+        '0|a': {
+          action_: 'copy',
+          nextState: '0'
+        }
+      },
+      '//': {
+        'd': {
+          action_: 'o=',
+          nextState: '/'
+        }
+      },
+      '/': {
+        'd': {
+          action_: 'o=',
+          nextState: '/'
+        }
+      },
+      '{...}|else': {
+        '0|d': {
+          action_: 'd=',
+          nextState: 'd'
+        },
+        'a': {
+          action_: ['space', 'd='],
+          nextState: 'd'
+        },
+        '/|q': {
+          action_: 'q=',
+          nextState: 'q'
+        }
+      }
+    }),
+    actions: {
+      'enumber': function enumber(buffer, m) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+
+        if (m[0] === "+-" || m[0] === "+/-") {
+          ret.push("\\pm ");
+        } else if (m[0]) {
+          ret.push(m[0]);
+        }
+
+        if (m[1]) {
+          mhchemParser.concatArray(ret, mhchemParser.go(m[1], 'pu-9,9'));
+
+          if (m[2]) {
+            if (m[2].match(/[,.]/)) {
+              mhchemParser.concatArray(ret, mhchemParser.go(m[2], 'pu-9,9'));
+            } else {
+              ret.push(m[2]);
+            }
+          }
+
+          m[3] = m[4] || m[3];
+
+          if (m[3]) {
+            m[3] = m[3].trim();
+
+            if (m[3] === "e" || m[3].substr(0, 1) === "*") {
+              ret.push({
+                type_: 'cdot'
+              });
+            } else {
+              ret.push({
+                type_: 'times'
+              });
+            }
+          }
+        }
+
+        if (m[3]) {
+          ret.push("10^{" + m[5] + "}");
+        }
+
+        return ret;
+      },
+      'number^': function number(buffer, m) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+
+        if (m[0] === "+-" || m[0] === "+/-") {
+          ret.push("\\pm ");
+        } else if (m[0]) {
+          ret.push(m[0]);
+        }
+
+        mhchemParser.concatArray(ret, mhchemParser.go(m[1], 'pu-9,9'));
+        ret.push("^{" + m[2] + "}");
+        return ret;
+      },
+      'operator': function operator(buffer, m, p1) {
+        return {
+          type_: 'operator',
+          kind_: p1 || m
+        };
+      },
+      'space': function space() {
+        return {
+          type_: 'pu-space-1'
+        };
+      },
+      'output': function output(buffer) {
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret;
+        var md = mhchemParser.patterns.match_('{(...)}', buffer.d || "");
+
+        if (md && md.remainder === '') {
+          buffer.d = md.match_;
+        }
+
+        var mq = mhchemParser.patterns.match_('{(...)}', buffer.q || "");
+
+        if (mq && mq.remainder === '') {
+          buffer.q = mq.match_;
+        }
+
+        if (buffer.d) {
+          buffer.d = buffer.d.replace(/\u00B0C|\^oC|\^{o}C/g, "{}^{\\circ}C");
+          buffer.d = buffer.d.replace(/\u00B0F|\^oF|\^{o}F/g, "{}^{\\circ}F");
+        }
+
+        if (buffer.q) {
+          // fraction
+          buffer.q = buffer.q.replace(/\u00B0C|\^oC|\^{o}C/g, "{}^{\\circ}C");
+          buffer.q = buffer.q.replace(/\u00B0F|\^oF|\^{o}F/g, "{}^{\\circ}F");
+          var b5 = {
+            d: mhchemParser.go(buffer.d, 'pu'),
+            q: mhchemParser.go(buffer.q, 'pu')
+          };
+
+          if (buffer.o === '//') {
+            ret = {
+              type_: 'pu-frac',
+              p1: b5.d,
+              p2: b5.q
+            };
+          } else {
+            ret = b5.d;
+
+            if (b5.d.length > 1 || b5.q.length > 1) {
+              ret.push({
+                type_: ' / '
+              });
+            } else {
+              ret.push({
+                type_: '/'
+              });
+            }
+
+            mhchemParser.concatArray(ret, b5.q);
+          }
+        } else {
+          // no fraction
+          ret = mhchemParser.go(buffer.d, 'pu-2');
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    }
+  },
+  'pu-2': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '*': {
+          action_: 'output'
+        }
+      },
+      '*': {
+        '*': {
+          action_: ['output', 'cdot'],
+          nextState: '0'
+        }
+      },
+      '\\x': {
+        '*': {
+          action_: 'rm='
+        }
+      },
+      'space': {
+        '*': {
+          action_: ['output', 'space'],
+          nextState: '0'
+        }
+      },
+      '^{(...)}|^(-1)': {
+        '1': {
+          action_: '^(-1)'
+        }
+      },
+      '-9.,9': {
+        '0': {
+          action_: 'rm=',
+          nextState: '0'
+        },
+        '1': {
+          action_: '^(-1)',
+          nextState: '0'
+        }
+      },
+      '{...}|else': {
+        '*': {
+          action_: 'rm=',
+          nextState: '1'
+        }
+      }
+    }),
+    actions: {
+      'cdot': function cdot() {
+        return {
+          type_: 'tight cdot'
+        };
+      },
+      '^(-1)': function _(buffer, m) {
+        buffer.rm += "^{" + m + "}";
+      },
+      'space': function space() {
+        return {
+          type_: 'pu-space-2'
+        };
+      },
+      'output': function output(buffer) {
+        /** @type {ParserOutput | ParserOutput[]} */
+        var ret = [];
+
+        if (buffer.rm) {
+          var mrm = mhchemParser.patterns.match_('{(...)}', buffer.rm || "");
+
+          if (mrm && mrm.remainder === '') {
+            ret = mhchemParser.go(mrm.match_, 'pu');
+          } else {
+            ret = {
+              type_: 'rm',
+              p1: buffer.rm
+            };
+          }
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    }
+  },
+  'pu-9,9': {
+    transitions: mhchemParser.createTransitions({
+      'empty': {
+        '0': {
+          action_: 'output-0'
+        },
+        'o': {
+          action_: 'output-o'
+        }
+      },
+      ',': {
+        '0': {
+          action_: ['output-0', 'comma'],
+          nextState: 'o'
+        }
+      },
+      '.': {
+        '0': {
+          action_: ['output-0', 'copy'],
+          nextState: 'o'
+        }
+      },
+      'else': {
+        '*': {
+          action_: 'text='
+        }
+      }
+    }),
+    actions: {
+      'comma': function comma() {
+        return {
+          type_: 'commaDecimal'
+        };
+      },
+      'output-0': function output0(buffer) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+        buffer.text_ = buffer.text_ || "";
+
+        if (buffer.text_.length > 4) {
+          var a = buffer.text_.length % 3;
+
+          if (a === 0) {
+            a = 3;
+          }
+
+          for (var i = buffer.text_.length - 3; i > 0; i -= 3) {
+            ret.push(buffer.text_.substr(i, 3));
+            ret.push({
+              type_: '1000 separator'
+            });
+          }
+
+          ret.push(buffer.text_.substr(0, a));
+          ret.reverse();
+        } else {
+          ret.push(buffer.text_);
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      },
+      'output-o': function outputO(buffer) {
+        /** @type {ParserOutput[]} */
+        var ret = [];
+        buffer.text_ = buffer.text_ || "";
+
+        if (buffer.text_.length > 4) {
+          var a = buffer.text_.length - 3;
+
+          for (var i = 0; i < a; i += 3) {
+            ret.push(buffer.text_.substr(i, 3));
+            ret.push({
+              type_: '1000 separator'
+            });
+          }
+
+          ret.push(buffer.text_.substr(i));
+        } else {
+          ret.push(buffer.text_);
+        }
+
+        for (var p in buffer) {
+          delete buffer[p];
+        }
+
+        return ret;
+      }
+    } //#endregion
+
+  }
+}; //
+// texify: Take MhchemParser output and convert it to TeX
+//
+
+/** @type {Texify} */
+
+var texify = {
+  go: function go(input, isInner) {
+    // (recursive, max 4 levels)
+    if (!input) {
+      return "";
+    }
+
+    var res = "";
+    var cee = false;
+
+    for (var i = 0; i < input.length; i++) {
+      var inputi = input[i];
+
+      if (typeof inputi === "string") {
+        res += inputi;
+      } else {
+        res += texify._go2(inputi);
+
+        if (inputi.type_ === '1st-level escape') {
+          cee = true;
+        }
+      }
+    }
+
+    if (!isInner && !cee && res) {
+      res = "{" + res + "}";
+    }
+
+    return res;
+  },
+  _goInner: function _goInner(input) {
+    if (!input) {
+      return input;
+    }
+
+    return texify.go(input, true);
+  },
+  _go2: function _go2(buf) {
+    /** @type {undefined | string} */
+    var res;
+
+    switch (buf.type_) {
+      case 'chemfive':
+        res = "";
+        var b5 = {
+          a: texify._goInner(buf.a),
+          b: texify._goInner(buf.b),
+          p: texify._goInner(buf.p),
+          o: texify._goInner(buf.o),
+          q: texify._goInner(buf.q),
+          d: texify._goInner(buf.d)
+        }; //
+        // a
+        //
+
+        if (b5.a) {
+          if (b5.a.match(/^[+\-]/)) {
+            b5.a = "{" + b5.a + "}";
+          }
+
+          res += b5.a + "\\,";
+        } //
+        // b and p
+        //
+
+
+        if (b5.b || b5.p) {
+          res += "{\\vphantom{X}}";
+          res += "^{\\hphantom{" + (b5.b || "") + "}}_{\\hphantom{" + (b5.p || "") + "}}";
+          res += "{\\vphantom{X}}";
+          res += "^{\\smash[t]{\\vphantom{2}}\\mathllap{" + (b5.b || "") + "}}";
+          res += "_{\\vphantom{2}\\mathllap{\\smash[t]{" + (b5.p || "") + "}}}";
+        } //
+        // o
+        //
+
+
+        if (b5.o) {
+          if (b5.o.match(/^[+\-]/)) {
+            b5.o = "{" + b5.o + "}";
+          }
+
+          res += b5.o;
+        } //
+        // q and d
+        //
+
+
+        if (buf.dType === 'kv') {
+          if (b5.d || b5.q) {
+            res += "{\\vphantom{X}}";
+          }
+
+          if (b5.d) {
+            res += "^{" + b5.d + "}";
+          }
+
+          if (b5.q) {
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+        } else if (buf.dType === 'oxidation') {
+          if (b5.d) {
+            res += "{\\vphantom{X}}";
+            res += "^{" + b5.d + "}";
+          }
+
+          if (b5.q) {
+            res += "{\\vphantom{X}}";
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+        } else {
+          if (b5.q) {
+            res += "{\\vphantom{X}}";
+            res += "_{\\smash[t]{" + b5.q + "}}";
+          }
+
+          if (b5.d) {
+            res += "{\\vphantom{X}}";
+            res += "^{" + b5.d + "}";
+          }
+        }
+
+        break;
+
+      case 'rm':
+        res = "\\mathrm{" + buf.p1 + "}";
+        break;
+
+      case 'text':
+        if (buf.p1.match(/[\^_]/)) {
+          buf.p1 = buf.p1.replace(" ", "~").replace("-", "\\text{-}");
+          res = "\\mathrm{" + buf.p1 + "}";
+        } else {
+          res = "\\text{" + buf.p1 + "}";
+        }
+
+        break;
+
+      case 'roman numeral':
+        res = "\\mathrm{" + buf.p1 + "}";
+        break;
+
+      case 'state of aggregation':
+        res = "\\mskip2mu " + texify._goInner(buf.p1);
+        break;
+
+      case 'state of aggregation subscript':
+        res = "\\mskip1mu " + texify._goInner(buf.p1);
+        break;
+
+      case 'bond':
+        res = texify._getBond(buf.kind_);
+
+        if (!res) {
+          throw ["MhchemErrorBond", "mhchem Error. Unknown bond type (" + buf.kind_ + ")"];
+        }
+
+        break;
+
+      case 'frac':
+        var c = "\\frac{" + buf.p1 + "}{" + buf.p2 + "}";
+        res = "\\mathchoice{\\textstyle" + c + "}{" + c + "}{" + c + "}{" + c + "}";
+        break;
+
+      case 'pu-frac':
+        var d = "\\frac{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        res = "\\mathchoice{\\textstyle" + d + "}{" + d + "}{" + d + "}{" + d + "}";
+        break;
+
+      case 'tex-math':
+        res = buf.p1 + " ";
+        break;
+
+      case 'frac-ce':
+        res = "\\frac{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'overset':
+        res = "\\overset{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'underset':
+        res = "\\underset{" + texify._goInner(buf.p1) + "}{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'underbrace':
+        res = "\\underbrace{" + texify._goInner(buf.p1) + "}_{" + texify._goInner(buf.p2) + "}";
+        break;
+
+      case 'color':
+        res = "{\\color{" + buf.color1 + "}{" + texify._goInner(buf.color2) + "}}";
+        break;
+
+      case 'color0':
+        res = "\\color{" + buf.color + "}";
+        break;
+
+      case 'arrow':
+        var b6 = {
+          rd: texify._goInner(buf.rd),
+          rq: texify._goInner(buf.rq)
+        };
+
+        var arrow = "\\x" + texify._getArrow(buf.r);
+
+        if (b6.rq) {
+          arrow += "[{" + b6.rq + "}]";
+        }
+
+        if (b6.rd) {
+          arrow += "{" + b6.rd + "}";
+        } else {
+          arrow += "{}";
+        }
+
+        res = arrow;
+        break;
+
+      case 'operator':
+        res = texify._getOperator(buf.kind_);
+        break;
+
+      case '1st-level escape':
+        res = buf.p1 + " "; // &, \\\\, \\hlin
+
+        break;
+
+      case 'space':
+        res = " ";
+        break;
+
+      case 'entitySkip':
+        res = "~";
+        break;
+
+      case 'pu-space-1':
+        res = "~";
+        break;
+
+      case 'pu-space-2':
+        res = "\\mkern3mu ";
+        break;
+
+      case '1000 separator':
+        res = "\\mkern2mu ";
+        break;
+
+      case 'commaDecimal':
+        res = "{,}";
+        break;
+
+      case 'comma enumeration L':
+        res = "{" + buf.p1 + "}\\mkern6mu ";
+        break;
+
+      case 'comma enumeration M':
+        res = "{" + buf.p1 + "}\\mkern3mu ";
+        break;
+
+      case 'comma enumeration S':
+        res = "{" + buf.p1 + "}\\mkern1mu ";
+        break;
+
+      case 'hyphen':
+        res = "\\text{-}";
+        break;
+
+      case 'addition compound':
+        res = "\\,{\\cdot}\\,";
+        break;
+
+      case 'electron dot':
+        res = "\\mkern1mu \\bullet\\mkern1mu ";
+        break;
+
+      case 'KV x':
+        res = "{\\times}";
+        break;
+
+      case 'prime':
+        res = "\\prime ";
+        break;
+
+      case 'cdot':
+        res = "\\cdot ";
+        break;
+
+      case 'tight cdot':
+        res = "\\mkern1mu{\\cdot}\\mkern1mu ";
+        break;
+
+      case 'times':
+        res = "\\times ";
+        break;
+
+      case 'circa':
+        res = "{\\sim}";
+        break;
+
+      case '^':
+        res = "uparrow";
+        break;
+
+      case 'v':
+        res = "downarrow";
+        break;
+
+      case 'ellipsis':
+        res = "\\ldots ";
+        break;
+
+      case '/':
+        res = "/";
+        break;
+
+      case ' / ':
+        res = "\\,/\\,";
+        break;
+
+      default:
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+      // Missing texify rule or unknown MhchemParser output
+    }
+    return res;
+  },
+  _getArrow: function _getArrow(a) {
+    switch (a) {
+      case "->":
+        return "rightarrow";
+
+      case "\u2192":
+        return "rightarrow";
+
+      case "\u27F6":
+        return "rightarrow";
+
+      case "<-":
+        return "leftarrow";
+
+      case "<->":
+        return "leftrightarrow";
+
+      case "<-->":
+        return "rightleftarrows";
+
+      case "<=>":
+        return "rightleftharpoons";
+
+      case "\u21CC":
+        return "rightleftharpoons";
+
+      case "<=>>":
+        return "rightequilibrium";
+
+      case "<<=>":
+        return "leftequilibrium";
+
+      default:
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  },
+  _getBond: function _getBond(a) {
+    switch (a) {
+      case "-":
+        return "{-}";
+
+      case "1":
+        return "{-}";
+
+      case "=":
+        return "{=}";
+
+      case "2":
+        return "{=}";
+
+      case "#":
+        return "{\\equiv}";
+
+      case "3":
+        return "{\\equiv}";
+
+      case "~":
+        return "{\\tripledash}";
+
+      case "~-":
+        return "{\\mathrlap{\\raisebox{-.1em}{$-$}}\\raisebox{.1em}{$\\tripledash$}}";
+
+      case "~=":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";
+
+      case "~--":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$\\tripledash$}}-}";
+
+      case "-~-":
+        return "{\\mathrlap{\\raisebox{-.2em}{$-$}}\\mathrlap{\\raisebox{.2em}{$-$}}\\tripledash}";
+
+      case "...":
+        return "{{\\cdot}{\\cdot}{\\cdot}}";
+
+      case "....":
+        return "{{\\cdot}{\\cdot}{\\cdot}{\\cdot}}";
+
+      case "->":
+        return "{\\rightarrow}";
+
+      case "<-":
+        return "{\\leftarrow}";
+
+      case "<":
+        return "{<}";
+
+      case ">":
+        return "{>}";
+
+      default:
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  },
+  _getOperator: function _getOperator(a) {
+    switch (a) {
+      case "+":
+        return " {}+{} ";
+
+      case "-":
+        return " {}-{} ";
+
+      case "=":
+        return " {}={} ";
+
+      case "<":
+        return " {}<{} ";
+
+      case ">":
+        return " {}>{} ";
+
+      case "<<":
+        return " {}\\ll{} ";
+
+      case ">>":
+        return " {}\\gg{} ";
+
+      case "\\pm":
+        return " {}\\pm{} ";
+
+      case "\\approx":
+        return " {}\\approx{} ";
+
+      case "$\\approx$":
+        return " {}\\approx{} ";
+
+      case "v":
+        return " \\downarrow{} ";
+
+      case "(v)":
+        return " \\downarrow{} ";
+
+      case "^":
+        return " \\uparrow{} ";
+
+      case "(^)":
+        return " \\uparrow{} ";
+
+      default:
+        throw ["MhchemBugT", "mhchem bug T. Please report."];
+    }
+  }
+}; //
diff --git a/katex.bk/contrib/render-a11y-string.js b/katex.bk/contrib/render-a11y-string.js
new file mode 100644
index 00000000..d8b27c77
--- /dev/null
+++ b/katex.bk/contrib/render-a11y-string.js
@@ -0,0 +1,870 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory(require("katex"));
+	else if(typeof define === 'function' && define.amd)
+		define(["katex"], factory);
+	else {
+		var a = typeof exports === 'object' ? factory(require("katex")) : factory(root["katex"]);
+		for(var i in a) (typeof exports === 'object' ? exports : root)[i] = a[i];
+	}
+})((typeof self !== 'undefined' ? self : this), function(__WEBPACK_EXTERNAL_MODULE__0__) {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
+/******/ 	function __webpack_require__(moduleId) {
+/******/
+/******/ 		// Check if module is in cache
+/******/ 		if(installedModules[moduleId]) {
+/******/ 			return installedModules[moduleId].exports;
+/******/ 		}
+/******/ 		// Create a new module (and put it into the cache)
+/******/ 		var module = installedModules[moduleId] = {
+/******/ 			i: moduleId,
+/******/ 			l: false,
+/******/ 			exports: {}
+/******/ 		};
+/******/
+/******/ 		// Execute the module function
+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
+/******/ 		// Flag the module as loaded
+/******/ 		module.l = true;
+/******/
+/******/ 		// Return the exports of the module
+/******/ 		return module.exports;
+/******/ 	}
+/******/
+/******/
+/******/ 	// expose the modules object (__webpack_modules__)
+/******/ 	__webpack_require__.m = modules;
+/******/
+/******/ 	// expose the module cache
+/******/ 	__webpack_require__.c = installedModules;
+/******/
+/******/ 	// define getter function for harmony exports
+/******/ 	__webpack_require__.d = function(exports, name, getter) {
+/******/ 		if(!__webpack_require__.o(exports, name)) {
+/******/ 			Object.defineProperty(exports, name, { enumerable: true, get: getter });
+/******/ 		}
+/******/ 	};
+/******/
+/******/ 	// define __esModule on exports
+/******/ 	__webpack_require__.r = function(exports) {
+/******/ 		if(typeof Symbol !== 'undefined' && Symbol.toStringTag) {
+/******/ 			Object.defineProperty(exports, Symbol.toStringTag, { value: 'Module' });
+/******/ 		}
+/******/ 		Object.defineProperty(exports, '__esModule', { value: true });
+/******/ 	};
+/******/
+/******/ 	// create a fake namespace object
+/******/ 	// mode & 1: value is a module id, require it
+/******/ 	// mode & 2: merge all properties of value into the ns
+/******/ 	// mode & 4: return value when already ns object
+/******/ 	// mode & 8|1: behave like require
+/******/ 	__webpack_require__.t = function(value, mode) {
+/******/ 		if(mode & 1) value = __webpack_require__(value);
+/******/ 		if(mode & 8) return value;
+/******/ 		if((mode & 4) && typeof value === 'object' && value && value.__esModule) return value;
+/******/ 		var ns = Object.create(null);
+/******/ 		__webpack_require__.r(ns);
+/******/ 		Object.defineProperty(ns, 'default', { enumerable: true, value: value });
+/******/ 		if(mode & 2 && typeof value != 'string') for(var key in value) __webpack_require__.d(ns, key, function(key) { return value[key]; }.bind(null, key));
+/******/ 		return ns;
+/******/ 	};
+/******/
+/******/ 	// getDefaultExport function for compatibility with non-harmony modules
+/******/ 	__webpack_require__.n = function(module) {
+/******/ 		var getter = module && module.__esModule ?
+/******/ 			function getDefault() { return module['default']; } :
+/******/ 			function getModuleExports() { return module; };
+/******/ 		__webpack_require__.d(getter, 'a', getter);
+/******/ 		return getter;
+/******/ 	};
+/******/
+/******/ 	// Object.prototype.hasOwnProperty.call
+/******/ 	__webpack_require__.o = function(object, property) { return Object.prototype.hasOwnProperty.call(object, property); };
+/******/
+/******/ 	// __webpack_public_path__
+/******/ 	__webpack_require__.p = "";
+/******/
+/******/
+/******/ 	// Load entry module and return exports
+/******/ 	return __webpack_require__(__webpack_require__.s = 1);
+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports) {
+
+module.exports = __WEBPACK_EXTERNAL_MODULE__0__;
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0__ = __webpack_require__(0);
+/* harmony import */ var katex__WEBPACK_IMPORTED_MODULE_0___default = /*#__PURE__*/__webpack_require__.n(katex__WEBPACK_IMPORTED_MODULE_0__);
+/**
+ * renderA11yString returns a readable string.
+ *
+ * In some cases the string will have the proper semantic math
+ * meaning,:
+ *   renderA11yString("\\frac{1}{2}"")
+ *   -> "start fraction, 1, divided by, 2, end fraction"
+ *
+ * However, other cases do not:
+ *   renderA11yString("f(x) = x^2")
+ *   -> "f, left parenthesis, x, right parenthesis, equals, x, squared"
+ *
+ * The commas in the string aim to increase ease of understanding
+ * when read by a screenreader.
+ */
+// NOTE: since we're importing types here these files won't actually be
+// included in the build.
+// $FlowIgnore: we import the types directly anyways
+
+var stringMap = {
+  "(": "left parenthesis",
+  ")": "right parenthesis",
+  "[": "open bracket",
+  "]": "close bracket",
+  "\\{": "left brace",
+  "\\}": "right brace",
+  "\\lvert": "open vertical bar",
+  "\\rvert": "close vertical bar",
+  "|": "vertical bar",
+  "\\uparrow": "up arrow",
+  "\\Uparrow": "up arrow",
+  "\\downarrow": "down arrow",
+  "\\Downarrow": "down arrow",
+  "\\updownarrow": "up down arrow",
+  "\\leftarrow": "left arrow",
+  "\\Leftarrow": "left arrow",
+  "\\rightarrow": "right arrow",
+  "\\Rightarrow": "right arrow",
+  "\\langle": "open angle",
+  "\\rangle": "close angle",
+  "\\lfloor": "open floor",
+  "\\rfloor": "close floor",
+  "\\int": "integral",
+  "\\intop": "integral",
+  "\\lim": "limit",
+  "\\ln": "natural log",
+  "\\log": "log",
+  "\\sin": "sine",
+  "\\cos": "cosine",
+  "\\tan": "tangent",
+  "\\cot": "cotangent",
+  "\\sum": "sum",
+  "/": "slash",
+  ",": "comma",
+  ".": "point",
+  "-": "negative",
+  "+": "plus",
+  "~": "tilde",
+  ":": "colon",
+  "?": "question mark",
+  "'": "apostrophe",
+  "\\%": "percent",
+  " ": "space",
+  "\\ ": "space",
+  "\\$": "dollar sign",
+  "\\angle": "angle",
+  "\\degree": "degree",
+  "\\circ": "circle",
+  "\\vec": "vector",
+  "\\triangle": "triangle",
+  "\\pi": "pi",
+  "\\prime": "prime",
+  "\\infty": "infinity",
+  "\\alpha": "alpha",
+  "\\beta": "beta",
+  "\\gamma": "gamma",
+  "\\omega": "omega",
+  "\\theta": "theta",
+  "\\sigma": "sigma",
+  "\\lambda": "lambda",
+  "\\tau": "tau",
+  "\\Delta": "delta",
+  "\\delta": "delta",
+  "\\mu": "mu",
+  "\\rho": "rho",
+  "\\nabla": "del",
+  "\\ell": "ell",
+  "\\ldots": "dots",
+  // TODO: add entries for all accents
+  "\\hat": "hat",
+  "\\acute": "acute"
+};
+var powerMap = {
+  "prime": "prime",
+  "degree": "degrees",
+  "circle": "degrees",
+  "2": "squared",
+  "3": "cubed"
+};
+var openMap = {
+  "|": "open vertical bar",
+  ".": ""
+};
+var closeMap = {
+  "|": "close vertical bar",
+  ".": ""
+};
+var binMap = {
+  "+": "plus",
+  "-": "minus",
+  "\\pm": "plus minus",
+  "\\cdot": "dot",
+  "*": "times",
+  "/": "divided by",
+  "\\times": "times",
+  "\\div": "divided by",
+  "\\circ": "circle",
+  "\\bullet": "bullet"
+};
+var relMap = {
+  "=": "equals",
+  "\\approx": "approximately equals",
+  "≠": "does not equal",
+  "\\geq": "is greater than or equal to",
+  "\\ge": "is greater than or equal to",
+  "\\leq": "is less than or equal to",
+  "\\le": "is less than or equal to",
+  ">": "is greater than",
+  "<": "is less than",
+  "\\leftarrow": "left arrow",
+  "\\Leftarrow": "left arrow",
+  "\\rightarrow": "right arrow",
+  "\\Rightarrow": "right arrow",
+  ":": "colon"
+};
+var accentUnderMap = {
+  "\\underleftarrow": "left arrow",
+  "\\underrightarrow": "right arrow",
+  "\\underleftrightarrow": "left-right arrow",
+  "\\undergroup": "group",
+  "\\underlinesegment": "line segment",
+  "\\utilde": "tilde"
+};
+
+var buildString = function buildString(str, type, a11yStrings) {
+  if (!str) {
+    return;
+  }
+
+  var ret;
+
+  if (type === "open") {
+    ret = str in openMap ? openMap[str] : stringMap[str] || str;
+  } else if (type === "close") {
+    ret = str in closeMap ? closeMap[str] : stringMap[str] || str;
+  } else if (type === "bin") {
+    ret = binMap[str] || str;
+  } else if (type === "rel") {
+    ret = relMap[str] || str;
+  } else {
+    ret = stringMap[str] || str;
+  } // If the text to add is a number and there is already a string
+  // in the list and the last string is a number then we should
+  // combine them into a single number
+
+
+  if (/^\d+$/.test(ret) && a11yStrings.length > 0 && // TODO(kevinb): check that the last item in a11yStrings is a string
+  // I think we might be able to drop the nested arrays, which would make
+  // this easier to type - $FlowFixMe
+  /^\d+$/.test(a11yStrings[a11yStrings.length - 1])) {
+    a11yStrings[a11yStrings.length - 1] += ret;
+  } else if (ret) {
+    a11yStrings.push(ret);
+  }
+};
+
+var buildRegion = function buildRegion(a11yStrings, callback) {
+  var regionStrings = [];
+  a11yStrings.push(regionStrings);
+  callback(regionStrings);
+};
+
+var handleObject = function handleObject(tree, a11yStrings, atomType) {
+  // Everything else is assumed to be an object...
+  switch (tree.type) {
+    case "accent":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          buildA11yStrings(tree.base, a11yStrings, atomType);
+          a11yStrings.push("with");
+          buildString(tree.label, "normal", a11yStrings);
+          a11yStrings.push("on top");
+        });
+        break;
+      }
+
+    case "accentUnder":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          buildA11yStrings(tree.base, a11yStrings, atomType);
+          a11yStrings.push("with");
+          buildString(accentUnderMap[tree.label], "normal", a11yStrings);
+          a11yStrings.push("underneath");
+        });
+        break;
+      }
+
+    case "accent-token":
+      {
+        // Used internally by accent symbols.
+        break;
+      }
+
+    case "atom":
+      {
+        var text = tree.text;
+
+        switch (tree.family) {
+          case "bin":
+            {
+              buildString(text, "bin", a11yStrings);
+              break;
+            }
+
+          case "close":
+            {
+              buildString(text, "close", a11yStrings);
+              break;
+            }
+          // TODO(kevinb): figure out what should be done for inner
+
+          case "inner":
+            {
+              buildString(tree.text, "inner", a11yStrings);
+              break;
+            }
+
+          case "open":
+            {
+              buildString(text, "open", a11yStrings);
+              break;
+            }
+
+          case "punct":
+            {
+              buildString(text, "punct", a11yStrings);
+              break;
+            }
+
+          case "rel":
+            {
+              buildString(text, "rel", a11yStrings);
+              break;
+            }
+
+          default:
+            {
+              tree.family;
+              throw new Error("\"" + tree.family + "\" is not a valid atom type");
+            }
+        }
+
+        break;
+      }
+
+    case "color":
+      {
+        var color = tree.color.replace(/katex-/, "");
+        buildRegion(a11yStrings, function (regionStrings) {
+          regionStrings.push("start color " + color);
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          regionStrings.push("end color " + color);
+        });
+        break;
+      }
+
+    case "color-token":
+      {
+        // Used by \color, \colorbox, and \fcolorbox but not directly rendered.
+        // It's a leaf node and has no children so just break.
+        break;
+      }
+
+    case "delimsizing":
+      {
+        if (tree.delim && tree.delim !== ".") {
+          buildString(tree.delim, "normal", a11yStrings);
+        }
+
+        break;
+      }
+
+    case "genfrac":
+      {
+        buildRegion(a11yStrings, function (regionStrings) {
+          // genfrac can have unbalanced delimiters
+          var leftDelim = tree.leftDelim,
+              rightDelim = tree.rightDelim; // NOTE: Not sure if this is a safe assumption
+          // hasBarLine true -> fraction, false -> binomial
+
+          if (tree.hasBarLine) {
+            regionStrings.push("start fraction");
+            leftDelim && buildString(leftDelim, "open", regionStrings);
+            buildA11yStrings(tree.numer, regionStrings, atomType);
+            regionStrings.push("divided by");
+            buildA11yStrings(tree.denom, regionStrings, atomType);
+            rightDelim && buildString(rightDelim, "close", regionStrings);
+            regionStrings.push("end fraction");
+          } else {
+            regionStrings.push("start binomial");
+            leftDelim && buildString(leftDelim, "open", regionStrings);
+            buildA11yStrings(tree.numer, regionStrings, atomType);
+            regionStrings.push("over");
+            buildA11yStrings(tree.denom, regionStrings, atomType);
+            rightDelim && buildString(rightDelim, "close", regionStrings);
+            regionStrings.push("end binomial");
+          }
+        });
+        break;
+      }
+
+    case "kern":
+      {
+        // No op: we don't attempt to present kerning information
+        // to the screen reader.
+        break;
+      }
+
+    case "leftright":
+      {
+        buildRegion(a11yStrings, function (regionStrings) {
+          buildString(tree.left, "open", regionStrings);
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          buildString(tree.right, "close", regionStrings);
+        });
+        break;
+      }
+
+    case "leftright-right":
+      {
+        // TODO: double check that this is a no-op
+        break;
+      }
+
+    case "lap":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "mathord":
+      {
+        buildString(tree.text, "normal", a11yStrings);
+        break;
+      }
+
+    case "op":
+      {
+        var body = tree.body,
+            name = tree.name;
+
+        if (body) {
+          buildA11yStrings(body, a11yStrings, atomType);
+        } else if (name) {
+          buildString(name, "normal", a11yStrings);
+        }
+
+        break;
+      }
+
+    case "op-token":
+      {
+        // Used internally by operator symbols.
+        buildString(tree.text, atomType, a11yStrings);
+        break;
+      }
+
+    case "ordgroup":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "overline":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          a11yStrings.push("start overline");
+          buildA11yStrings(tree.body, a11yStrings, atomType);
+          a11yStrings.push("end overline");
+        });
+        break;
+      }
+
+    case "phantom":
+      {
+        a11yStrings.push("empty space");
+        break;
+      }
+
+    case "raisebox":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "rule":
+      {
+        a11yStrings.push("rectangle");
+        break;
+      }
+
+    case "sizing":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "spacing":
+      {
+        a11yStrings.push("space");
+        break;
+      }
+
+    case "styling":
+      {
+        // We ignore the styling and just pass through the contents
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "sqrt":
+      {
+        buildRegion(a11yStrings, function (regionStrings) {
+          var body = tree.body,
+              index = tree.index;
+
+          if (index) {
+            var indexString = flatten(buildA11yStrings(index, [], atomType)).join(",");
+
+            if (indexString === "3") {
+              regionStrings.push("cube root of");
+              buildA11yStrings(body, regionStrings, atomType);
+              regionStrings.push("end cube root");
+              return;
+            }
+
+            regionStrings.push("root");
+            regionStrings.push("start index");
+            buildA11yStrings(index, regionStrings, atomType);
+            regionStrings.push("end index");
+            return;
+          }
+
+          regionStrings.push("square root of");
+          buildA11yStrings(body, regionStrings, atomType);
+          regionStrings.push("end square root");
+        });
+        break;
+      }
+
+    case "supsub":
+      {
+        var base = tree.base,
+            sub = tree.sub,
+            sup = tree.sup;
+        var isLog = false;
+
+        if (base) {
+          buildA11yStrings(base, a11yStrings, atomType);
+          isLog = base.type === "op" && base.name === "\\log";
+        }
+
+        if (sub) {
+          var regionName = isLog ? "base" : "subscript";
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start " + regionName);
+            buildA11yStrings(sub, regionStrings, atomType);
+            regionStrings.push("end " + regionName);
+          });
+        }
+
+        if (sup) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            var supString = flatten(buildA11yStrings(sup, [], atomType)).join(",");
+
+            if (supString in powerMap) {
+              regionStrings.push(powerMap[supString]);
+              return;
+            }
+
+            regionStrings.push("start superscript");
+            buildA11yStrings(sup, regionStrings, atomType);
+            regionStrings.push("end superscript");
+          });
+        }
+
+        break;
+      }
+
+    case "text":
+      {
+        // TODO: handle other fonts
+        if (tree.font === "\\textbf") {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start bold text");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end bold text");
+          });
+          break;
+        }
+
+        buildRegion(a11yStrings, function (regionStrings) {
+          regionStrings.push("start text");
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          regionStrings.push("end text");
+        });
+        break;
+      }
+
+    case "textord":
+      {
+        buildString(tree.text, atomType, a11yStrings);
+        break;
+      }
+
+    case "smash":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "enclose":
+      {
+        // TODO: create a map for these.
+        // TODO: differentiate between a body with a single atom, e.g.
+        // "cancel a" instead of "start cancel, a, end cancel"
+        if (/cancel/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start cancel");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end cancel");
+          });
+          break;
+        } else if (/box/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start box");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end box");
+          });
+          break;
+        } else if (/sout/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start strikeout");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end strikeout");
+          });
+          break;
+        }
+
+        throw new Error("KaTeX-a11y: enclose node with " + tree.label + " not supported yet");
+      }
+
+    case "vphantom":
+      {
+        throw new Error("KaTeX-a11y: vphantom not implemented yet");
+      }
+
+    case "hphantom":
+      {
+        throw new Error("KaTeX-a11y: hphantom not implemented yet");
+      }
+
+    case "operatorname":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "array":
+      {
+        throw new Error("KaTeX-a11y: array not implemented yet");
+      }
+
+    case "raw":
+      {
+        throw new Error("KaTeX-a11y: raw not implemented yet");
+      }
+
+    case "size":
+      {
+        // Although there are nodes of type "size" in the parse tree, they have
+        // no semantic meaning and should be ignored.
+        break;
+      }
+
+    case "url":
+      {
+        throw new Error("KaTeX-a11y: url not implemented yet");
+      }
+
+    case "tag":
+      {
+        throw new Error("KaTeX-a11y: tag not implemented yet");
+      }
+
+    case "verb":
+      {
+        buildString("start verbatim", "normal", a11yStrings);
+        buildString(tree.body, "normal", a11yStrings);
+        buildString("end verbatim", "normal", a11yStrings);
+        break;
+      }
+
+    case "environment":
+      {
+        throw new Error("KaTeX-a11y: environment not implemented yet");
+      }
+
+    case "horizBrace":
+      {
+        buildString("start " + tree.label.slice(1), "normal", a11yStrings);
+        buildA11yStrings(tree.base, a11yStrings, atomType);
+        buildString("end " + tree.label.slice(1), "normal", a11yStrings);
+        break;
+      }
+
+    case "infix":
+      {
+        // All infix nodes are replace with other nodes.
+        break;
+      }
+
+    case "includegraphics":
+      {
+        throw new Error("KaTeX-a11y: includegraphics not implemented yet");
+      }
+
+    case "font":
+      {
+        // TODO: callout the start/end of specific fonts
+        // TODO: map \BBb{N} to "the naturals" or something like that
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "href":
+      {
+        throw new Error("KaTeX-a11y: href not implemented yet");
+      }
+
+    case "cr":
+      {
+        // This is used by environments.
+        throw new Error("KaTeX-a11y: cr not implemented yet");
+      }
+
+    case "underline":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          a11yStrings.push("start underline");
+          buildA11yStrings(tree.body, a11yStrings, atomType);
+          a11yStrings.push("end underline");
+        });
+        break;
+      }
+
+    case "xArrow":
+      {
+        throw new Error("KaTeX-a11y: xArrow not implemented yet");
+      }
+
+    case "mclass":
+      {
+        // \neq and \ne are macros so we let "htmlmathml" render the mathmal
+        // side of things and extract the text from that.
+        var _atomType = tree.mclass.slice(1); // $FlowFixMe: drop the leading "m" from the values in mclass
+
+
+        buildA11yStrings(tree.body, a11yStrings, _atomType);
+        break;
+      }
+
+    case "mathchoice":
+      {
+        // TODO: track which which style we're using, e.g. dispaly, text, etc.
+        // default to text style if even that may not be the correct style
+        buildA11yStrings(tree.text, a11yStrings, atomType);
+        break;
+      }
+
+    case "htmlmathml":
+      {
+        buildA11yStrings(tree.mathml, a11yStrings, atomType);
+        break;
+      }
+
+    case "middle":
+      {
+        buildString(tree.delim, atomType, a11yStrings);
+        break;
+      }
+
+    case "internal":
+      {
+        // internal nodes are never included in the parse tree
+        break;
+      }
+
+    case "html":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    default:
+      tree.type;
+      throw new Error("KaTeX a11y un-recognized type: " + tree.type);
+  }
+};
+
+var buildA11yStrings = function buildA11yStrings(tree, a11yStrings, atomType) {
+  if (a11yStrings === void 0) {
+    a11yStrings = [];
+  }
+
+  if (tree instanceof Array) {
+    for (var i = 0; i < tree.length; i++) {
+      buildA11yStrings(tree[i], a11yStrings, atomType);
+    }
+  } else {
+    handleObject(tree, a11yStrings, atomType);
+  }
+
+  return a11yStrings;
+};
+
+var flatten = function flatten(array) {
+  var result = [];
+  array.forEach(function (item) {
+    if (item instanceof Array) {
+      result = result.concat(flatten(item));
+    } else {
+      result.push(item);
+    }
+  });
+  return result;
+};
+
+var renderA11yString = function renderA11yString(text, settings) {
+  var tree = katex__WEBPACK_IMPORTED_MODULE_0___default.a.__parse(text, settings);
+
+  var a11yStrings = buildA11yStrings(tree, [], "normal");
+  return flatten(a11yStrings).join(", ");
+};
+
+/* harmony default export */ __webpack_exports__["default"] = (renderA11yString);
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/katex.bk/contrib/render-a11y-string.min.js b/katex.bk/contrib/render-a11y-string.min.js
new file mode 100644
index 00000000..5d406a6c
--- /dev/null
+++ b/katex.bk/contrib/render-a11y-string.min.js
@@ -0,0 +1 @@
+!function(e,r){if("object"==typeof exports&&"object"==typeof module)module.exports=r(require("katex"));else if("function"==typeof define&&define.amd)define(["katex"],r);else{var t="object"==typeof exports?r(require("katex")):r(e.katex);for(var a in t)("object"==typeof exports?exports:e)[a]=t[a]}}("undefined"!=typeof self?self:this,function(e){return function(e){var r={};function t(a){if(r[a])return r[a].exports;var o=r[a]={i:a,l:!1,exports:{}};return e[a].call(o.exports,o,o.exports,t),o.l=!0,o.exports}return t.m=e,t.c=r,t.d=function(e,r,a){t.o(e,r)||Object.defineProperty(e,r,{enumerable:!0,get:a})},t.r=function(e){"undefined"!=typeof Symbol&&Symbol.toStringTag&&Object.defineProperty(e,Symbol.toStringTag,{value:"Module"}),Object.defineProperty(e,"__esModule",{value:!0})},t.t=function(e,r){if(1&r&&(e=t(e)),8&r)return e;if(4&r&&"object"==typeof e&&e&&e.__esModule)return e;var a=Object.create(null);if(t.r(a),Object.defineProperty(a,"default",{enumerable:!0,value:e}),2&r&&"string"!=typeof e)for(var o in e)t.d(a,o,function(r){return e[r]}.bind(null,o));return a},t.n=function(e){var r=e&&e.__esModule?function(){return e.default}:function(){return e};return t.d(r,"a",r),r},t.o=function(e,r){return Object.prototype.hasOwnProperty.call(e,r)},t.p="",t(t.s=1)}([function(r,t){r.exports=e},function(e,r,t){"use strict";t.r(r);var a=t(0),o=t.n(a),n={"(":"left parenthesis",")":"right parenthesis","[":"open bracket","]":"close bracket","\\{":"left brace","\\}":"right brace","\\lvert":"open vertical bar","\\rvert":"close vertical bar","|":"vertical bar","\\uparrow":"up arrow","\\Uparrow":"up arrow","\\downarrow":"down arrow","\\Downarrow":"down arrow","\\updownarrow":"up down arrow","\\leftarrow":"left arrow","\\Leftarrow":"left arrow","\\rightarrow":"right arrow","\\Rightarrow":"right arrow","\\langle":"open angle","\\rangle":"close angle","\\lfloor":"open floor","\\rfloor":"close floor","\\int":"integral","\\intop":"integral","\\lim":"limit","\\ln":"natural log","\\log":"log","\\sin":"sine","\\cos":"cosine","\\tan":"tangent","\\cot":"cotangent","\\sum":"sum","/":"slash",",":"comma",".":"point","-":"negative","+":"plus","~":"tilde",":":"colon","?":"question mark","'":"apostrophe","\\%":"percent"," ":"space","\\ ":"space","\\$":"dollar sign","\\angle":"angle","\\degree":"degree","\\circ":"circle","\\vec":"vector","\\triangle":"triangle","\\pi":"pi","\\prime":"prime","\\infty":"infinity","\\alpha":"alpha","\\beta":"beta","\\gamma":"gamma","\\omega":"omega","\\theta":"theta","\\sigma":"sigma","\\lambda":"lambda","\\tau":"tau","\\Delta":"delta","\\delta":"delta","\\mu":"mu","\\rho":"rho","\\nabla":"del","\\ell":"ell","\\ldots":"dots","\\hat":"hat","\\acute":"acute"},s={prime:"prime",degree:"degrees",circle:"degrees",2:"squared",3:"cubed"},i={"|":"open vertical bar",".":""},l={"|":"close vertical bar",".":""},c={"+":"plus","-":"minus","\\pm":"plus minus","\\cdot":"dot","*":"times","/":"divided by","\\times":"times","\\div":"divided by","\\circ":"circle","\\bullet":"bullet"},u={"=":"equals","\\approx":"approximately equals","\u2260":"does not equal","\\geq":"is greater than or equal to","\\ge":"is greater than or equal to","\\leq":"is less than or equal to","\\le":"is less than or equal to",">":"is greater than","<":"is less than","\\leftarrow":"left arrow","\\Leftarrow":"left arrow","\\rightarrow":"right arrow","\\Rightarrow":"right arrow",":":"colon"},p={"\\underleftarrow":"left arrow","\\underrightarrow":"right arrow","\\underleftrightarrow":"left-right arrow","\\undergroup":"group","\\underlinesegment":"line segment","\\utilde":"tilde"},d=function(e,r,t){var a;e&&(/^\d+$/.test(a="open"===r?e in i?i[e]:n[e]||e:"close"===r?e in l?l[e]:n[e]||e:"bin"===r?c[e]||e:"rel"===r?u[e]||e:n[e]||e)&&t.length>0&&/^\d+$/.test(t[t.length-1])?t[t.length-1]+=a:a&&t.push(a))},b=function(e,r){var t=[];e.push(t),r(t)},h=function(e,r,t){switch(e.type){case"accent":b(r,function(r){f(e.base,r,t),r.push("with"),d(e.label,"normal",r),r.push("on top")});break;case"accentUnder":b(r,function(r){f(e.base,r,t),r.push("with"),d(p[e.label],"normal",r),r.push("underneath")});break;case"accent-token":break;case"atom":var a=e.text;switch(e.family){case"bin":d(a,"bin",r);break;case"close":d(a,"close",r);break;case"inner":d(e.text,"inner",r);break;case"open":d(a,"open",r);break;case"punct":d(a,"punct",r);break;case"rel":d(a,"rel",r);break;default:throw e.family,new Error('"'+e.family+'" is not a valid atom type')}break;case"color":var o=e.color.replace(/katex-/,"");b(r,function(r){r.push("start color "+o),f(e.body,r,t),r.push("end color "+o)});break;case"color-token":break;case"delimsizing":e.delim&&"."!==e.delim&&d(e.delim,"normal",r);break;case"genfrac":b(r,function(r){var a=e.leftDelim,o=e.rightDelim;e.hasBarLine?(r.push("start fraction"),a&&d(a,"open",r),f(e.numer,r,t),r.push("divided by"),f(e.denom,r,t),o&&d(o,"close",r),r.push("end fraction")):(r.push("start binomial"),a&&d(a,"open",r),f(e.numer,r,t),r.push("over"),f(e.denom,r,t),o&&d(o,"close",r),r.push("end binomial"))});break;case"kern":break;case"leftright":b(r,function(r){d(e.left,"open",r),f(e.body,r,t),d(e.right,"close",r)});break;case"leftright-right":break;case"lap":f(e.body,r,t);break;case"mathord":d(e.text,"normal",r);break;case"op":var n=e.body,i=e.name;n?f(n,r,t):i&&d(i,"normal",r);break;case"op-token":d(e.text,t,r);break;case"ordgroup":f(e.body,r,t);break;case"overline":b(r,function(r){r.push("start overline"),f(e.body,r,t),r.push("end overline")});break;case"phantom":r.push("empty space");break;case"raisebox":f(e.body,r,t);break;case"rule":r.push("rectangle");break;case"sizing":f(e.body,r,t);break;case"spacing":r.push("space");break;case"styling":f(e.body,r,t);break;case"sqrt":b(r,function(r){var a=e.body,o=e.index;if(o)return"3"===m(f(o,[],t)).join(",")?(r.push("cube root of"),f(a,r,t),void r.push("end cube root")):(r.push("root"),r.push("start index"),f(o,r,t),void r.push("end index"));r.push("square root of"),f(a,r,t),r.push("end square root")});break;case"supsub":var l=e.base,c=e.sub,u=e.sup,h=!1;if(l&&(f(l,r,t),h="op"===l.type&&"\\log"===l.name),c){var y=h?"base":"subscript";b(r,function(e){e.push("start "+y),f(c,e,t),e.push("end "+y)})}u&&b(r,function(e){var r=m(f(u,[],t)).join(",");r in s?e.push(s[r]):(e.push("start superscript"),f(u,e,t),e.push("end superscript"))});break;case"text":if("\\textbf"===e.font){b(r,function(r){r.push("start bold text"),f(e.body,r,t),r.push("end bold text")});break}b(r,function(r){r.push("start text"),f(e.body,r,t),r.push("end text")});break;case"textord":d(e.text,t,r);break;case"smash":f(e.body,r,t);break;case"enclose":if(/cancel/.test(e.label)){b(r,function(r){r.push("start cancel"),f(e.body,r,t),r.push("end cancel")});break}if(/box/.test(e.label)){b(r,function(r){r.push("start box"),f(e.body,r,t),r.push("end box")});break}if(/sout/.test(e.label)){b(r,function(r){r.push("start strikeout"),f(e.body,r,t),r.push("end strikeout")});break}throw new Error("KaTeX-a11y: enclose node with "+e.label+" not supported yet");case"vphantom":throw new Error("KaTeX-a11y: vphantom not implemented yet");case"hphantom":throw new Error("KaTeX-a11y: hphantom not implemented yet");case"operatorname":f(e.body,r,t);break;case"array":throw new Error("KaTeX-a11y: array not implemented yet");case"raw":throw new Error("KaTeX-a11y: raw not implemented yet");case"size":break;case"url":throw new Error("KaTeX-a11y: url not implemented yet");case"tag":throw new Error("KaTeX-a11y: tag not implemented yet");case"verb":d("start verbatim","normal",r),d(e.body,"normal",r),d("end verbatim","normal",r);break;case"environment":throw new Error("KaTeX-a11y: environment not implemented yet");case"horizBrace":d("start "+e.label.slice(1),"normal",r),f(e.base,r,t),d("end "+e.label.slice(1),"normal",r);break;case"infix":break;case"includegraphics":throw new Error("KaTeX-a11y: includegraphics not implemented yet");case"font":f(e.body,r,t);break;case"href":throw new Error("KaTeX-a11y: href not implemented yet");case"cr":throw new Error("KaTeX-a11y: cr not implemented yet");case"underline":b(r,function(r){r.push("start underline"),f(e.body,r,t),r.push("end underline")});break;case"xArrow":throw new Error("KaTeX-a11y: xArrow not implemented yet");case"mclass":var g=e.mclass.slice(1);f(e.body,r,g);break;case"mathchoice":f(e.text,r,t);break;case"htmlmathml":f(e.mathml,r,t);break;case"middle":d(e.delim,t,r);break;case"internal":break;case"html":f(e.body,r,t);break;default:throw e.type,new Error("KaTeX a11y un-recognized type: "+e.type)}},f=function e(r,t,a){if(void 0===t&&(t=[]),r instanceof Array)for(var o=0;o<r.length;o++)e(r[o],t,a);else h(r,t,a);return t},m=function e(r){var t=[];return r.forEach(function(r){r instanceof Array?t=t.concat(e(r)):t.push(r)}),t};r.default=function(e,r){var t=o.a.__parse(e,r),a=f(t,[],"normal");return m(a).join(", ")}}]).default});
\ No newline at end of file
diff --git a/katex.bk/contrib/render-a11y-string.mjs b/katex.bk/contrib/render-a11y-string.mjs
new file mode 100644
index 00000000..4eabb7ba
--- /dev/null
+++ b/katex.bk/contrib/render-a11y-string.mjs
@@ -0,0 +1,753 @@
+import katex from '../katex.mjs';
+
+/**
+ * renderA11yString returns a readable string.
+ *
+ * In some cases the string will have the proper semantic math
+ * meaning,:
+ *   renderA11yString("\\frac{1}{2}"")
+ *   -> "start fraction, 1, divided by, 2, end fraction"
+ *
+ * However, other cases do not:
+ *   renderA11yString("f(x) = x^2")
+ *   -> "f, left parenthesis, x, right parenthesis, equals, x, squared"
+ *
+ * The commas in the string aim to increase ease of understanding
+ * when read by a screenreader.
+ */
+const stringMap = {
+  "(": "left parenthesis",
+  ")": "right parenthesis",
+  "[": "open bracket",
+  "]": "close bracket",
+  "\\{": "left brace",
+  "\\}": "right brace",
+  "\\lvert": "open vertical bar",
+  "\\rvert": "close vertical bar",
+  "|": "vertical bar",
+  "\\uparrow": "up arrow",
+  "\\Uparrow": "up arrow",
+  "\\downarrow": "down arrow",
+  "\\Downarrow": "down arrow",
+  "\\updownarrow": "up down arrow",
+  "\\leftarrow": "left arrow",
+  "\\Leftarrow": "left arrow",
+  "\\rightarrow": "right arrow",
+  "\\Rightarrow": "right arrow",
+  "\\langle": "open angle",
+  "\\rangle": "close angle",
+  "\\lfloor": "open floor",
+  "\\rfloor": "close floor",
+  "\\int": "integral",
+  "\\intop": "integral",
+  "\\lim": "limit",
+  "\\ln": "natural log",
+  "\\log": "log",
+  "\\sin": "sine",
+  "\\cos": "cosine",
+  "\\tan": "tangent",
+  "\\cot": "cotangent",
+  "\\sum": "sum",
+  "/": "slash",
+  ",": "comma",
+  ".": "point",
+  "-": "negative",
+  "+": "plus",
+  "~": "tilde",
+  ":": "colon",
+  "?": "question mark",
+  "'": "apostrophe",
+  "\\%": "percent",
+  " ": "space",
+  "\\ ": "space",
+  "\\$": "dollar sign",
+  "\\angle": "angle",
+  "\\degree": "degree",
+  "\\circ": "circle",
+  "\\vec": "vector",
+  "\\triangle": "triangle",
+  "\\pi": "pi",
+  "\\prime": "prime",
+  "\\infty": "infinity",
+  "\\alpha": "alpha",
+  "\\beta": "beta",
+  "\\gamma": "gamma",
+  "\\omega": "omega",
+  "\\theta": "theta",
+  "\\sigma": "sigma",
+  "\\lambda": "lambda",
+  "\\tau": "tau",
+  "\\Delta": "delta",
+  "\\delta": "delta",
+  "\\mu": "mu",
+  "\\rho": "rho",
+  "\\nabla": "del",
+  "\\ell": "ell",
+  "\\ldots": "dots",
+  // TODO: add entries for all accents
+  "\\hat": "hat",
+  "\\acute": "acute"
+};
+const powerMap = {
+  "prime": "prime",
+  "degree": "degrees",
+  "circle": "degrees",
+  "2": "squared",
+  "3": "cubed"
+};
+const openMap = {
+  "|": "open vertical bar",
+  ".": ""
+};
+const closeMap = {
+  "|": "close vertical bar",
+  ".": ""
+};
+const binMap = {
+  "+": "plus",
+  "-": "minus",
+  "\\pm": "plus minus",
+  "\\cdot": "dot",
+  "*": "times",
+  "/": "divided by",
+  "\\times": "times",
+  "\\div": "divided by",
+  "\\circ": "circle",
+  "\\bullet": "bullet"
+};
+const relMap = {
+  "=": "equals",
+  "\\approx": "approximately equals",
+  "≠": "does not equal",
+  "\\geq": "is greater than or equal to",
+  "\\ge": "is greater than or equal to",
+  "\\leq": "is less than or equal to",
+  "\\le": "is less than or equal to",
+  ">": "is greater than",
+  "<": "is less than",
+  "\\leftarrow": "left arrow",
+  "\\Leftarrow": "left arrow",
+  "\\rightarrow": "right arrow",
+  "\\Rightarrow": "right arrow",
+  ":": "colon"
+};
+const accentUnderMap = {
+  "\\underleftarrow": "left arrow",
+  "\\underrightarrow": "right arrow",
+  "\\underleftrightarrow": "left-right arrow",
+  "\\undergroup": "group",
+  "\\underlinesegment": "line segment",
+  "\\utilde": "tilde"
+};
+
+const buildString = (str, type, a11yStrings) => {
+  if (!str) {
+    return;
+  }
+
+  let ret;
+
+  if (type === "open") {
+    ret = str in openMap ? openMap[str] : stringMap[str] || str;
+  } else if (type === "close") {
+    ret = str in closeMap ? closeMap[str] : stringMap[str] || str;
+  } else if (type === "bin") {
+    ret = binMap[str] || str;
+  } else if (type === "rel") {
+    ret = relMap[str] || str;
+  } else {
+    ret = stringMap[str] || str;
+  } // If the text to add is a number and there is already a string
+  // in the list and the last string is a number then we should
+  // combine them into a single number
+
+
+  if (/^\d+$/.test(ret) && a11yStrings.length > 0 && // TODO(kevinb): check that the last item in a11yStrings is a string
+  // I think we might be able to drop the nested arrays, which would make
+  // this easier to type - $FlowFixMe
+  /^\d+$/.test(a11yStrings[a11yStrings.length - 1])) {
+    a11yStrings[a11yStrings.length - 1] += ret;
+  } else if (ret) {
+    a11yStrings.push(ret);
+  }
+};
+
+const buildRegion = (a11yStrings, callback) => {
+  const regionStrings = [];
+  a11yStrings.push(regionStrings);
+  callback(regionStrings);
+};
+
+const handleObject = (tree, a11yStrings, atomType) => {
+  // Everything else is assumed to be an object...
+  switch (tree.type) {
+    case "accent":
+      {
+        buildRegion(a11yStrings, a11yStrings => {
+          buildA11yStrings(tree.base, a11yStrings, atomType);
+          a11yStrings.push("with");
+          buildString(tree.label, "normal", a11yStrings);
+          a11yStrings.push("on top");
+        });
+        break;
+      }
+
+    case "accentUnder":
+      {
+        buildRegion(a11yStrings, a11yStrings => {
+          buildA11yStrings(tree.base, a11yStrings, atomType);
+          a11yStrings.push("with");
+          buildString(accentUnderMap[tree.label], "normal", a11yStrings);
+          a11yStrings.push("underneath");
+        });
+        break;
+      }
+
+    case "accent-token":
+      {
+        // Used internally by accent symbols.
+        break;
+      }
+
+    case "atom":
+      {
+        const text = tree.text;
+
+        switch (tree.family) {
+          case "bin":
+            {
+              buildString(text, "bin", a11yStrings);
+              break;
+            }
+
+          case "close":
+            {
+              buildString(text, "close", a11yStrings);
+              break;
+            }
+          // TODO(kevinb): figure out what should be done for inner
+
+          case "inner":
+            {
+              buildString(tree.text, "inner", a11yStrings);
+              break;
+            }
+
+          case "open":
+            {
+              buildString(text, "open", a11yStrings);
+              break;
+            }
+
+          case "punct":
+            {
+              buildString(text, "punct", a11yStrings);
+              break;
+            }
+
+          case "rel":
+            {
+              buildString(text, "rel", a11yStrings);
+              break;
+            }
+
+          default:
+            {
+              tree.family;
+              throw new Error(`"${tree.family}" is not a valid atom type`);
+            }
+        }
+
+        break;
+      }
+
+    case "color":
+      {
+        const color = tree.color.replace(/katex-/, "");
+        buildRegion(a11yStrings, regionStrings => {
+          regionStrings.push("start color " + color);
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          regionStrings.push("end color " + color);
+        });
+        break;
+      }
+
+    case "color-token":
+      {
+        // Used by \color, \colorbox, and \fcolorbox but not directly rendered.
+        // It's a leaf node and has no children so just break.
+        break;
+      }
+
+    case "delimsizing":
+      {
+        if (tree.delim && tree.delim !== ".") {
+          buildString(tree.delim, "normal", a11yStrings);
+        }
+
+        break;
+      }
+
+    case "genfrac":
+      {
+        buildRegion(a11yStrings, regionStrings => {
+          // genfrac can have unbalanced delimiters
+          const leftDelim = tree.leftDelim,
+                rightDelim = tree.rightDelim; // NOTE: Not sure if this is a safe assumption
+          // hasBarLine true -> fraction, false -> binomial
+
+          if (tree.hasBarLine) {
+            regionStrings.push("start fraction");
+            leftDelim && buildString(leftDelim, "open", regionStrings);
+            buildA11yStrings(tree.numer, regionStrings, atomType);
+            regionStrings.push("divided by");
+            buildA11yStrings(tree.denom, regionStrings, atomType);
+            rightDelim && buildString(rightDelim, "close", regionStrings);
+            regionStrings.push("end fraction");
+          } else {
+            regionStrings.push("start binomial");
+            leftDelim && buildString(leftDelim, "open", regionStrings);
+            buildA11yStrings(tree.numer, regionStrings, atomType);
+            regionStrings.push("over");
+            buildA11yStrings(tree.denom, regionStrings, atomType);
+            rightDelim && buildString(rightDelim, "close", regionStrings);
+            regionStrings.push("end binomial");
+          }
+        });
+        break;
+      }
+
+    case "kern":
+      {
+        // No op: we don't attempt to present kerning information
+        // to the screen reader.
+        break;
+      }
+
+    case "leftright":
+      {
+        buildRegion(a11yStrings, regionStrings => {
+          buildString(tree.left, "open", regionStrings);
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          buildString(tree.right, "close", regionStrings);
+        });
+        break;
+      }
+
+    case "leftright-right":
+      {
+        // TODO: double check that this is a no-op
+        break;
+      }
+
+    case "lap":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "mathord":
+      {
+        buildString(tree.text, "normal", a11yStrings);
+        break;
+      }
+
+    case "op":
+      {
+        const body = tree.body,
+              name = tree.name;
+
+        if (body) {
+          buildA11yStrings(body, a11yStrings, atomType);
+        } else if (name) {
+          buildString(name, "normal", a11yStrings);
+        }
+
+        break;
+      }
+
+    case "op-token":
+      {
+        // Used internally by operator symbols.
+        buildString(tree.text, atomType, a11yStrings);
+        break;
+      }
+
+    case "ordgroup":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "overline":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          a11yStrings.push("start overline");
+          buildA11yStrings(tree.body, a11yStrings, atomType);
+          a11yStrings.push("end overline");
+        });
+        break;
+      }
+
+    case "phantom":
+      {
+        a11yStrings.push("empty space");
+        break;
+      }
+
+    case "raisebox":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "rule":
+      {
+        a11yStrings.push("rectangle");
+        break;
+      }
+
+    case "sizing":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "spacing":
+      {
+        a11yStrings.push("space");
+        break;
+      }
+
+    case "styling":
+      {
+        // We ignore the styling and just pass through the contents
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "sqrt":
+      {
+        buildRegion(a11yStrings, regionStrings => {
+          const body = tree.body,
+                index = tree.index;
+
+          if (index) {
+            const indexString = flatten(buildA11yStrings(index, [], atomType)).join(",");
+
+            if (indexString === "3") {
+              regionStrings.push("cube root of");
+              buildA11yStrings(body, regionStrings, atomType);
+              regionStrings.push("end cube root");
+              return;
+            }
+
+            regionStrings.push("root");
+            regionStrings.push("start index");
+            buildA11yStrings(index, regionStrings, atomType);
+            regionStrings.push("end index");
+            return;
+          }
+
+          regionStrings.push("square root of");
+          buildA11yStrings(body, regionStrings, atomType);
+          regionStrings.push("end square root");
+        });
+        break;
+      }
+
+    case "supsub":
+      {
+        const base = tree.base,
+              sub = tree.sub,
+              sup = tree.sup;
+        let isLog = false;
+
+        if (base) {
+          buildA11yStrings(base, a11yStrings, atomType);
+          isLog = base.type === "op" && base.name === "\\log";
+        }
+
+        if (sub) {
+          const regionName = isLog ? "base" : "subscript";
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push(`start ${regionName}`);
+            buildA11yStrings(sub, regionStrings, atomType);
+            regionStrings.push(`end ${regionName}`);
+          });
+        }
+
+        if (sup) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            const supString = flatten(buildA11yStrings(sup, [], atomType)).join(",");
+
+            if (supString in powerMap) {
+              regionStrings.push(powerMap[supString]);
+              return;
+            }
+
+            regionStrings.push("start superscript");
+            buildA11yStrings(sup, regionStrings, atomType);
+            regionStrings.push("end superscript");
+          });
+        }
+
+        break;
+      }
+
+    case "text":
+      {
+        // TODO: handle other fonts
+        if (tree.font === "\\textbf") {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start bold text");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end bold text");
+          });
+          break;
+        }
+
+        buildRegion(a11yStrings, function (regionStrings) {
+          regionStrings.push("start text");
+          buildA11yStrings(tree.body, regionStrings, atomType);
+          regionStrings.push("end text");
+        });
+        break;
+      }
+
+    case "textord":
+      {
+        buildString(tree.text, atomType, a11yStrings);
+        break;
+      }
+
+    case "smash":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "enclose":
+      {
+        // TODO: create a map for these.
+        // TODO: differentiate between a body with a single atom, e.g.
+        // "cancel a" instead of "start cancel, a, end cancel"
+        if (/cancel/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start cancel");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end cancel");
+          });
+          break;
+        } else if (/box/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start box");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end box");
+          });
+          break;
+        } else if (/sout/.test(tree.label)) {
+          buildRegion(a11yStrings, function (regionStrings) {
+            regionStrings.push("start strikeout");
+            buildA11yStrings(tree.body, regionStrings, atomType);
+            regionStrings.push("end strikeout");
+          });
+          break;
+        }
+
+        throw new Error(`KaTeX-a11y: enclose node with ${tree.label} not supported yet`);
+      }
+
+    case "vphantom":
+      {
+        throw new Error("KaTeX-a11y: vphantom not implemented yet");
+      }
+
+    case "hphantom":
+      {
+        throw new Error("KaTeX-a11y: hphantom not implemented yet");
+      }
+
+    case "operatorname":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "array":
+      {
+        throw new Error("KaTeX-a11y: array not implemented yet");
+      }
+
+    case "raw":
+      {
+        throw new Error("KaTeX-a11y: raw not implemented yet");
+      }
+
+    case "size":
+      {
+        // Although there are nodes of type "size" in the parse tree, they have
+        // no semantic meaning and should be ignored.
+        break;
+      }
+
+    case "url":
+      {
+        throw new Error("KaTeX-a11y: url not implemented yet");
+      }
+
+    case "tag":
+      {
+        throw new Error("KaTeX-a11y: tag not implemented yet");
+      }
+
+    case "verb":
+      {
+        buildString(`start verbatim`, "normal", a11yStrings);
+        buildString(tree.body, "normal", a11yStrings);
+        buildString(`end verbatim`, "normal", a11yStrings);
+        break;
+      }
+
+    case "environment":
+      {
+        throw new Error("KaTeX-a11y: environment not implemented yet");
+      }
+
+    case "horizBrace":
+      {
+        buildString(`start ${tree.label.slice(1)}`, "normal", a11yStrings);
+        buildA11yStrings(tree.base, a11yStrings, atomType);
+        buildString(`end ${tree.label.slice(1)}`, "normal", a11yStrings);
+        break;
+      }
+
+    case "infix":
+      {
+        // All infix nodes are replace with other nodes.
+        break;
+      }
+
+    case "includegraphics":
+      {
+        throw new Error("KaTeX-a11y: includegraphics not implemented yet");
+      }
+
+    case "font":
+      {
+        // TODO: callout the start/end of specific fonts
+        // TODO: map \BBb{N} to "the naturals" or something like that
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "href":
+      {
+        throw new Error("KaTeX-a11y: href not implemented yet");
+      }
+
+    case "cr":
+      {
+        // This is used by environments.
+        throw new Error("KaTeX-a11y: cr not implemented yet");
+      }
+
+    case "underline":
+      {
+        buildRegion(a11yStrings, function (a11yStrings) {
+          a11yStrings.push("start underline");
+          buildA11yStrings(tree.body, a11yStrings, atomType);
+          a11yStrings.push("end underline");
+        });
+        break;
+      }
+
+    case "xArrow":
+      {
+        throw new Error("KaTeX-a11y: xArrow not implemented yet");
+      }
+
+    case "mclass":
+      {
+        // \neq and \ne are macros so we let "htmlmathml" render the mathmal
+        // side of things and extract the text from that.
+        const atomType = tree.mclass.slice(1); // $FlowFixMe: drop the leading "m" from the values in mclass
+
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    case "mathchoice":
+      {
+        // TODO: track which which style we're using, e.g. dispaly, text, etc.
+        // default to text style if even that may not be the correct style
+        buildA11yStrings(tree.text, a11yStrings, atomType);
+        break;
+      }
+
+    case "htmlmathml":
+      {
+        buildA11yStrings(tree.mathml, a11yStrings, atomType);
+        break;
+      }
+
+    case "middle":
+      {
+        buildString(tree.delim, atomType, a11yStrings);
+        break;
+      }
+
+    case "internal":
+      {
+        // internal nodes are never included in the parse tree
+        break;
+      }
+
+    case "html":
+      {
+        buildA11yStrings(tree.body, a11yStrings, atomType);
+        break;
+      }
+
+    default:
+      tree.type;
+      throw new Error("KaTeX a11y un-recognized type: " + tree.type);
+  }
+};
+
+const buildA11yStrings = function buildA11yStrings(tree, a11yStrings, atomType) {
+  if (a11yStrings === void 0) {
+    a11yStrings = [];
+  }
+
+  if (tree instanceof Array) {
+    for (let i = 0; i < tree.length; i++) {
+      buildA11yStrings(tree[i], a11yStrings, atomType);
+    }
+  } else {
+    handleObject(tree, a11yStrings, atomType);
+  }
+
+  return a11yStrings;
+};
+
+const flatten = function flatten(array) {
+  let result = [];
+  array.forEach(function (item) {
+    if (item instanceof Array) {
+      result = result.concat(flatten(item));
+    } else {
+      result.push(item);
+    }
+  });
+  return result;
+};
+
+const renderA11yString = function renderA11yString(text, settings) {
+  const tree = katex.__parse(text, settings);
+
+  const a11yStrings = buildA11yStrings(tree, [], "normal");
+  return flatten(a11yStrings).join(", ");
+};
+
+export default renderA11yString;
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diff --git a/katex.bk/fonts/KaTeX_Main-Regular.woff2 b/katex.bk/fonts/KaTeX_Main-Regular.woff2
new file mode 100644
index 00000000..18647fa6
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diff --git a/katex.bk/fonts/KaTeX_Math-BoldItalic.ttf b/katex.bk/fonts/KaTeX_Math-BoldItalic.ttf
new file mode 100644
index 00000000..326b523b
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diff --git a/katex.bk/fonts/KaTeX_Math-BoldItalic.woff b/katex.bk/fonts/KaTeX_Math-BoldItalic.woff
new file mode 100644
index 00000000..5b4041aa
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diff --git a/katex.bk/fonts/KaTeX_Math-BoldItalic.woff2 b/katex.bk/fonts/KaTeX_Math-BoldItalic.woff2
new file mode 100644
index 00000000..ba55276d
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diff --git a/katex.bk/fonts/KaTeX_Math-Italic.ttf b/katex.bk/fonts/KaTeX_Math-Italic.ttf
new file mode 100644
index 00000000..f148fcee
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diff --git a/katex.bk/fonts/KaTeX_Math-Italic.woff b/katex.bk/fonts/KaTeX_Math-Italic.woff
new file mode 100644
index 00000000..31d00384
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diff --git a/katex.bk/fonts/KaTeX_Math-Italic.woff2 b/katex.bk/fonts/KaTeX_Math-Italic.woff2
new file mode 100644
index 00000000..9871ab6b
Binary files /dev/null and b/katex.bk/fonts/KaTeX_Math-Italic.woff2 differ
diff --git a/katex.bk/fonts/KaTeX_SansSerif-Bold.ttf b/katex.bk/fonts/KaTeX_SansSerif-Bold.ttf
new file mode 100644
index 00000000..dce35c8f
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diff --git a/katex.bk/fonts/KaTeX_SansSerif-Bold.woff b/katex.bk/fonts/KaTeX_SansSerif-Bold.woff
new file mode 100644
index 00000000..992cb3d6
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diff --git a/katex.bk/fonts/KaTeX_SansSerif-Bold.woff2 b/katex.bk/fonts/KaTeX_SansSerif-Bold.woff2
new file mode 100644
index 00000000..6dd10388
Binary files /dev/null and b/katex.bk/fonts/KaTeX_SansSerif-Bold.woff2 differ
diff --git a/katex.bk/fonts/KaTeX_SansSerif-Italic.ttf b/katex.bk/fonts/KaTeX_SansSerif-Italic.ttf
new file mode 100644
index 00000000..a3eb86c3
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diff --git a/katex.bk/fonts/KaTeX_SansSerif-Italic.woff b/katex.bk/fonts/KaTeX_SansSerif-Italic.woff
new file mode 100644
index 00000000..f4fa252a
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diff --git a/katex.bk/fonts/KaTeX_SansSerif-Italic.woff2 b/katex.bk/fonts/KaTeX_SansSerif-Italic.woff2
new file mode 100644
index 00000000..9f2501a3
Binary files /dev/null and b/katex.bk/fonts/KaTeX_SansSerif-Italic.woff2 differ
diff --git a/katex.bk/fonts/KaTeX_SansSerif-Regular.ttf b/katex.bk/fonts/KaTeX_SansSerif-Regular.ttf
new file mode 100644
index 00000000..3be73ce1
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diff --git a/katex.bk/fonts/KaTeX_SansSerif-Regular.woff b/katex.bk/fonts/KaTeX_SansSerif-Regular.woff
new file mode 100644
index 00000000..ec283f41
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diff --git a/katex.bk/fonts/KaTeX_SansSerif-Regular.woff2 b/katex.bk/fonts/KaTeX_SansSerif-Regular.woff2
new file mode 100644
index 00000000..e46094fb
Binary files /dev/null and b/katex.bk/fonts/KaTeX_SansSerif-Regular.woff2 differ
diff --git a/katex.bk/fonts/KaTeX_Script-Regular.ttf b/katex.bk/fonts/KaTeX_Script-Regular.ttf
new file mode 100644
index 00000000..40c8a997
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diff --git a/katex.bk/fonts/KaTeX_Script-Regular.woff b/katex.bk/fonts/KaTeX_Script-Regular.woff
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index 00000000..4eafae75
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diff --git a/katex.bk/fonts/KaTeX_Script-Regular.woff2 b/katex.bk/fonts/KaTeX_Script-Regular.woff2
new file mode 100644
index 00000000..69b1754d
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diff --git a/katex.bk/fonts/KaTeX_Size1-Regular.ttf b/katex.bk/fonts/KaTeX_Size1-Regular.ttf
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index 00000000..f0aff83e
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diff --git a/katex.bk/fonts/KaTeX_Size1-Regular.woff b/katex.bk/fonts/KaTeX_Size1-Regular.woff
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index 00000000..0358ee4a
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diff --git a/katex.bk/fonts/KaTeX_Size1-Regular.woff2 b/katex.bk/fonts/KaTeX_Size1-Regular.woff2
new file mode 100644
index 00000000..f951ed01
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diff --git a/katex.bk/fonts/KaTeX_Size2-Regular.ttf b/katex.bk/fonts/KaTeX_Size2-Regular.ttf
new file mode 100644
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diff --git a/katex.bk/fonts/KaTeX_Size2-Regular.woff b/katex.bk/fonts/KaTeX_Size2-Regular.woff
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diff --git a/katex.bk/fonts/KaTeX_Size2-Regular.woff2 b/katex.bk/fonts/KaTeX_Size2-Regular.woff2
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index 00000000..181d9625
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diff --git a/katex.bk/fonts/KaTeX_Size3-Regular.ttf b/katex.bk/fonts/KaTeX_Size3-Regular.ttf
new file mode 100644
index 00000000..56d2dc6c
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diff --git a/katex.bk/fonts/KaTeX_Size3-Regular.woff b/katex.bk/fonts/KaTeX_Size3-Regular.woff
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index 00000000..0ec99ad1
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diff --git a/katex.bk/fonts/KaTeX_Size3-Regular.woff2 b/katex.bk/fonts/KaTeX_Size3-Regular.woff2
new file mode 100644
index 00000000..c2985cd3
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diff --git a/katex.bk/fonts/KaTeX_Size4-Regular.ttf b/katex.bk/fonts/KaTeX_Size4-Regular.ttf
new file mode 100644
index 00000000..baf02091
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diff --git a/katex.bk/fonts/KaTeX_Size4-Regular.woff b/katex.bk/fonts/KaTeX_Size4-Regular.woff
new file mode 100644
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diff --git a/katex.bk/fonts/KaTeX_Size4-Regular.woff2 b/katex.bk/fonts/KaTeX_Size4-Regular.woff2
new file mode 100644
index 00000000..a4e810da
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diff --git a/katex.bk/fonts/KaTeX_Typewriter-Regular.ttf b/katex.bk/fonts/KaTeX_Typewriter-Regular.ttf
new file mode 100644
index 00000000..e66c218d
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diff --git a/katex.bk/fonts/KaTeX_Typewriter-Regular.woff b/katex.bk/fonts/KaTeX_Typewriter-Regular.woff
new file mode 100644
index 00000000..c66d149d
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diff --git a/katex.bk/fonts/KaTeX_Typewriter-Regular.woff2 b/katex.bk/fonts/KaTeX_Typewriter-Regular.woff2
new file mode 100644
index 00000000..e5bf2ce1
Binary files /dev/null and b/katex.bk/fonts/KaTeX_Typewriter-Regular.woff2 differ
diff --git a/katex.bk/katex.css b/katex.bk/katex.css
new file mode 100644
index 00000000..ba887582
--- /dev/null
+++ b/katex.bk/katex.css
@@ -0,0 +1,1035 @@
+/* stylelint-disable font-family-no-missing-generic-family-keyword */
+@font-face {
+  font-family: 'KaTeX_AMS';
+  src: url(fonts/KaTeX_AMS-Regular.woff2) format('woff2'), url(fonts/KaTeX_AMS-Regular.woff) format('woff'), url(fonts/KaTeX_AMS-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Caligraphic';
+  src: url(fonts/KaTeX_Caligraphic-Bold.woff2) format('woff2'), url(fonts/KaTeX_Caligraphic-Bold.woff) format('woff'), url(fonts/KaTeX_Caligraphic-Bold.ttf) format('truetype');
+  font-weight: bold;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Caligraphic';
+  src: url(fonts/KaTeX_Caligraphic-Regular.woff2) format('woff2'), url(fonts/KaTeX_Caligraphic-Regular.woff) format('woff'), url(fonts/KaTeX_Caligraphic-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Fraktur';
+  src: url(fonts/KaTeX_Fraktur-Bold.woff2) format('woff2'), url(fonts/KaTeX_Fraktur-Bold.woff) format('woff'), url(fonts/KaTeX_Fraktur-Bold.ttf) format('truetype');
+  font-weight: bold;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Fraktur';
+  src: url(fonts/KaTeX_Fraktur-Regular.woff2) format('woff2'), url(fonts/KaTeX_Fraktur-Regular.woff) format('woff'), url(fonts/KaTeX_Fraktur-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Main';
+  src: url(fonts/KaTeX_Main-Bold.woff2) format('woff2'), url(fonts/KaTeX_Main-Bold.woff) format('woff'), url(fonts/KaTeX_Main-Bold.ttf) format('truetype');
+  font-weight: bold;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Main';
+  src: url(fonts/KaTeX_Main-BoldItalic.woff2) format('woff2'), url(fonts/KaTeX_Main-BoldItalic.woff) format('woff'), url(fonts/KaTeX_Main-BoldItalic.ttf) format('truetype');
+  font-weight: bold;
+  font-style: italic;
+}
+@font-face {
+  font-family: 'KaTeX_Main';
+  src: url(fonts/KaTeX_Main-Italic.woff2) format('woff2'), url(fonts/KaTeX_Main-Italic.woff) format('woff'), url(fonts/KaTeX_Main-Italic.ttf) format('truetype');
+  font-weight: normal;
+  font-style: italic;
+}
+@font-face {
+  font-family: 'KaTeX_Main';
+  src: url(fonts/KaTeX_Main-Regular.woff2) format('woff2'), url(fonts/KaTeX_Main-Regular.woff) format('woff'), url(fonts/KaTeX_Main-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Math';
+  src: url(fonts/KaTeX_Math-BoldItalic.woff2) format('woff2'), url(fonts/KaTeX_Math-BoldItalic.woff) format('woff'), url(fonts/KaTeX_Math-BoldItalic.ttf) format('truetype');
+  font-weight: bold;
+  font-style: italic;
+}
+@font-face {
+  font-family: 'KaTeX_Math';
+  src: url(fonts/KaTeX_Math-Italic.woff2) format('woff2'), url(fonts/KaTeX_Math-Italic.woff) format('woff'), url(fonts/KaTeX_Math-Italic.ttf) format('truetype');
+  font-weight: normal;
+  font-style: italic;
+}
+@font-face {
+  font-family: 'KaTeX_SansSerif';
+  src: url(fonts/KaTeX_SansSerif-Bold.woff2) format('woff2'), url(fonts/KaTeX_SansSerif-Bold.woff) format('woff'), url(fonts/KaTeX_SansSerif-Bold.ttf) format('truetype');
+  font-weight: bold;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_SansSerif';
+  src: url(fonts/KaTeX_SansSerif-Italic.woff2) format('woff2'), url(fonts/KaTeX_SansSerif-Italic.woff) format('woff'), url(fonts/KaTeX_SansSerif-Italic.ttf) format('truetype');
+  font-weight: normal;
+  font-style: italic;
+}
+@font-face {
+  font-family: 'KaTeX_SansSerif';
+  src: url(fonts/KaTeX_SansSerif-Regular.woff2) format('woff2'), url(fonts/KaTeX_SansSerif-Regular.woff) format('woff'), url(fonts/KaTeX_SansSerif-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Script';
+  src: url(fonts/KaTeX_Script-Regular.woff2) format('woff2'), url(fonts/KaTeX_Script-Regular.woff) format('woff'), url(fonts/KaTeX_Script-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Size1';
+  src: url(fonts/KaTeX_Size1-Regular.woff2) format('woff2'), url(fonts/KaTeX_Size1-Regular.woff) format('woff'), url(fonts/KaTeX_Size1-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Size2';
+  src: url(fonts/KaTeX_Size2-Regular.woff2) format('woff2'), url(fonts/KaTeX_Size2-Regular.woff) format('woff'), url(fonts/KaTeX_Size2-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Size3';
+  src: url(fonts/KaTeX_Size3-Regular.woff2) format('woff2'), url(fonts/KaTeX_Size3-Regular.woff) format('woff'), url(fonts/KaTeX_Size3-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Size4';
+  src: url(fonts/KaTeX_Size4-Regular.woff2) format('woff2'), url(fonts/KaTeX_Size4-Regular.woff) format('woff'), url(fonts/KaTeX_Size4-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+@font-face {
+  font-family: 'KaTeX_Typewriter';
+  src: url(fonts/KaTeX_Typewriter-Regular.woff2) format('woff2'), url(fonts/KaTeX_Typewriter-Regular.woff) format('woff'), url(fonts/KaTeX_Typewriter-Regular.ttf) format('truetype');
+  font-weight: normal;
+  font-style: normal;
+}
+.katex {
+  font: normal 1.21em KaTeX_Main, Times New Roman, serif;
+  line-height: 1.2;
+  text-indent: 0;
+  text-rendering: auto;
+  border-color: currentColor;
+}
+.katex * {
+  -ms-high-contrast-adjust: none !important;
+}
+.katex .katex-version::after {
+  content: "0.12.0";
+}
+.katex .katex-mathml {
+  position: absolute;
+  clip: rect(1px, 1px, 1px, 1px);
+  padding: 0;
+  border: 0;
+  height: 1px;
+  width: 1px;
+  overflow: hidden;
+}
+.katex .katex-html {
+  /* \newline is an empty block at top level, between .base elements */
+}
+.katex .katex-html > .newline {
+  display: block;
+}
+.katex .base {
+  position: relative;
+  display: inline-block;
+  white-space: nowrap;
+  width: min-content;
+}
+.katex .strut {
+  display: inline-block;
+}
+.katex .textbf {
+  font-weight: bold;
+}
+.katex .textit {
+  font-style: italic;
+}
+.katex .textrm {
+  font-family: KaTeX_Main;
+}
+.katex .textsf {
+  font-family: KaTeX_SansSerif;
+}
+.katex .texttt {
+  font-family: KaTeX_Typewriter;
+}
+.katex .mathnormal {
+  font-family: KaTeX_Math;
+  font-style: italic;
+}
+.katex .mathit {
+  font-family: KaTeX_Main;
+  font-style: italic;
+}
+.katex .mathrm {
+  font-style: normal;
+}
+.katex .mathbf {
+  font-family: KaTeX_Main;
+  font-weight: bold;
+}
+.katex .boldsymbol {
+  font-family: KaTeX_Math;
+  font-weight: bold;
+  font-style: italic;
+}
+.katex .amsrm {
+  font-family: KaTeX_AMS;
+}
+.katex .mathbb,
+.katex .textbb {
+  font-family: KaTeX_AMS;
+}
+.katex .mathcal {
+  font-family: KaTeX_Caligraphic;
+}
+.katex .mathfrak,
+.katex .textfrak {
+  font-family: KaTeX_Fraktur;
+}
+.katex .mathtt {
+  font-family: KaTeX_Typewriter;
+}
+.katex .mathscr,
+.katex .textscr {
+  font-family: KaTeX_Script;
+}
+.katex .mathsf,
+.katex .textsf {
+  font-family: KaTeX_SansSerif;
+}
+.katex .mathboldsf,
+.katex .textboldsf {
+  font-family: KaTeX_SansSerif;
+  font-weight: bold;
+}
+.katex .mathitsf,
+.katex .textitsf {
+  font-family: KaTeX_SansSerif;
+  font-style: italic;
+}
+.katex .mainrm {
+  font-family: KaTeX_Main;
+  font-style: normal;
+}
+.katex .vlist-t {
+  display: inline-table;
+  table-layout: fixed;
+  border-collapse: collapse;
+}
+.katex .vlist-r {
+  display: table-row;
+}
+.katex .vlist {
+  display: table-cell;
+  vertical-align: bottom;
+  position: relative;
+}
+.katex .vlist > span {
+  display: block;
+  height: 0;
+  position: relative;
+}
+.katex .vlist > span > span {
+  display: inline-block;
+}
+.katex .vlist > span > .pstrut {
+  overflow: hidden;
+  width: 0;
+}
+.katex .vlist-t2 {
+  margin-right: -2px;
+}
+.katex .vlist-s {
+  display: table-cell;
+  vertical-align: bottom;
+  font-size: 1px;
+  width: 2px;
+  min-width: 2px;
+}
+.katex .vbox {
+  display: -ms-inline-flexbox;
+  display: inline-flex;
+  -ms-flex-direction: column;
+  flex-direction: column;
+  align-items: baseline;
+}
+.katex .hbox {
+  display: -ms-inline-flexbox;
+  display: inline-flex;
+  -ms-flex-direction: row;
+  flex-direction: row;
+  width: 100%;
+}
+.katex .thinbox {
+  display: inline-flex;
+  flex-direction: row;
+  width: 0;
+  max-width: 0;
+}
+.katex .msupsub {
+  text-align: left;
+}
+.katex .mfrac > span > span {
+  text-align: center;
+}
+.katex .mfrac .frac-line {
+  display: inline-block;
+  width: 100%;
+  border-bottom-style: solid;
+}
+.katex .mfrac .frac-line,
+.katex .overline .overline-line,
+.katex .underline .underline-line,
+.katex .hline,
+.katex .hdashline,
+.katex .rule {
+  min-height: 1px;
+}
+.katex .mspace {
+  display: inline-block;
+}
+.katex .llap,
+.katex .rlap,
+.katex .clap {
+  width: 0;
+  position: relative;
+}
+.katex .llap > .inner,
+.katex .rlap > .inner,
+.katex .clap > .inner {
+  position: absolute;
+}
+.katex .llap > .fix,
+.katex .rlap > .fix,
+.katex .clap > .fix {
+  display: inline-block;
+}
+.katex .llap > .inner {
+  right: 0;
+}
+.katex .rlap > .inner,
+.katex .clap > .inner {
+  left: 0;
+}
+.katex .clap > .inner > span {
+  margin-left: -50%;
+  margin-right: 50%;
+}
+.katex .rule {
+  display: inline-block;
+  border: solid 0;
+  position: relative;
+}
+.katex .overline .overline-line,
+.katex .underline .underline-line,
+.katex .hline {
+  display: inline-block;
+  width: 100%;
+  border-bottom-style: solid;
+}
+.katex .hdashline {
+  display: inline-block;
+  width: 100%;
+  border-bottom-style: dashed;
+}
+.katex .sqrt > .root {
+  margin-left: 0.27777778em;
+  margin-right: -0.55555556em;
+}
+.katex .sizing.reset-size1.size1,
+.katex .fontsize-ensurer.reset-size1.size1 {
+  font-size: 1em;
+}
+.katex .sizing.reset-size1.size2,
+.katex .fontsize-ensurer.reset-size1.size2 {
+  font-size: 1.2em;
+}
+.katex .sizing.reset-size1.size3,
+.katex .fontsize-ensurer.reset-size1.size3 {
+  font-size: 1.4em;
+}
+.katex .sizing.reset-size1.size4,
+.katex .fontsize-ensurer.reset-size1.size4 {
+  font-size: 1.6em;
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+  display: block;
+  text-align: center;
+  white-space: nowrap;
+}
+.katex-display > .katex > .katex-html {
+  display: block;
+  position: relative;
+}
+.katex-display > .katex > .katex-html > .tag {
+  position: absolute;
+  right: 0;
+}
+.katex-display.leqno > .katex > .katex-html > .tag {
+  left: 0;
+  right: auto;
+}
+.katex-display.fleqn > .katex {
+  text-align: left;
+  padding-left: 2em;
+}
+
diff --git a/katex.bk/katex.js b/katex.bk/katex.js
new file mode 100644
index 00000000..37fb0fa8
--- /dev/null
+++ b/katex.bk/katex.js
@@ -0,0 +1,17308 @@
+(function webpackUniversalModuleDefinition(root, factory) {
+	if(typeof exports === 'object' && typeof module === 'object')
+		module.exports = factory();
+	else if(typeof define === 'function' && define.amd)
+		define([], factory);
+	else if(typeof exports === 'object')
+		exports["katex"] = factory();
+	else
+		root["katex"] = factory();
+})((typeof self !== 'undefined' ? self : this), function() {
+return /******/ (function(modules) { // webpackBootstrap
+/******/ 	// The module cache
+/******/ 	var installedModules = {};
+/******/
+/******/ 	// The require function
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+/******/ 		modules[moduleId].call(module.exports, module, module.exports, __webpack_require__);
+/******/
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+/******/ 		module.l = true;
+/******/
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+/******/ 		return module.exports;
+/******/ 	}
+/******/
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+/******/ 	__webpack_require__.m = modules;
+/******/
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+/******/
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+/******/ 		if(!__webpack_require__.o(exports, name)) {
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+/******/ 	__webpack_require__.r = function(exports) {
+/******/ 		if(typeof Symbol !== 'undefined' && Symbol.toStringTag) {
+/******/ 			Object.defineProperty(exports, Symbol.toStringTag, { value: 'Module' });
+/******/ 		}
+/******/ 		Object.defineProperty(exports, '__esModule', { value: true });
+/******/ 	};
+/******/
+/******/ 	// create a fake namespace object
+/******/ 	// mode & 1: value is a module id, require it
+/******/ 	// mode & 2: merge all properties of value into the ns
+/******/ 	// mode & 4: return value when already ns object
+/******/ 	// mode & 8|1: behave like require
+/******/ 	__webpack_require__.t = function(value, mode) {
+/******/ 		if(mode & 1) value = __webpack_require__(value);
+/******/ 		if(mode & 8) return value;
+/******/ 		if((mode & 4) && typeof value === 'object' && value && value.__esModule) return value;
+/******/ 		var ns = Object.create(null);
+/******/ 		__webpack_require__.r(ns);
+/******/ 		Object.defineProperty(ns, 'default', { enumerable: true, value: value });
+/******/ 		if(mode & 2 && typeof value != 'string') for(var key in value) __webpack_require__.d(ns, key, function(key) { return value[key]; }.bind(null, key));
+/******/ 		return ns;
+/******/ 	};
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+/******/ 	__webpack_require__.n = function(module) {
+/******/ 		var getter = module && module.__esModule ?
+/******/ 			function getDefault() { return module['default']; } :
+/******/ 			function getModuleExports() { return module; };
+/******/ 		__webpack_require__.d(getter, 'a', getter);
+/******/ 		return getter;
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+/******/ 	__webpack_require__.p = "";
+/******/
+/******/
+/******/ 	// Load entry module and return exports
+/******/ 	return __webpack_require__(__webpack_require__.s = 1);
+/******/ })
+/************************************************************************/
+/******/ ([
+/* 0 */
+/***/ (function(module, exports, __webpack_require__) {
+
+// extracted by mini-css-extract-plugin
+
+/***/ }),
+/* 1 */
+/***/ (function(module, __webpack_exports__, __webpack_require__) {
+
+"use strict";
+__webpack_require__.r(__webpack_exports__);
+
+// EXTERNAL MODULE: ./src/katex.less
+var katex = __webpack_require__(0);
+
+// CONCATENATED MODULE: ./src/SourceLocation.js
+/**
+ * Lexing or parsing positional information for error reporting.
+ * This object is immutable.
+ */
+var SourceLocation =
+/*#__PURE__*/
+function () {
+  // The + prefix indicates that these fields aren't writeable
+  // Lexer holding the input string.
+  // Start offset, zero-based inclusive.
+  // End offset, zero-based exclusive.
+  function SourceLocation(lexer, start, end) {
+    this.lexer = void 0;
+    this.start = void 0;
+    this.end = void 0;
+    this.lexer = lexer;
+    this.start = start;
+    this.end = end;
+  }
+  /**
+   * Merges two `SourceLocation`s from location providers, given they are
+   * provided in order of appearance.
+   * - Returns the first one's location if only the first is provided.
+   * - Returns a merged range of the first and the last if both are provided
+   *   and their lexers match.
+   * - Otherwise, returns null.
+   */
+
+
+  SourceLocation.range = function range(first, second) {
+    if (!second) {
+      return first && first.loc;
+    } else if (!first || !first.loc || !second.loc || first.loc.lexer !== second.loc.lexer) {
+      return null;
+    } else {
+      return new SourceLocation(first.loc.lexer, first.loc.start, second.loc.end);
+    }
+  };
+
+  return SourceLocation;
+}();
+
+
+// CONCATENATED MODULE: ./src/Token.js
+
+/**
+ * Interface required to break circular dependency between Token, Lexer, and
+ * ParseError.
+ */
+
+/**
+ * The resulting token returned from `lex`.
+ *
+ * It consists of the token text plus some position information.
+ * The position information is essentially a range in an input string,
+ * but instead of referencing the bare input string, we refer to the lexer.
+ * That way it is possible to attach extra metadata to the input string,
+ * like for example a file name or similar.
+ *
+ * The position information is optional, so it is OK to construct synthetic
+ * tokens if appropriate. Not providing available position information may
+ * lead to degraded error reporting, though.
+ */
+var Token_Token =
+/*#__PURE__*/
+function () {
+  // don't expand the token
+  // used in \noexpand
+  function Token(text, // the text of this token
+  loc) {
+    this.text = void 0;
+    this.loc = void 0;
+    this.noexpand = void 0;
+    this.treatAsRelax = void 0;
+    this.text = text;
+    this.loc = loc;
+  }
+  /**
+   * Given a pair of tokens (this and endToken), compute a `Token` encompassing
+   * the whole input range enclosed by these two.
+   */
+
+
+  var _proto = Token.prototype;
+
+  _proto.range = function range(endToken, // last token of the range, inclusive
+  text) // the text of the newly constructed token
+  {
+    return new Token(text, SourceLocation.range(this, endToken));
+  };
+
+  return Token;
+}();
+// CONCATENATED MODULE: ./src/ParseError.js
+
+
+/**
+ * This is the ParseError class, which is the main error thrown by KaTeX
+ * functions when something has gone wrong. This is used to distinguish internal
+ * errors from errors in the expression that the user provided.
+ *
+ * If possible, a caller should provide a Token or ParseNode with information
+ * about where in the source string the problem occurred.
+ */
+var ParseError = // Error position based on passed-in Token or ParseNode.
+function ParseError(message, // The error message
+token) // An object providing position information
+{
+  this.position = void 0;
+  var error = "KaTeX parse error: " + message;
+  var start;
+  var loc = token && token.loc;
+
+  if (loc && loc.start <= loc.end) {
+    // If we have the input and a position, make the error a bit fancier
+    // Get the input
+    var input = loc.lexer.input; // Prepend some information
+
+    start = loc.start;
+    var end = loc.end;
+
+    if (start === input.length) {
+      error += " at end of input: ";
+    } else {
+      error += " at position " + (start + 1) + ": ";
+    } // Underline token in question using combining underscores
+
+
+    var underlined = input.slice(start, end).replace(/[^]/g, "$&\u0332"); // Extract some context from the input and add it to the error
+
+    var left;
+
+    if (start > 15) {
+      left = "…" + input.slice(start - 15, start);
+    } else {
+      left = input.slice(0, start);
+    }
+
+    var right;
+
+    if (end + 15 < input.length) {
+      right = input.slice(end, end + 15) + "…";
+    } else {
+      right = input.slice(end);
+    }
+
+    error += left + underlined + right;
+  } // Some hackery to make ParseError a prototype of Error
+  // See http://stackoverflow.com/a/8460753
+
+
+  var self = new Error(error);
+  self.name = "ParseError"; // $FlowFixMe
+
+  self.__proto__ = ParseError.prototype; // $FlowFixMe
+
+  self.position = start;
+  return self;
+}; // $FlowFixMe More hackery
+
+
+ParseError.prototype.__proto__ = Error.prototype;
+/* harmony default export */ var src_ParseError = (ParseError);
+// CONCATENATED MODULE: ./src/utils.js
+/**
+ * This file contains a list of utility functions which are useful in other
+ * files.
+ */
+
+/**
+ * Return whether an element is contained in a list
+ */
+var contains = function contains(list, elem) {
+  return list.indexOf(elem) !== -1;
+};
+/**
+ * Provide a default value if a setting is undefined
+ * NOTE: Couldn't use `T` as the output type due to facebook/flow#5022.
+ */
+
+
+var deflt = function deflt(setting, defaultIfUndefined) {
+  return setting === undefined ? defaultIfUndefined : setting;
+}; // hyphenate and escape adapted from Facebook's React under Apache 2 license
+
+
+var uppercase = /([A-Z])/g;
+
+var hyphenate = function hyphenate(str) {
+  return str.replace(uppercase, "-$1").toLowerCase();
+};
+
+var ESCAPE_LOOKUP = {
+  "&": "&amp;",
+  ">": "&gt;",
+  "<": "&lt;",
+  "\"": "&quot;",
+  "'": "&#x27;"
+};
+var ESCAPE_REGEX = /[&><"']/g;
+/**
+ * Escapes text to prevent scripting attacks.
+ */
+
+function utils_escape(text) {
+  return String(text).replace(ESCAPE_REGEX, function (match) {
+    return ESCAPE_LOOKUP[match];
+  });
+}
+/**
+ * Sometimes we want to pull out the innermost element of a group. In most
+ * cases, this will just be the group itself, but when ordgroups and colors have
+ * a single element, we want to pull that out.
+ */
+
+
+var getBaseElem = function getBaseElem(group) {
+  if (group.type === "ordgroup") {
+    if (group.body.length === 1) {
+      return getBaseElem(group.body[0]);
+    } else {
+      return group;
+    }
+  } else if (group.type === "color") {
+    if (group.body.length === 1) {
+      return getBaseElem(group.body[0]);
+    } else {
+      return group;
+    }
+  } else if (group.type === "font") {
+    return getBaseElem(group.body);
+  } else {
+    return group;
+  }
+};
+/**
+ * TeXbook algorithms often reference "character boxes", which are simply groups
+ * with a single character in them. To decide if something is a character box,
+ * we find its innermost group, and see if it is a single character.
+ */
+
+
+var utils_isCharacterBox = function isCharacterBox(group) {
+  var baseElem = getBaseElem(group); // These are all they types of groups which hold single characters
+
+  return baseElem.type === "mathord" || baseElem.type === "textord" || baseElem.type === "atom";
+};
+
+var assert = function assert(value) {
+  if (!value) {
+    throw new Error('Expected non-null, but got ' + String(value));
+  }
+
+  return value;
+};
+/**
+ * Return the protocol of a URL, or "_relative" if the URL does not specify a
+ * protocol (and thus is relative).
+ */
+
+var protocolFromUrl = function protocolFromUrl(url) {
+  var protocol = /^\s*([^\\/#]*?)(?::|&#0*58|&#x0*3a)/i.exec(url);
+  return protocol != null ? protocol[1] : "_relative";
+};
+/* harmony default export */ var utils = ({
+  contains: contains,
+  deflt: deflt,
+  escape: utils_escape,
+  hyphenate: hyphenate,
+  getBaseElem: getBaseElem,
+  isCharacterBox: utils_isCharacterBox,
+  protocolFromUrl: protocolFromUrl
+});
+// CONCATENATED MODULE: ./src/Settings.js
+/* eslint no-console:0 */
+
+/**
+ * This is a module for storing settings passed into KaTeX. It correctly handles
+ * default settings.
+ */
+
+
+
+
+/**
+ * The main Settings object
+ *
+ * The current options stored are:
+ *  - displayMode: Whether the expression should be typeset as inline math
+ *                 (false, the default), meaning that the math starts in
+ *                 \textstyle and is placed in an inline-block); or as display
+ *                 math (true), meaning that the math starts in \displaystyle
+ *                 and is placed in a block with vertical margin.
+ */
+var Settings_Settings =
+/*#__PURE__*/
+function () {
+  function Settings(options) {
+    this.displayMode = void 0;
+    this.output = void 0;
+    this.leqno = void 0;
+    this.fleqn = void 0;
+    this.throwOnError = void 0;
+    this.errorColor = void 0;
+    this.macros = void 0;
+    this.minRuleThickness = void 0;
+    this.colorIsTextColor = void 0;
+    this.strict = void 0;
+    this.trust = void 0;
+    this.maxSize = void 0;
+    this.maxExpand = void 0;
+    this.globalGroup = void 0;
+    // allow null options
+    options = options || {};
+    this.displayMode = utils.deflt(options.displayMode, false);
+    this.output = utils.deflt(options.output, "htmlAndMathml");
+    this.leqno = utils.deflt(options.leqno, false);
+    this.fleqn = utils.deflt(options.fleqn, false);
+    this.throwOnError = utils.deflt(options.throwOnError, true);
+    this.errorColor = utils.deflt(options.errorColor, "#cc0000");
+    this.macros = options.macros || {};
+    this.minRuleThickness = Math.max(0, utils.deflt(options.minRuleThickness, 0));
+    this.colorIsTextColor = utils.deflt(options.colorIsTextColor, false);
+    this.strict = utils.deflt(options.strict, "warn");
+    this.trust = utils.deflt(options.trust, false);
+    this.maxSize = Math.max(0, utils.deflt(options.maxSize, Infinity));
+    this.maxExpand = Math.max(0, utils.deflt(options.maxExpand, 1000));
+    this.globalGroup = utils.deflt(options.globalGroup, false);
+  }
+  /**
+   * Report nonstrict (non-LaTeX-compatible) input.
+   * Can safely not be called if `this.strict` is false in JavaScript.
+   */
+
+
+  var _proto = Settings.prototype;
+
+  _proto.reportNonstrict = function reportNonstrict(errorCode, errorMsg, token) {
+    var strict = this.strict;
+
+    if (typeof strict === "function") {
+      // Allow return value of strict function to be boolean or string
+      // (or null/undefined, meaning no further processing).
+      strict = strict(errorCode, errorMsg, token);
+    }
+
+    if (!strict || strict === "ignore") {
+      return;
+    } else if (strict === true || strict === "error") {
+      throw new src_ParseError("LaTeX-incompatible input and strict mode is set to 'error': " + (errorMsg + " [" + errorCode + "]"), token);
+    } else if (strict === "warn") {
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to 'warn': " + (errorMsg + " [" + errorCode + "]"));
+    } else {
+      // won't happen in type-safe code
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to " + ("unrecognized '" + strict + "': " + errorMsg + " [" + errorCode + "]"));
+    }
+  }
+  /**
+   * Check whether to apply strict (LaTeX-adhering) behavior for unusual
+   * input (like `\\`).  Unlike `nonstrict`, will not throw an error;
+   * instead, "error" translates to a return value of `true`, while "ignore"
+   * translates to a return value of `false`.  May still print a warning:
+   * "warn" prints a warning and returns `false`.
+   * This is for the second category of `errorCode`s listed in the README.
+   */
+  ;
+
+  _proto.useStrictBehavior = function useStrictBehavior(errorCode, errorMsg, token) {
+    var strict = this.strict;
+
+    if (typeof strict === "function") {
+      // Allow return value of strict function to be boolean or string
+      // (or null/undefined, meaning no further processing).
+      // But catch any exceptions thrown by function, treating them
+      // like "error".
+      try {
+        strict = strict(errorCode, errorMsg, token);
+      } catch (error) {
+        strict = "error";
+      }
+    }
+
+    if (!strict || strict === "ignore") {
+      return false;
+    } else if (strict === true || strict === "error") {
+      return true;
+    } else if (strict === "warn") {
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to 'warn': " + (errorMsg + " [" + errorCode + "]"));
+      return false;
+    } else {
+      // won't happen in type-safe code
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to " + ("unrecognized '" + strict + "': " + errorMsg + " [" + errorCode + "]"));
+      return false;
+    }
+  }
+  /**
+   * Check whether to test potentially dangerous input, and return
+   * `true` (trusted) or `false` (untrusted).  The sole argument `context`
+   * should be an object with `command` field specifying the relevant LaTeX
+   * command (as a string starting with `\`), and any other arguments, etc.
+   * If `context` has a `url` field, a `protocol` field will automatically
+   * get added by this function (changing the specified object).
+   */
+  ;
+
+  _proto.isTrusted = function isTrusted(context) {
+    if (context.url && !context.protocol) {
+      context.protocol = utils.protocolFromUrl(context.url);
+    }
+
+    var trust = typeof this.trust === "function" ? this.trust(context) : this.trust;
+    return Boolean(trust);
+  };
+
+  return Settings;
+}();
+
+
+// CONCATENATED MODULE: ./src/Style.js
+/**
+ * This file contains information and classes for the various kinds of styles
+ * used in TeX. It provides a generic `Style` class, which holds information
+ * about a specific style. It then provides instances of all the different kinds
+ * of styles possible, and provides functions to move between them and get
+ * information about them.
+ */
+
+/**
+ * The main style class. Contains a unique id for the style, a size (which is
+ * the same for cramped and uncramped version of a style), and a cramped flag.
+ */
+var Style =
+/*#__PURE__*/
+function () {
+  function Style(id, size, cramped) {
+    this.id = void 0;
+    this.size = void 0;
+    this.cramped = void 0;
+    this.id = id;
+    this.size = size;
+    this.cramped = cramped;
+  }
+  /**
+   * Get the style of a superscript given a base in the current style.
+   */
+
+
+  var _proto = Style.prototype;
+
+  _proto.sup = function sup() {
+    return Style_styles[_sup[this.id]];
+  }
+  /**
+   * Get the style of a subscript given a base in the current style.
+   */
+  ;
+
+  _proto.sub = function sub() {
+    return Style_styles[_sub[this.id]];
+  }
+  /**
+   * Get the style of a fraction numerator given the fraction in the current
+   * style.
+   */
+  ;
+
+  _proto.fracNum = function fracNum() {
+    return Style_styles[_fracNum[this.id]];
+  }
+  /**
+   * Get the style of a fraction denominator given the fraction in the current
+   * style.
+   */
+  ;
+
+  _proto.fracDen = function fracDen() {
+    return Style_styles[_fracDen[this.id]];
+  }
+  /**
+   * Get the cramped version of a style (in particular, cramping a cramped style
+   * doesn't change the style).
+   */
+  ;
+
+  _proto.cramp = function cramp() {
+    return Style_styles[_cramp[this.id]];
+  }
+  /**
+   * Get a text or display version of this style.
+   */
+  ;
+
+  _proto.text = function text() {
+    return Style_styles[_text[this.id]];
+  }
+  /**
+   * Return true if this style is tightly spaced (scriptstyle/scriptscriptstyle)
+   */
+  ;
+
+  _proto.isTight = function isTight() {
+    return this.size >= 2;
+  };
+
+  return Style;
+}(); // Export an interface for type checking, but don't expose the implementation.
+// This way, no more styles can be generated.
+
+
+// IDs of the different styles
+var D = 0;
+var Dc = 1;
+var T = 2;
+var Tc = 3;
+var S = 4;
+var Sc = 5;
+var SS = 6;
+var SSc = 7; // Instances of the different styles
+
+var Style_styles = [new Style(D, 0, false), new Style(Dc, 0, true), new Style(T, 1, false), new Style(Tc, 1, true), new Style(S, 2, false), new Style(Sc, 2, true), new Style(SS, 3, false), new Style(SSc, 3, true)]; // Lookup tables for switching from one style to another
+
+var _sup = [S, Sc, S, Sc, SS, SSc, SS, SSc];
+var _sub = [Sc, Sc, Sc, Sc, SSc, SSc, SSc, SSc];
+var _fracNum = [T, Tc, S, Sc, SS, SSc, SS, SSc];
+var _fracDen = [Tc, Tc, Sc, Sc, SSc, SSc, SSc, SSc];
+var _cramp = [Dc, Dc, Tc, Tc, Sc, Sc, SSc, SSc];
+var _text = [D, Dc, T, Tc, T, Tc, T, Tc]; // We only export some of the styles.
+
+/* harmony default export */ var src_Style = ({
+  DISPLAY: Style_styles[D],
+  TEXT: Style_styles[T],
+  SCRIPT: Style_styles[S],
+  SCRIPTSCRIPT: Style_styles[SS]
+});
+// CONCATENATED MODULE: ./src/unicodeScripts.js
+/*
+ * This file defines the Unicode scripts and script families that we
+ * support. To add new scripts or families, just add a new entry to the
+ * scriptData array below. Adding scripts to the scriptData array allows
+ * characters from that script to appear in \text{} environments.
+ */
+
+/**
+ * Each script or script family has a name and an array of blocks.
+ * Each block is an array of two numbers which specify the start and
+ * end points (inclusive) of a block of Unicode codepoints.
+ */
+
+/**
+ * Unicode block data for the families of scripts we support in \text{}.
+ * Scripts only need to appear here if they do not have font metrics.
+ */
+var scriptData = [{
+  // Latin characters beyond the Latin-1 characters we have metrics for.
+  // Needed for Czech, Hungarian and Turkish text, for example.
+  name: 'latin',
+  blocks: [[0x0100, 0x024f], // Latin Extended-A and Latin Extended-B
+  [0x0300, 0x036f]]
+}, {
+  // The Cyrillic script used by Russian and related languages.
+  // A Cyrillic subset used to be supported as explicitly defined
+  // symbols in symbols.js
+  name: 'cyrillic',
+  blocks: [[0x0400, 0x04ff]]
+}, {
+  // The Brahmic scripts of South and Southeast Asia
+  // Devanagari (0900–097F)
+  // Bengali (0980–09FF)
+  // Gurmukhi (0A00–0A7F)
+  // Gujarati (0A80–0AFF)
+  // Oriya (0B00–0B7F)
+  // Tamil (0B80–0BFF)
+  // Telugu (0C00–0C7F)
+  // Kannada (0C80–0CFF)
+  // Malayalam (0D00–0D7F)
+  // Sinhala (0D80–0DFF)
+  // Thai (0E00–0E7F)
+  // Lao (0E80–0EFF)
+  // Tibetan (0F00–0FFF)
+  // Myanmar (1000–109F)
+  name: 'brahmic',
+  blocks: [[0x0900, 0x109F]]
+}, {
+  name: 'georgian',
+  blocks: [[0x10A0, 0x10ff]]
+}, {
+  // Chinese and Japanese.
+  // The "k" in cjk is for Korean, but we've separated Korean out
+  name: "cjk",
+  blocks: [[0x3000, 0x30FF], // CJK symbols and punctuation, Hiragana, Katakana
+  [0x4E00, 0x9FAF], // CJK ideograms
+  [0xFF00, 0xFF60]]
+}, {
+  // Korean
+  name: 'hangul',
+  blocks: [[0xAC00, 0xD7AF]]
+}];
+/**
+ * Given a codepoint, return the name of the script or script family
+ * it is from, or null if it is not part of a known block
+ */
+
+function scriptFromCodepoint(codepoint) {
+  for (var i = 0; i < scriptData.length; i++) {
+    var script = scriptData[i];
+
+    for (var _i = 0; _i < script.blocks.length; _i++) {
+      var block = script.blocks[_i];
+
+      if (codepoint >= block[0] && codepoint <= block[1]) {
+        return script.name;
+      }
+    }
+  }
+
+  return null;
+}
+/**
+ * A flattened version of all the supported blocks in a single array.
+ * This is an optimization to make supportedCodepoint() fast.
+ */
+
+var allBlocks = [];
+scriptData.forEach(function (s) {
+  return s.blocks.forEach(function (b) {
+    return allBlocks.push.apply(allBlocks, b);
+  });
+});
+/**
+ * Given a codepoint, return true if it falls within one of the
+ * scripts or script families defined above and false otherwise.
+ *
+ * Micro benchmarks shows that this is faster than
+ * /[\u3000-\u30FF\u4E00-\u9FAF\uFF00-\uFF60\uAC00-\uD7AF\u0900-\u109F]/.test()
+ * in Firefox, Chrome and Node.
+ */
+
+function supportedCodepoint(codepoint) {
+  for (var i = 0; i < allBlocks.length; i += 2) {
+    if (codepoint >= allBlocks[i] && codepoint <= allBlocks[i + 1]) {
+      return true;
+    }
+  }
+
+  return false;
+}
+// CONCATENATED MODULE: ./src/svgGeometry.js
+/**
+ * This file provides support to domTree.js and delimiter.js.
+ * It's a storehouse of path geometry for SVG images.
+ */
+// In all paths below, the viewBox-to-em scale is 1000:1.
+var hLinePad = 80; // padding above a sqrt viniculum. Prevents image cropping.
+// The viniculum of a \sqrt can be made thicker by a KaTeX rendering option.
+// Think of variable extraViniculum as two detours in the SVG path.
+// The detour begins at the lower left of the area labeled extraViniculum below.
+// The detour proceeds one extraViniculum distance up and slightly to the right,
+// displacing the radiused corner between surd and viniculum. The radius is
+// traversed as usual, then the detour resumes. It goes right, to the end of
+// the very long viniculumn, then down one extraViniculum distance,
+// after which it resumes regular path geometry for the radical.
+
+/*                                                  viniculum
+                                                   /
+         /▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒←extraViniculum
+        / █████████████████████←0.04em (40 unit) std viniculum thickness
+       / /
+      / /
+     / /\
+    / / surd
+*/
+
+var sqrtMain = function sqrtMain(extraViniculum, hLinePad) {
+  // sqrtMain path geometry is from glyph U221A in the font KaTeX Main
+  return "M95," + (622 + extraViniculum + hLinePad) + "\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl" + extraViniculum / 2.075 + " -" + extraViniculum + "\nc5.3,-9.3,12,-14,20,-14\nH400000v" + (40 + extraViniculum) + "H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM" + (834 + extraViniculum) + " " + hLinePad + "h400000v" + (40 + extraViniculum) + "h-400000z";
+};
+
+var sqrtSize1 = function sqrtSize1(extraViniculum, hLinePad) {
+  // size1 is from glyph U221A in the font KaTeX_Size1-Regular
+  return "M263," + (601 + extraViniculum + hLinePad) + "c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl" + extraViniculum / 2.084 + " -" + extraViniculum + "\nc4.7,-7.3,11,-11,19,-11\nH40000v" + (40 + extraViniculum) + "H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM" + (1001 + extraViniculum) + " " + hLinePad + "h400000v" + (40 + extraViniculum) + "h-400000z";
+};
+
+var sqrtSize2 = function sqrtSize2(extraViniculum, hLinePad) {
+  // size2 is from glyph U221A in the font KaTeX_Size2-Regular
+  return "M983 " + (10 + extraViniculum + hLinePad) + "\nl" + extraViniculum / 3.13 + " -" + extraViniculum + "\nc4,-6.7,10,-10,18,-10 H400000v" + (40 + extraViniculum) + "\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM" + (1001 + extraViniculum) + " " + hLinePad + "h400000v" + (40 + extraViniculum) + "h-400000z";
+};
+
+var sqrtSize3 = function sqrtSize3(extraViniculum, hLinePad) {
+  // size3 is from glyph U221A in the font KaTeX_Size3-Regular
+  return "M424," + (2398 + extraViniculum + hLinePad) + "\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl" + extraViniculum / 4.223 + " -" + extraViniculum + "c4,-6.7,10,-10,18,-10 H400000\nv" + (40 + extraViniculum) + "H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M" + (1001 + extraViniculum) + " " + hLinePad + "\nh400000v" + (40 + extraViniculum) + "h-400000z";
+};
+
+var sqrtSize4 = function sqrtSize4(extraViniculum, hLinePad) {
+  // size4 is from glyph U221A in the font KaTeX_Size4-Regular
+  return "M473," + (2713 + extraViniculum + hLinePad) + "\nc339.3,-1799.3,509.3,-2700,510,-2702 l" + extraViniculum / 5.298 + " -" + extraViniculum + "\nc3.3,-7.3,9.3,-11,18,-11 H400000v" + (40 + extraViniculum) + "H1017.7\ns-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200\nc0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26\ns76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,\n606zM" + (1001 + extraViniculum) + " " + hLinePad + "h400000v" + (40 + extraViniculum) + "H1017.7z";
+};
+
+var sqrtTall = function sqrtTall(extraViniculum, hLinePad, viewBoxHeight) {
+  // sqrtTall is from glyph U23B7 in the font KaTeX_Size4-Regular
+  // One path edge has a variable length. It runs vertically from the viniculumn
+  // to a point near (14 units) the bottom of the surd. The viniculum
+  // is normally 40 units thick. So the length of the line in question is:
+  var vertSegment = viewBoxHeight - 54 - hLinePad - extraViniculum;
+  return "M702 " + (extraViniculum + hLinePad) + "H400000" + (40 + extraViniculum) + "\nH742v" + vertSegment + "l-4 4-4 4c-.667.7 -2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1\nh-12l-28-84c-16.667-52-96.667 -294.333-240-727l-212 -643 -85 170\nc-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667\n219 661 l218 661zM702 " + hLinePad + "H400000v" + (40 + extraViniculum) + "H742z";
+};
+
+var sqrtPath = function sqrtPath(size, extraViniculum, viewBoxHeight) {
+  extraViniculum = 1000 * extraViniculum; // Convert from document ems to viewBox.
+
+  var path = "";
+
+  switch (size) {
+    case "sqrtMain":
+      path = sqrtMain(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize1":
+      path = sqrtSize1(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize2":
+      path = sqrtSize2(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize3":
+      path = sqrtSize3(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize4":
+      path = sqrtSize4(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtTall":
+      path = sqrtTall(extraViniculum, hLinePad, viewBoxHeight);
+  }
+
+  return path;
+};
+var svgGeometry_path = {
+  // Two paths that cover gaps in built-up parentheses.
+  leftParenInner: "M291 0 H417 V300 H291 z",
+  rightParenInner: "M457 0 H583 V300 H457 z",
+  // The doubleleftarrow geometry is from glyph U+21D0 in the font KaTeX Main
+  doubleleftarrow: "M262 157\nl10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3\n 0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28\n 14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5\nc2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5\n 157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87\n-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7\n-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z\nm8 0v40h399730v-40zm0 194v40h399730v-40z",
+  // doublerightarrow is from glyph U+21D2 in font KaTeX Main
+  doublerightarrow: "M399738 392l\n-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5\n 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88\n-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68\n-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18\n-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782\nc-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3\n-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z",
+  // leftarrow is from glyph U+2190 in font KaTeX Main
+  leftarrow: "M400000 241H110l3-3c68.7-52.7 113.7-120\n 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8\n-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247\nc-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208\n 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3\n 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202\n l-3-3h399890zM100 241v40h399900v-40z",
+  // overbrace is from glyphs U+23A9/23A8/23A7 in font KaTeX_Size4-Regular
+  leftbrace: "M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117\n-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7\n 5-6 9-10 13-.7 1-7.3 1-20 1H6z",
+  leftbraceunder: "M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",
+  // overgroup is from the MnSymbol package (public domain)
+  leftgroup: "M400000 80\nH435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0\n 435 0h399565z",
+  leftgroupunder: "M400000 262\nH435C64 262 168.3 112.6 21 82c-5.9-1.2-18 0-18 0-2 0-3 1-3 3v38c76 158 257 219\n 435 219h399565z",
+  // Harpoons are from glyph U+21BD in font KaTeX Main
+  leftharpoon: "M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3\n-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5\n-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7\n-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z",
+  leftharpoonplus: "M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3-3.3 10.2-9.5\n 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5-18.3 3-21-1.3\n-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7-196 228-6.7 4.7\n-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40zM0 435v40h400000v-40z\nm0 0v40h400000v-40z",
+  leftharpoondown: "M7 241c-4 4-6.333 8.667-7 14 0 5.333.667 9 2 11s5.333\n 5.333 12 10c90.667 54 156 130 196 228 3.333 10.667 6.333 16.333 9 17 2 .667 5\n 1 9 1h5c10.667 0 16.667-2 18-6 2-2.667 1-9.667-3-21-32-87.333-82.667-157.667\n-152-211l-3-3h399907v-40zM93 281 H400000 v-40L7 241z",
+  leftharpoondownplus: "M7 435c-4 4-6.3 8.7-7 14 0 5.3.7 9 2 11s5.3 5.3 12\n 10c90.7 54 156 130 196 228 3.3 10.7 6.3 16.3 9 17 2 .7 5 1 9 1h5c10.7 0 16.7\n-2 18-6 2-2.7 1-9.7-3-21-32-87.3-82.7-157.7-152-211l-3-3h399907v-40H7zm93 0\nv40h399900v-40zM0 241v40h399900v-40zm0 0v40h399900v-40z",
+  // hook is from glyph U+21A9 in font KaTeX Main
+  lefthook: "M400000 281 H103s-33-11.2-61-33.5S0 197.3 0 164s14.2-61.2 42.5\n-83.5C70.8 58.2 104 47 142 47 c16.7 0 25 6.7 25 20 0 12-8.7 18.7-26 20-40 3.3\n-68.7 15.7-86 37-10 12-15 25.3-15 40 0 22.7 9.8 40.7 29.5 54 19.7 13.3 43.5 21\n 71.5 23h399859zM103 281v-40h399897v40z",
+  leftlinesegment: "M40 281 V428 H0 V94 H40 V241 H400000 v40z\nM40 281 V428 H0 V94 H40 V241 H400000 v40z",
+  leftmapsto: "M40 281 V448H0V74H40V241H400000v40z\nM40 281 V448H0V74H40V241H400000v40z",
+  // tofrom is from glyph U+21C4 in font KaTeX AMS Regular
+  leftToFrom: "M0 147h400000v40H0zm0 214c68 40 115.7 95.7 143 167h22c15.3 0 23\n-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69-70-101l-7-8h399905v-40H95l7-8\nc28.7-32 52-65.7 70-101 10.7-23.3 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 265.3\n 68 321 0 361zm0-174v-40h399900v40zm100 154v40h399900v-40z",
+  longequal: "M0 50 h400000 v40H0z m0 194h40000v40H0z\nM0 50 h400000 v40H0z m0 194h40000v40H0z",
+  midbrace: "M200428 334\nc-100.7-8.3-195.3-44-280-108-55.3-42-101.7-93-139-153l-9-14c-2.7 4-5.7 8.7-9 14\n-53.3 86.7-123.7 153-211 199-66.7 36-137.3 56.3-212 62H0V214h199568c178.3-11.7\n 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11\n 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z",
+  midbraceunder: "M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",
+  oiintSize1: "M512.6 71.6c272.6 0 320.3 106.8 320.3 178.2 0 70.8-47.7 177.6\n-320.3 177.6S193.1 320.6 193.1 249.8c0-71.4 46.9-178.2 319.5-178.2z\nm368.1 178.2c0-86.4-60.9-215.4-368.1-215.4-306.4 0-367.3 129-367.3 215.4 0 85.8\n60.9 214.8 367.3 214.8 307.2 0 368.1-129 368.1-214.8z",
+  oiintSize2: "M757.8 100.1c384.7 0 451.1 137.6 451.1 230 0 91.3-66.4 228.8\n-451.1 228.8-386.3 0-452.7-137.5-452.7-228.8 0-92.4 66.4-230 452.7-230z\nm502.4 230c0-111.2-82.4-277.2-502.4-277.2s-504 166-504 277.2\nc0 110 84 276 504 276s502.4-166 502.4-276z",
+  oiiintSize1: "M681.4 71.6c408.9 0 480.5 106.8 480.5 178.2 0 70.8-71.6 177.6\n-480.5 177.6S202.1 320.6 202.1 249.8c0-71.4 70.5-178.2 479.3-178.2z\nm525.8 178.2c0-86.4-86.8-215.4-525.7-215.4-437.9 0-524.7 129-524.7 215.4 0\n85.8 86.8 214.8 524.7 214.8 438.9 0 525.7-129 525.7-214.8z",
+  oiiintSize2: "M1021.2 53c603.6 0 707.8 165.8 707.8 277.2 0 110-104.2 275.8\n-707.8 275.8-606 0-710.2-165.8-710.2-275.8C311 218.8 415.2 53 1021.2 53z\nm770.4 277.1c0-131.2-126.4-327.6-770.5-327.6S248.4 198.9 248.4 330.1\nc0 130 128.8 326.4 772.7 326.4s770.5-196.4 770.5-326.4z",
+  rightarrow: "M0 241v40h399891c-47.3 35.3-84 78-110 128\n-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20\n 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7\n 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85\n-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n 151.7 139 205zm0 0v40h399900v-40z",
+  rightbrace: "M400000 542l\n-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5\ns-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1\nc124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z",
+  rightbraceunder: "M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",
+  rightgroup: "M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0\n 3-1 3-3v-38c-76-158-257-219-435-219H0z",
+  rightgroupunder: "M0 262h399565c371 0 266.7-149.4 414-180 5.9-1.2 18 0 18\n 0 2 0 3 1 3 3v38c-76 158-257 219-435 219H0z",
+  rightharpoon: "M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3\n-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2\n-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58\n 69.2 92 94.5zm0 0v40h399900v-40z",
+  rightharpoonplus: "M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3-3.7-15.3-11\n-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2-10.7 0-16.7\n 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58 69.2 92 94.5z\nm0 0v40h399900v-40z m100 194v40h399900v-40zm0 0v40h399900v-40z",
+  rightharpoondown: "M399747 511c0 7.3 6.7 11 20 11 8 0 13-.8 15-2.5s4.7-6.8\n 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3 8.5-5.8 9.5\n-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3-64.7 57-92 95\n-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 241v40h399900v-40z",
+  rightharpoondownplus: "M399747 705c0 7.3 6.7 11 20 11 8 0 13-.8\n 15-2.5s4.7-6.8 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3\n 8.5-5.8 9.5-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3\n-64.7 57-92 95-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 435v40h399900v-40z\nm0-194v40h400000v-40zm0 0v40h400000v-40z",
+  righthook: "M399859 241c-764 0 0 0 0 0 40-3.3 68.7-15.7 86-37 10-12 15-25.3\n 15-40 0-22.7-9.8-40.7-29.5-54-19.7-13.3-43.5-21-71.5-23-17.3-1.3-26-8-26-20 0\n-13.3 8.7-20 26-20 38 0 71 11.2 99 33.5 0 0 7 5.6 21 16.7 14 11.2 21 33.5 21\n 66.8s-14 61.2-42 83.5c-28 22.3-61 33.5-99 33.5L0 241z M0 281v-40h399859v40z",
+  rightlinesegment: "M399960 241 V94 h40 V428 h-40 V281 H0 v-40z\nM399960 241 V94 h40 V428 h-40 V281 H0 v-40z",
+  rightToFrom: "M400000 167c-70.7-42-118-97.7-142-167h-23c-15.3 0-23 .3-23\n 1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8H0v40h399905l-7 8c-28.7 32\n-52 65.7-70 101-10.7 23.3-16 35.7-16 37 0 .7 7.7 1 23 1h23c24-69.3 71.3-125 142\n-167z M100 147v40h399900v-40zM0 341v40h399900v-40z",
+  // twoheadleftarrow is from glyph U+219E in font KaTeX AMS Regular
+  twoheadleftarrow: "M0 167c68 40\n 115.7 95.7 143 167h22c15.3 0 23-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69\n-70-101l-7-8h125l9 7c50.7 39.3 85 86 103 140h46c0-4.7-6.3-18.7-19-42-18-35.3\n-40-67.3-66-96l-9-9h399716v-40H284l9-9c26-28.7 48-60.7 66-96 12.7-23.333 19\n-37.333 19-42h-46c-18 54-52.3 100.7-103 140l-9 7H95l7-8c28.7-32 52-65.7 70-101\n 10.7-23.333 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 71.3 68 127 0 167z",
+  twoheadrightarrow: "M400000 167\nc-68-40-115.7-95.7-143-167h-22c-15.3 0-23 .3-23 1 0 1.3 5.3 13.7 16 37 18 35.3\n 41.3 69 70 101l7 8h-125l-9-7c-50.7-39.3-85-86-103-140h-46c0 4.7 6.3 18.7 19 42\n 18 35.3 40 67.3 66 96l9 9H0v40h399716l-9 9c-26 28.7-48 60.7-66 96-12.7 23.333\n-19 37.333-19 42h46c18-54 52.3-100.7 103-140l9-7h125l-7 8c-28.7 32-52 65.7-70\n 101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z",
+  // tilde1 is a modified version of a glyph from the MnSymbol package
+  tilde1: "M200 55.538c-77 0-168 73.953-177 73.953-3 0-7\n-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0\n 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0\n 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128\n-68.267.847-113-73.952-191-73.952z",
+  // ditto tilde2, tilde3, & tilde4
+  tilde2: "M344 55.266c-142 0-300.638 81.316-311.5 86.418\n-8.01 3.762-22.5 10.91-23.5 5.562L1 120c-1-2-1-3-1-4 0-5 3-9 8-10l18.4-9C160.9\n 31.9 283 0 358 0c148 0 188 122 331 122s314-97 326-97c4 0 8 2 10 7l7 21.114\nc1 2.14 1 3.21 1 4.28 0 5.347-3 9.626-7 10.696l-22.3 12.622C852.6 158.372 751\n 181.476 676 181.476c-149 0-189-126.21-332-126.21z",
+  tilde3: "M786 59C457 59 32 175.242 13 175.242c-6 0-10-3.457\n-11-10.37L.15 138c-1-7 3-12 10-13l19.2-6.4C378.4 40.7 634.3 0 804.3 0c337 0\n 411.8 157 746.8 157 328 0 754-112 773-112 5 0 10 3 11 9l1 14.075c1 8.066-.697\n 16.595-6.697 17.492l-21.052 7.31c-367.9 98.146-609.15 122.696-778.15 122.696\n -338 0-409-156.573-744-156.573z",
+  tilde4: "M786 58C457 58 32 177.487 13 177.487c-6 0-10-3.345\n-11-10.035L.15 143c-1-7 3-12 10-13l22-6.7C381.2 35 637.15 0 807.15 0c337 0 409\n 177 744 177 328 0 754-127 773-127 5 0 10 3 11 9l1 14.794c1 7.805-3 13.38-9\n 14.495l-20.7 5.574c-366.85 99.79-607.3 139.372-776.3 139.372-338 0-409\n -175.236-744-175.236z",
+  // vec is from glyph U+20D7 in font KaTeX Main
+  vec: "M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z",
+  // widehat1 is a modified version of a glyph from the MnSymbol package
+  widehat1: "M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22\nc-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z",
+  // ditto widehat2, widehat3, & widehat4
+  widehat2: "M1181 0h2l1171 176c6 0 10 5 10 11l-2 23c-1 6-5 10\n-11 10h-1L1182 67 15 220h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z",
+  widehat3: "M1181 0h2l1171 236c6 0 10 5 10 11l-2 23c-1 6-5 10\n-11 10h-1L1182 67 15 280h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z",
+  widehat4: "M1181 0h2l1171 296c6 0 10 5 10 11l-2 23c-1 6-5 10\n-11 10h-1L1182 67 15 340h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z",
+  // widecheck paths are all inverted versions of widehat
+  widecheck1: "M529,159h5l519,-115c5,-1,9,-5,9,-10c0,-1,-1,-2,-1,-3l-4,-22c-1,\n-5,-5,-9,-11,-9h-2l-512,92l-513,-92h-2c-5,0,-9,4,-11,9l-5,22c-1,6,2,12,8,13z",
+  widecheck2: "M1181,220h2l1171,-176c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,\n-11,-10h-1l-1168,153l-1167,-153h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z",
+  widecheck3: "M1181,280h2l1171,-236c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,\n-11,-10h-1l-1168,213l-1167,-213h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z",
+  widecheck4: "M1181,340h2l1171,-296c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,\n-11,-10h-1l-1168,273l-1167,-273h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z",
+  // The next ten paths support reaction arrows from the mhchem package.
+  // Arrows for \ce{<-->} are offset from xAxis by 0.22ex, per mhchem in LaTeX
+  // baraboveleftarrow is mostly from from glyph U+2190 in font KaTeX Main
+  baraboveleftarrow: "M400000 620h-399890l3 -3c68.7 -52.7 113.7 -120 135 -202\nc4 -14.7 6 -23 6 -25c0 -7.3 -7 -11 -21 -11c-8 0 -13.2 0.8 -15.5 2.5\nc-2.3 1.7 -4.2 5.8 -5.5 12.5c-1.3 4.7 -2.7 10.3 -4 17c-12 48.7 -34.8 92 -68.5 130\ns-74.2 66.3 -121.5 85c-10 4 -16 7.7 -18 11c0 8.7 6 14.3 18 17c47.3 18.7 87.8 47\n121.5 85s56.5 81.3 68.5 130c0.7 2 1.3 5 2 9s1.2 6.7 1.5 8c0.3 1.3 1 3.3 2 6\ns2.2 4.5 3.5 5.5c1.3 1 3.3 1.8 6 2.5s6 1 10 1c14 0 21 -3.7 21 -11\nc0 -2 -2 -10.3 -6 -25c-20 -79.3 -65 -146.7 -135 -202l-3 -3h399890z\nM100 620v40h399900v-40z M0 241v40h399900v-40zM0 241v40h399900v-40z",
+  // rightarrowabovebar is mostly from glyph U+2192, KaTeX Main
+  rightarrowabovebar: "M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32\n-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0\n13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39\n-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5\n-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5\n-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67\n151.7 139 205zm96 379h399894v40H0zm0 0h399904v40H0z",
+  // The short left harpoon has 0.5em (i.e. 500 units) kern on the left end.
+  // Ref from mhchem.sty: \rlap{\raisebox{-.22ex}{$\kern0.5em
+  baraboveshortleftharpoon: "M507,435c-4,4,-6.3,8.7,-7,14c0,5.3,0.7,9,2,11\nc1.3,2,5.3,5.3,12,10c90.7,54,156,130,196,228c3.3,10.7,6.3,16.3,9,17\nc2,0.7,5,1,9,1c0,0,5,0,5,0c10.7,0,16.7,-2,18,-6c2,-2.7,1,-9.7,-3,-21\nc-32,-87.3,-82.7,-157.7,-152,-211c0,0,-3,-3,-3,-3l399351,0l0,-40\nc-398570,0,-399437,0,-399437,0z M593 435 v40 H399500 v-40z\nM0 281 v-40 H399908 v40z M0 281 v-40 H399908 v40z",
+  rightharpoonaboveshortbar: "M0,241 l0,40c399126,0,399993,0,399993,0\nc4.7,-4.7,7,-9.3,7,-14c0,-9.3,-3.7,-15.3,-11,-18c-92.7,-56.7,-159,-133.7,-199,\n-231c-3.3,-9.3,-6,-14.7,-8,-16c-2,-1.3,-7,-2,-15,-2c-10.7,0,-16.7,2,-18,6\nc-2,2.7,-1,9.7,3,21c15.3,42,36.7,81.8,64,119.5c27.3,37.7,58,69.2,92,94.5z\nM0 241 v40 H399908 v-40z M0 475 v-40 H399500 v40z M0 475 v-40 H399500 v40z",
+  shortbaraboveleftharpoon: "M7,435c-4,4,-6.3,8.7,-7,14c0,5.3,0.7,9,2,11\nc1.3,2,5.3,5.3,12,10c90.7,54,156,130,196,228c3.3,10.7,6.3,16.3,9,17c2,0.7,5,1,9,\n1c0,0,5,0,5,0c10.7,0,16.7,-2,18,-6c2,-2.7,1,-9.7,-3,-21c-32,-87.3,-82.7,-157.7,\n-152,-211c0,0,-3,-3,-3,-3l399907,0l0,-40c-399126,0,-399993,0,-399993,0z\nM93 435 v40 H400000 v-40z M500 241 v40 H400000 v-40z M500 241 v40 H400000 v-40z",
+  shortrightharpoonabovebar: "M53,241l0,40c398570,0,399437,0,399437,0\nc4.7,-4.7,7,-9.3,7,-14c0,-9.3,-3.7,-15.3,-11,-18c-92.7,-56.7,-159,-133.7,-199,\n-231c-3.3,-9.3,-6,-14.7,-8,-16c-2,-1.3,-7,-2,-15,-2c-10.7,0,-16.7,2,-18,6\nc-2,2.7,-1,9.7,3,21c15.3,42,36.7,81.8,64,119.5c27.3,37.7,58,69.2,92,94.5z\nM500 241 v40 H399408 v-40z M500 435 v40 H400000 v-40z"
+};
+// CONCATENATED MODULE: ./src/tree.js
+
+
+/**
+ * This node represents a document fragment, which contains elements, but when
+ * placed into the DOM doesn't have any representation itself. It only contains
+ * children and doesn't have any DOM node properties.
+ */
+var tree_DocumentFragment =
+/*#__PURE__*/
+function () {
+  // HtmlDomNode
+  // Never used; needed for satisfying interface.
+  function DocumentFragment(children) {
+    this.children = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    this.children = children;
+    this.classes = [];
+    this.height = 0;
+    this.depth = 0;
+    this.maxFontSize = 0;
+    this.style = {};
+  }
+
+  var _proto = DocumentFragment.prototype;
+
+  _proto.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+  /** Convert the fragment into a node. */
+  ;
+
+  _proto.toNode = function toNode() {
+    var frag = document.createDocumentFragment();
+
+    for (var i = 0; i < this.children.length; i++) {
+      frag.appendChild(this.children[i].toNode());
+    }
+
+    return frag;
+  }
+  /** Convert the fragment into HTML markup. */
+  ;
+
+  _proto.toMarkup = function toMarkup() {
+    var markup = ""; // Simply concatenate the markup for the children together.
+
+    for (var i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    return markup;
+  }
+  /**
+   * Converts the math node into a string, similar to innerText. Applies to
+   * MathDomNode's only.
+   */
+  ;
+
+  _proto.toText = function toText() {
+    // To avoid this, we would subclass documentFragment separately for
+    // MathML, but polyfills for subclassing is expensive per PR 1469.
+    // $FlowFixMe: Only works for ChildType = MathDomNode.
+    var toText = function toText(child) {
+      return child.toText();
+    };
+
+    return this.children.map(toText).join("");
+  };
+
+  return DocumentFragment;
+}();
+// CONCATENATED MODULE: ./src/domTree.js
+/**
+ * These objects store the data about the DOM nodes we create, as well as some
+ * extra data. They can then be transformed into real DOM nodes with the
+ * `toNode` function or HTML markup using `toMarkup`. They are useful for both
+ * storing extra properties on the nodes, as well as providing a way to easily
+ * work with the DOM.
+ *
+ * Similar functions for working with MathML nodes exist in mathMLTree.js.
+ *
+ * TODO: refactor `span` and `anchor` into common superclass when
+ * target environments support class inheritance
+ */
+
+
+
+
+
+/**
+ * Create an HTML className based on a list of classes. In addition to joining
+ * with spaces, we also remove empty classes.
+ */
+var createClass = function createClass(classes) {
+  return classes.filter(function (cls) {
+    return cls;
+  }).join(" ");
+};
+
+var initNode = function initNode(classes, options, style) {
+  this.classes = classes || [];
+  this.attributes = {};
+  this.height = 0;
+  this.depth = 0;
+  this.maxFontSize = 0;
+  this.style = style || {};
+
+  if (options) {
+    if (options.style.isTight()) {
+      this.classes.push("mtight");
+    }
+
+    var color = options.getColor();
+
+    if (color) {
+      this.style.color = color;
+    }
+  }
+};
+/**
+ * Convert into an HTML node
+ */
+
+
+var _toNode = function toNode(tagName) {
+  var node = document.createElement(tagName); // Apply the class
+
+  node.className = createClass(this.classes); // Apply inline styles
+
+  for (var style in this.style) {
+    if (this.style.hasOwnProperty(style)) {
+      // $FlowFixMe Flow doesn't seem to understand span.style's type.
+      node.style[style] = this.style[style];
+    }
+  } // Apply attributes
+
+
+  for (var attr in this.attributes) {
+    if (this.attributes.hasOwnProperty(attr)) {
+      node.setAttribute(attr, this.attributes[attr]);
+    }
+  } // Append the children, also as HTML nodes
+
+
+  for (var i = 0; i < this.children.length; i++) {
+    node.appendChild(this.children[i].toNode());
+  }
+
+  return node;
+};
+/**
+ * Convert into an HTML markup string
+ */
+
+
+var _toMarkup = function toMarkup(tagName) {
+  var markup = "<" + tagName; // Add the class
+
+  if (this.classes.length) {
+    markup += " class=\"" + utils.escape(createClass(this.classes)) + "\"";
+  }
+
+  var styles = ""; // Add the styles, after hyphenation
+
+  for (var style in this.style) {
+    if (this.style.hasOwnProperty(style)) {
+      styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+    }
+  }
+
+  if (styles) {
+    markup += " style=\"" + utils.escape(styles) + "\"";
+  } // Add the attributes
+
+
+  for (var attr in this.attributes) {
+    if (this.attributes.hasOwnProperty(attr)) {
+      markup += " " + attr + "=\"" + utils.escape(this.attributes[attr]) + "\"";
+    }
+  }
+
+  markup += ">"; // Add the markup of the children, also as markup
+
+  for (var i = 0; i < this.children.length; i++) {
+    markup += this.children[i].toMarkup();
+  }
+
+  markup += "</" + tagName + ">";
+  return markup;
+}; // Making the type below exact with all optional fields doesn't work due to
+// - https://github.com/facebook/flow/issues/4582
+// - https://github.com/facebook/flow/issues/5688
+// However, since *all* fields are optional, $Shape<> works as suggested in 5688
+// above.
+// This type does not include all CSS properties. Additional properties should
+// be added as needed.
+
+
+/**
+ * This node represents a span node, with a className, a list of children, and
+ * an inline style. It also contains information about its height, depth, and
+ * maxFontSize.
+ *
+ * Represents two types with different uses: SvgSpan to wrap an SVG and DomSpan
+ * otherwise. This typesafety is important when HTML builders access a span's
+ * children.
+ */
+var domTree_Span =
+/*#__PURE__*/
+function () {
+  function Span(classes, children, options, style) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.width = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    initNode.call(this, classes, options, style);
+    this.children = children || [];
+  }
+  /**
+   * Sets an arbitrary attribute on the span. Warning: use this wisely. Not
+   * all browsers support attributes the same, and having too many custom
+   * attributes is probably bad.
+   */
+
+
+  var _proto = Span.prototype;
+
+  _proto.setAttribute = function setAttribute(attribute, value) {
+    this.attributes[attribute] = value;
+  };
+
+  _proto.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  };
+
+  _proto.toNode = function toNode() {
+    return _toNode.call(this, "span");
+  };
+
+  _proto.toMarkup = function toMarkup() {
+    return _toMarkup.call(this, "span");
+  };
+
+  return Span;
+}();
+/**
+ * This node represents an anchor (<a>) element with a hyperlink.  See `span`
+ * for further details.
+ */
+
+var domTree_Anchor =
+/*#__PURE__*/
+function () {
+  function Anchor(href, classes, children, options) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    initNode.call(this, classes, options);
+    this.children = children || [];
+    this.setAttribute('href', href);
+  }
+
+  var _proto2 = Anchor.prototype;
+
+  _proto2.setAttribute = function setAttribute(attribute, value) {
+    this.attributes[attribute] = value;
+  };
+
+  _proto2.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  };
+
+  _proto2.toNode = function toNode() {
+    return _toNode.call(this, "a");
+  };
+
+  _proto2.toMarkup = function toMarkup() {
+    return _toMarkup.call(this, "a");
+  };
+
+  return Anchor;
+}();
+/**
+ * This node represents an image embed (<img>) element.
+ */
+
+var domTree_Img =
+/*#__PURE__*/
+function () {
+  function Img(src, alt, style) {
+    this.src = void 0;
+    this.alt = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    this.alt = alt;
+    this.src = src;
+    this.classes = ["mord"];
+    this.style = style;
+  }
+
+  var _proto3 = Img.prototype;
+
+  _proto3.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  };
+
+  _proto3.toNode = function toNode() {
+    var node = document.createElement("img");
+    node.src = this.src;
+    node.alt = this.alt;
+    node.className = "mord"; // Apply inline styles
+
+    for (var style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        // $FlowFixMe
+        node.style[style] = this.style[style];
+      }
+    }
+
+    return node;
+  };
+
+  _proto3.toMarkup = function toMarkup() {
+    var markup = "<img  src='" + this.src + " 'alt='" + this.alt + "' "; // Add the styles, after hyphenation
+
+    var styles = "";
+
+    for (var style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+      }
+    }
+
+    if (styles) {
+      markup += " style=\"" + utils.escape(styles) + "\"";
+    }
+
+    markup += "'/>";
+    return markup;
+  };
+
+  return Img;
+}();
+var iCombinations = {
+  'î': "\u0131\u0302",
+  'ï': "\u0131\u0308",
+  'í': "\u0131\u0301",
+  // 'ī': '\u0131\u0304', // enable when we add Extended Latin
+  'ì': "\u0131\u0300"
+};
+/**
+ * A symbol node contains information about a single symbol. It either renders
+ * to a single text node, or a span with a single text node in it, depending on
+ * whether it has CSS classes, styles, or needs italic correction.
+ */
+
+var domTree_SymbolNode =
+/*#__PURE__*/
+function () {
+  function SymbolNode(text, height, depth, italic, skew, width, classes, style) {
+    this.text = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.italic = void 0;
+    this.skew = void 0;
+    this.width = void 0;
+    this.maxFontSize = void 0;
+    this.classes = void 0;
+    this.style = void 0;
+    this.text = text;
+    this.height = height || 0;
+    this.depth = depth || 0;
+    this.italic = italic || 0;
+    this.skew = skew || 0;
+    this.width = width || 0;
+    this.classes = classes || [];
+    this.style = style || {};
+    this.maxFontSize = 0; // Mark text from non-Latin scripts with specific classes so that we
+    // can specify which fonts to use.  This allows us to render these
+    // characters with a serif font in situations where the browser would
+    // either default to a sans serif or render a placeholder character.
+    // We use CSS class names like cjk_fallback, hangul_fallback and
+    // brahmic_fallback. See ./unicodeScripts.js for the set of possible
+    // script names
+
+    var script = scriptFromCodepoint(this.text.charCodeAt(0));
+
+    if (script) {
+      this.classes.push(script + "_fallback");
+    }
+
+    if (/[îïíì]/.test(this.text)) {
+      // add ī when we add Extended Latin
+      this.text = iCombinations[this.text];
+    }
+  }
+
+  var _proto4 = SymbolNode.prototype;
+
+  _proto4.hasClass = function hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+  /**
+   * Creates a text node or span from a symbol node. Note that a span is only
+   * created if it is needed.
+   */
+  ;
+
+  _proto4.toNode = function toNode() {
+    var node = document.createTextNode(this.text);
+    var span = null;
+
+    if (this.italic > 0) {
+      span = document.createElement("span");
+      span.style.marginRight = this.italic + "em";
+    }
+
+    if (this.classes.length > 0) {
+      span = span || document.createElement("span");
+      span.className = createClass(this.classes);
+    }
+
+    for (var style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        span = span || document.createElement("span"); // $FlowFixMe Flow doesn't seem to understand span.style's type.
+
+        span.style[style] = this.style[style];
+      }
+    }
+
+    if (span) {
+      span.appendChild(node);
+      return span;
+    } else {
+      return node;
+    }
+  }
+  /**
+   * Creates markup for a symbol node.
+   */
+  ;
+
+  _proto4.toMarkup = function toMarkup() {
+    // TODO(alpert): More duplication than I'd like from
+    // span.prototype.toMarkup and symbolNode.prototype.toNode...
+    var needsSpan = false;
+    var markup = "<span";
+
+    if (this.classes.length) {
+      needsSpan = true;
+      markup += " class=\"";
+      markup += utils.escape(createClass(this.classes));
+      markup += "\"";
+    }
+
+    var styles = "";
+
+    if (this.italic > 0) {
+      styles += "margin-right:" + this.italic + "em;";
+    }
+
+    for (var style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+      }
+    }
+
+    if (styles) {
+      needsSpan = true;
+      markup += " style=\"" + utils.escape(styles) + "\"";
+    }
+
+    var escaped = utils.escape(this.text);
+
+    if (needsSpan) {
+      markup += ">";
+      markup += escaped;
+      markup += "</span>";
+      return markup;
+    } else {
+      return escaped;
+    }
+  };
+
+  return SymbolNode;
+}();
+/**
+ * SVG nodes are used to render stretchy wide elements.
+ */
+
+var SvgNode =
+/*#__PURE__*/
+function () {
+  function SvgNode(children, attributes) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.children = children || [];
+    this.attributes = attributes || {};
+  }
+
+  var _proto5 = SvgNode.prototype;
+
+  _proto5.toNode = function toNode() {
+    var svgNS = "http://www.w3.org/2000/svg";
+    var node = document.createElementNS(svgNS, "svg"); // Apply attributes
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    for (var i = 0; i < this.children.length; i++) {
+      node.appendChild(this.children[i].toNode());
+    }
+
+    return node;
+  };
+
+  _proto5.toMarkup = function toMarkup() {
+    var markup = "<svg"; // Apply attributes
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += " " + attr + "='" + this.attributes[attr] + "'";
+      }
+    }
+
+    markup += ">";
+
+    for (var i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    markup += "</svg>";
+    return markup;
+  };
+
+  return SvgNode;
+}();
+var domTree_PathNode =
+/*#__PURE__*/
+function () {
+  function PathNode(pathName, alternate) {
+    this.pathName = void 0;
+    this.alternate = void 0;
+    this.pathName = pathName;
+    this.alternate = alternate; // Used only for \sqrt
+  }
+
+  var _proto6 = PathNode.prototype;
+
+  _proto6.toNode = function toNode() {
+    var svgNS = "http://www.w3.org/2000/svg";
+    var node = document.createElementNS(svgNS, "path");
+
+    if (this.alternate) {
+      node.setAttribute("d", this.alternate);
+    } else {
+      node.setAttribute("d", svgGeometry_path[this.pathName]);
+    }
+
+    return node;
+  };
+
+  _proto6.toMarkup = function toMarkup() {
+    if (this.alternate) {
+      return "<path d='" + this.alternate + "'/>";
+    } else {
+      return "<path d='" + svgGeometry_path[this.pathName] + "'/>";
+    }
+  };
+
+  return PathNode;
+}();
+var LineNode =
+/*#__PURE__*/
+function () {
+  function LineNode(attributes) {
+    this.attributes = void 0;
+    this.attributes = attributes || {};
+  }
+
+  var _proto7 = LineNode.prototype;
+
+  _proto7.toNode = function toNode() {
+    var svgNS = "http://www.w3.org/2000/svg";
+    var node = document.createElementNS(svgNS, "line"); // Apply attributes
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    return node;
+  };
+
+  _proto7.toMarkup = function toMarkup() {
+    var markup = "<line";
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += " " + attr + "='" + this.attributes[attr] + "'";
+      }
+    }
+
+    markup += "/>";
+    return markup;
+  };
+
+  return LineNode;
+}();
+function assertSymbolDomNode(group) {
+  if (group instanceof domTree_SymbolNode) {
+    return group;
+  } else {
+    throw new Error("Expected symbolNode but got " + String(group) + ".");
+  }
+}
+function assertSpan(group) {
+  if (group instanceof domTree_Span) {
+    return group;
+  } else {
+    throw new Error("Expected span<HtmlDomNode> but got " + String(group) + ".");
+  }
+}
+// CONCATENATED MODULE: ./submodules/katex-fonts/fontMetricsData.js
+// This file is GENERATED by buildMetrics.sh. DO NOT MODIFY.
+/* harmony default export */ var fontMetricsData = ({
+  "AMS-Regular": {
+    "32": [0, 0, 0, 0, 0.25],
+    "65": [0, 0.68889, 0, 0, 0.72222],
+    "66": [0, 0.68889, 0, 0, 0.66667],
+    "67": [0, 0.68889, 0, 0, 0.72222],
+    "68": [0, 0.68889, 0, 0, 0.72222],
+    "69": [0, 0.68889, 0, 0, 0.66667],
+    "70": [0, 0.68889, 0, 0, 0.61111],
+    "71": [0, 0.68889, 0, 0, 0.77778],
+    "72": [0, 0.68889, 0, 0, 0.77778],
+    "73": [0, 0.68889, 0, 0, 0.38889],
+    "74": [0.16667, 0.68889, 0, 0, 0.5],
+    "75": [0, 0.68889, 0, 0, 0.77778],
+    "76": [0, 0.68889, 0, 0, 0.66667],
+    "77": [0, 0.68889, 0, 0, 0.94445],
+    "78": [0, 0.68889, 0, 0, 0.72222],
+    "79": [0.16667, 0.68889, 0, 0, 0.77778],
+    "80": [0, 0.68889, 0, 0, 0.61111],
+    "81": [0.16667, 0.68889, 0, 0, 0.77778],
+    "82": [0, 0.68889, 0, 0, 0.72222],
+    "83": [0, 0.68889, 0, 0, 0.55556],
+    "84": [0, 0.68889, 0, 0, 0.66667],
+    "85": [0, 0.68889, 0, 0, 0.72222],
+    "86": [0, 0.68889, 0, 0, 0.72222],
+    "87": [0, 0.68889, 0, 0, 1.0],
+    "88": [0, 0.68889, 0, 0, 0.72222],
+    "89": [0, 0.68889, 0, 0, 0.72222],
+    "90": [0, 0.68889, 0, 0, 0.66667],
+    "107": [0, 0.68889, 0, 0, 0.55556],
+    "160": [0, 0, 0, 0, 0.25],
+    "165": [0, 0.675, 0.025, 0, 0.75],
+    "174": [0.15559, 0.69224, 0, 0, 0.94666],
+    "240": [0, 0.68889, 0, 0, 0.55556],
+    "295": [0, 0.68889, 0, 0, 0.54028],
+    "710": [0, 0.825, 0, 0, 2.33334],
+    "732": [0, 0.9, 0, 0, 2.33334],
+    "770": [0, 0.825, 0, 0, 2.33334],
+    "771": [0, 0.9, 0, 0, 2.33334],
+    "989": [0.08167, 0.58167, 0, 0, 0.77778],
+    "1008": [0, 0.43056, 0.04028, 0, 0.66667],
+    "8245": [0, 0.54986, 0, 0, 0.275],
+    "8463": [0, 0.68889, 0, 0, 0.54028],
+    "8487": [0, 0.68889, 0, 0, 0.72222],
+    "8498": [0, 0.68889, 0, 0, 0.55556],
+    "8502": [0, 0.68889, 0, 0, 0.66667],
+    "8503": [0, 0.68889, 0, 0, 0.44445],
+    "8504": [0, 0.68889, 0, 0, 0.66667],
+    "8513": [0, 0.68889, 0, 0, 0.63889],
+    "8592": [-0.03598, 0.46402, 0, 0, 0.5],
+    "8594": [-0.03598, 0.46402, 0, 0, 0.5],
+    "8602": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8603": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8606": [0.01354, 0.52239, 0, 0, 1.0],
+    "8608": [0.01354, 0.52239, 0, 0, 1.0],
+    "8610": [0.01354, 0.52239, 0, 0, 1.11111],
+    "8611": [0.01354, 0.52239, 0, 0, 1.11111],
+    "8619": [0, 0.54986, 0, 0, 1.0],
+    "8620": [0, 0.54986, 0, 0, 1.0],
+    "8621": [-0.13313, 0.37788, 0, 0, 1.38889],
+    "8622": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8624": [0, 0.69224, 0, 0, 0.5],
+    "8625": [0, 0.69224, 0, 0, 0.5],
+    "8630": [0, 0.43056, 0, 0, 1.0],
+    "8631": [0, 0.43056, 0, 0, 1.0],
+    "8634": [0.08198, 0.58198, 0, 0, 0.77778],
+    "8635": [0.08198, 0.58198, 0, 0, 0.77778],
+    "8638": [0.19444, 0.69224, 0, 0, 0.41667],
+    "8639": [0.19444, 0.69224, 0, 0, 0.41667],
+    "8642": [0.19444, 0.69224, 0, 0, 0.41667],
+    "8643": [0.19444, 0.69224, 0, 0, 0.41667],
+    "8644": [0.1808, 0.675, 0, 0, 1.0],
+    "8646": [0.1808, 0.675, 0, 0, 1.0],
+    "8647": [0.1808, 0.675, 0, 0, 1.0],
+    "8648": [0.19444, 0.69224, 0, 0, 0.83334],
+    "8649": [0.1808, 0.675, 0, 0, 1.0],
+    "8650": [0.19444, 0.69224, 0, 0, 0.83334],
+    "8651": [0.01354, 0.52239, 0, 0, 1.0],
+    "8652": [0.01354, 0.52239, 0, 0, 1.0],
+    "8653": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8654": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8655": [-0.13313, 0.36687, 0, 0, 1.0],
+    "8666": [0.13667, 0.63667, 0, 0, 1.0],
+    "8667": [0.13667, 0.63667, 0, 0, 1.0],
+    "8669": [-0.13313, 0.37788, 0, 0, 1.0],
+    "8672": [-0.064, 0.437, 0, 0, 1.334],
+    "8674": [-0.064, 0.437, 0, 0, 1.334],
+    "8705": [0, 0.825, 0, 0, 0.5],
+    "8708": [0, 0.68889, 0, 0, 0.55556],
+    "8709": [0.08167, 0.58167, 0, 0, 0.77778],
+    "8717": [0, 0.43056, 0, 0, 0.42917],
+    "8722": [-0.03598, 0.46402, 0, 0, 0.5],
+    "8724": [0.08198, 0.69224, 0, 0, 0.77778],
+    "8726": [0.08167, 0.58167, 0, 0, 0.77778],
+    "8733": [0, 0.69224, 0, 0, 0.77778],
+    "8736": [0, 0.69224, 0, 0, 0.72222],
+    "8737": [0, 0.69224, 0, 0, 0.72222],
+    "8738": [0.03517, 0.52239, 0, 0, 0.72222],
+    "8739": [0.08167, 0.58167, 0, 0, 0.22222],
+    "8740": [0.25142, 0.74111, 0, 0, 0.27778],
+    "8741": [0.08167, 0.58167, 0, 0, 0.38889],
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+    "73": [0, 0.61111, 0, 0, 0.525],
+    "74": [0, 0.61111, 0, 0, 0.525],
+    "75": [0, 0.61111, 0, 0, 0.525],
+    "76": [0, 0.61111, 0, 0, 0.525],
+    "77": [0, 0.61111, 0, 0, 0.525],
+    "78": [0, 0.61111, 0, 0, 0.525],
+    "79": [0, 0.61111, 0, 0, 0.525],
+    "80": [0, 0.61111, 0, 0, 0.525],
+    "81": [0.13889, 0.61111, 0, 0, 0.525],
+    "82": [0, 0.61111, 0, 0, 0.525],
+    "83": [0, 0.61111, 0, 0, 0.525],
+    "84": [0, 0.61111, 0, 0, 0.525],
+    "85": [0, 0.61111, 0, 0, 0.525],
+    "86": [0, 0.61111, 0, 0, 0.525],
+    "87": [0, 0.61111, 0, 0, 0.525],
+    "88": [0, 0.61111, 0, 0, 0.525],
+    "89": [0, 0.61111, 0, 0, 0.525],
+    "90": [0, 0.61111, 0, 0, 0.525],
+    "91": [0.08333, 0.69444, 0, 0, 0.525],
+    "92": [0.08333, 0.69444, 0, 0, 0.525],
+    "93": [0.08333, 0.69444, 0, 0, 0.525],
+    "94": [0, 0.61111, 0, 0, 0.525],
+    "95": [0.09514, 0, 0, 0, 0.525],
+    "96": [0, 0.61111, 0, 0, 0.525],
+    "97": [0, 0.43056, 0, 0, 0.525],
+    "98": [0, 0.61111, 0, 0, 0.525],
+    "99": [0, 0.43056, 0, 0, 0.525],
+    "100": [0, 0.61111, 0, 0, 0.525],
+    "101": [0, 0.43056, 0, 0, 0.525],
+    "102": [0, 0.61111, 0, 0, 0.525],
+    "103": [0.22222, 0.43056, 0, 0, 0.525],
+    "104": [0, 0.61111, 0, 0, 0.525],
+    "105": [0, 0.61111, 0, 0, 0.525],
+    "106": [0.22222, 0.61111, 0, 0, 0.525],
+    "107": [0, 0.61111, 0, 0, 0.525],
+    "108": [0, 0.61111, 0, 0, 0.525],
+    "109": [0, 0.43056, 0, 0, 0.525],
+    "110": [0, 0.43056, 0, 0, 0.525],
+    "111": [0, 0.43056, 0, 0, 0.525],
+    "112": [0.22222, 0.43056, 0, 0, 0.525],
+    "113": [0.22222, 0.43056, 0, 0, 0.525],
+    "114": [0, 0.43056, 0, 0, 0.525],
+    "115": [0, 0.43056, 0, 0, 0.525],
+    "116": [0, 0.55358, 0, 0, 0.525],
+    "117": [0, 0.43056, 0, 0, 0.525],
+    "118": [0, 0.43056, 0, 0, 0.525],
+    "119": [0, 0.43056, 0, 0, 0.525],
+    "120": [0, 0.43056, 0, 0, 0.525],
+    "121": [0.22222, 0.43056, 0, 0, 0.525],
+    "122": [0, 0.43056, 0, 0, 0.525],
+    "123": [0.08333, 0.69444, 0, 0, 0.525],
+    "124": [0.08333, 0.69444, 0, 0, 0.525],
+    "125": [0.08333, 0.69444, 0, 0, 0.525],
+    "126": [0, 0.61111, 0, 0, 0.525],
+    "127": [0, 0.61111, 0, 0, 0.525],
+    "160": [0, 0, 0, 0, 0.525],
+    "176": [0, 0.61111, 0, 0, 0.525],
+    "184": [0.19445, 0, 0, 0, 0.525],
+    "305": [0, 0.43056, 0, 0, 0.525],
+    "567": [0.22222, 0.43056, 0, 0, 0.525],
+    "711": [0, 0.56597, 0, 0, 0.525],
+    "713": [0, 0.56555, 0, 0, 0.525],
+    "714": [0, 0.61111, 0, 0, 0.525],
+    "715": [0, 0.61111, 0, 0, 0.525],
+    "728": [0, 0.61111, 0, 0, 0.525],
+    "730": [0, 0.61111, 0, 0, 0.525],
+    "770": [0, 0.61111, 0, 0, 0.525],
+    "771": [0, 0.61111, 0, 0, 0.525],
+    "776": [0, 0.61111, 0, 0, 0.525],
+    "915": [0, 0.61111, 0, 0, 0.525],
+    "916": [0, 0.61111, 0, 0, 0.525],
+    "920": [0, 0.61111, 0, 0, 0.525],
+    "923": [0, 0.61111, 0, 0, 0.525],
+    "926": [0, 0.61111, 0, 0, 0.525],
+    "928": [0, 0.61111, 0, 0, 0.525],
+    "931": [0, 0.61111, 0, 0, 0.525],
+    "933": [0, 0.61111, 0, 0, 0.525],
+    "934": [0, 0.61111, 0, 0, 0.525],
+    "936": [0, 0.61111, 0, 0, 0.525],
+    "937": [0, 0.61111, 0, 0, 0.525],
+    "8216": [0, 0.61111, 0, 0, 0.525],
+    "8217": [0, 0.61111, 0, 0, 0.525],
+    "8242": [0, 0.61111, 0, 0, 0.525],
+    "9251": [0.11111, 0.21944, 0, 0, 0.525]
+  }
+});
+// CONCATENATED MODULE: ./src/fontMetrics.js
+
+
+/**
+ * This file contains metrics regarding fonts and individual symbols. The sigma
+ * and xi variables, as well as the metricMap map contain data extracted from
+ * TeX, TeX font metrics, and the TTF files. These data are then exposed via the
+ * `metrics` variable and the getCharacterMetrics function.
+ */
+// In TeX, there are actually three sets of dimensions, one for each of
+// textstyle (size index 5 and higher: >=9pt), scriptstyle (size index 3 and 4:
+// 7-8pt), and scriptscriptstyle (size index 1 and 2: 5-6pt).  These are
+// provided in the the arrays below, in that order.
+//
+// The font metrics are stored in fonts cmsy10, cmsy7, and cmsy5 respsectively.
+// This was determined by running the following script:
+//
+//     latex -interaction=nonstopmode \
+//     '\documentclass{article}\usepackage{amsmath}\begin{document}' \
+//     '$a$ \expandafter\show\the\textfont2' \
+//     '\expandafter\show\the\scriptfont2' \
+//     '\expandafter\show\the\scriptscriptfont2' \
+//     '\stop'
+//
+// The metrics themselves were retreived using the following commands:
+//
+//     tftopl cmsy10
+//     tftopl cmsy7
+//     tftopl cmsy5
+//
+// The output of each of these commands is quite lengthy.  The only part we
+// care about is the FONTDIMEN section. Each value is measured in EMs.
+var sigmasAndXis = {
+  slant: [0.250, 0.250, 0.250],
+  // sigma1
+  space: [0.000, 0.000, 0.000],
+  // sigma2
+  stretch: [0.000, 0.000, 0.000],
+  // sigma3
+  shrink: [0.000, 0.000, 0.000],
+  // sigma4
+  xHeight: [0.431, 0.431, 0.431],
+  // sigma5
+  quad: [1.000, 1.171, 1.472],
+  // sigma6
+  extraSpace: [0.000, 0.000, 0.000],
+  // sigma7
+  num1: [0.677, 0.732, 0.925],
+  // sigma8
+  num2: [0.394, 0.384, 0.387],
+  // sigma9
+  num3: [0.444, 0.471, 0.504],
+  // sigma10
+  denom1: [0.686, 0.752, 1.025],
+  // sigma11
+  denom2: [0.345, 0.344, 0.532],
+  // sigma12
+  sup1: [0.413, 0.503, 0.504],
+  // sigma13
+  sup2: [0.363, 0.431, 0.404],
+  // sigma14
+  sup3: [0.289, 0.286, 0.294],
+  // sigma15
+  sub1: [0.150, 0.143, 0.200],
+  // sigma16
+  sub2: [0.247, 0.286, 0.400],
+  // sigma17
+  supDrop: [0.386, 0.353, 0.494],
+  // sigma18
+  subDrop: [0.050, 0.071, 0.100],
+  // sigma19
+  delim1: [2.390, 1.700, 1.980],
+  // sigma20
+  delim2: [1.010, 1.157, 1.420],
+  // sigma21
+  axisHeight: [0.250, 0.250, 0.250],
+  // sigma22
+  // These font metrics are extracted from TeX by using tftopl on cmex10.tfm;
+  // they correspond to the font parameters of the extension fonts (family 3).
+  // See the TeXbook, page 441. In AMSTeX, the extension fonts scale; to
+  // match cmex7, we'd use cmex7.tfm values for script and scriptscript
+  // values.
+  defaultRuleThickness: [0.04, 0.049, 0.049],
+  // xi8; cmex7: 0.049
+  bigOpSpacing1: [0.111, 0.111, 0.111],
+  // xi9
+  bigOpSpacing2: [0.166, 0.166, 0.166],
+  // xi10
+  bigOpSpacing3: [0.2, 0.2, 0.2],
+  // xi11
+  bigOpSpacing4: [0.6, 0.611, 0.611],
+  // xi12; cmex7: 0.611
+  bigOpSpacing5: [0.1, 0.143, 0.143],
+  // xi13; cmex7: 0.143
+  // The \sqrt rule width is taken from the height of the surd character.
+  // Since we use the same font at all sizes, this thickness doesn't scale.
+  sqrtRuleThickness: [0.04, 0.04, 0.04],
+  // This value determines how large a pt is, for metrics which are defined
+  // in terms of pts.
+  // This value is also used in katex.less; if you change it make sure the
+  // values match.
+  ptPerEm: [10.0, 10.0, 10.0],
+  // The space between adjacent `|` columns in an array definition. From
+  // `\showthe\doublerulesep` in LaTeX. Equals 2.0 / ptPerEm.
+  doubleRuleSep: [0.2, 0.2, 0.2],
+  // The width of separator lines in {array} environments. From
+  // `\showthe\arrayrulewidth` in LaTeX. Equals 0.4 / ptPerEm.
+  arrayRuleWidth: [0.04, 0.04, 0.04],
+  // Two values from LaTeX source2e:
+  fboxsep: [0.3, 0.3, 0.3],
+  //        3 pt / ptPerEm
+  fboxrule: [0.04, 0.04, 0.04] // 0.4 pt / ptPerEm
+
+}; // This map contains a mapping from font name and character code to character
+// metrics, including height, depth, italic correction, and skew (kern from the
+// character to the corresponding \skewchar)
+// This map is generated via `make metrics`. It should not be changed manually.
+
+ // These are very rough approximations.  We default to Times New Roman which
+// should have Latin-1 and Cyrillic characters, but may not depending on the
+// operating system.  The metrics do not account for extra height from the
+// accents.  In the case of Cyrillic characters which have both ascenders and
+// descenders we prefer approximations with ascenders, primarily to prevent
+// the fraction bar or root line from intersecting the glyph.
+// TODO(kevinb) allow union of multiple glyph metrics for better accuracy.
+
+var extraCharacterMap = {
+  // Latin-1
+  'Å': 'A',
+  'Ç': 'C',
+  'Ð': 'D',
+  'Þ': 'o',
+  'å': 'a',
+  'ç': 'c',
+  'ð': 'd',
+  'þ': 'o',
+  // Cyrillic
+  'А': 'A',
+  'Б': 'B',
+  'В': 'B',
+  'Г': 'F',
+  'Д': 'A',
+  'Е': 'E',
+  'Ж': 'K',
+  'З': '3',
+  'И': 'N',
+  'Й': 'N',
+  'К': 'K',
+  'Л': 'N',
+  'М': 'M',
+  'Н': 'H',
+  'О': 'O',
+  'П': 'N',
+  'Р': 'P',
+  'С': 'C',
+  'Т': 'T',
+  'У': 'y',
+  'Ф': 'O',
+  'Х': 'X',
+  'Ц': 'U',
+  'Ч': 'h',
+  'Ш': 'W',
+  'Щ': 'W',
+  'Ъ': 'B',
+  'Ы': 'X',
+  'Ь': 'B',
+  'Э': '3',
+  'Ю': 'X',
+  'Я': 'R',
+  'а': 'a',
+  'б': 'b',
+  'в': 'a',
+  'г': 'r',
+  'д': 'y',
+  'е': 'e',
+  'ж': 'm',
+  'з': 'e',
+  'и': 'n',
+  'й': 'n',
+  'к': 'n',
+  'л': 'n',
+  'м': 'm',
+  'н': 'n',
+  'о': 'o',
+  'п': 'n',
+  'р': 'p',
+  'с': 'c',
+  'т': 'o',
+  'у': 'y',
+  'ф': 'b',
+  'х': 'x',
+  'ц': 'n',
+  'ч': 'n',
+  'ш': 'w',
+  'щ': 'w',
+  'ъ': 'a',
+  'ы': 'm',
+  'ь': 'a',
+  'э': 'e',
+  'ю': 'm',
+  'я': 'r'
+};
+
+/**
+ * This function adds new font metrics to default metricMap
+ * It can also override existing metrics
+ */
+function setFontMetrics(fontName, metrics) {
+  fontMetricsData[fontName] = metrics;
+}
+/**
+ * This function is a convenience function for looking up information in the
+ * metricMap table. It takes a character as a string, and a font.
+ *
+ * Note: the `width` property may be undefined if fontMetricsData.js wasn't
+ * built using `Make extended_metrics`.
+ */
+
+function getCharacterMetrics(character, font, mode) {
+  if (!fontMetricsData[font]) {
+    throw new Error("Font metrics not found for font: " + font + ".");
+  }
+
+  var ch = character.charCodeAt(0);
+  var metrics = fontMetricsData[font][ch];
+
+  if (!metrics && character[0] in extraCharacterMap) {
+    ch = extraCharacterMap[character[0]].charCodeAt(0);
+    metrics = fontMetricsData[font][ch];
+  }
+
+  if (!metrics && mode === 'text') {
+    // We don't typically have font metrics for Asian scripts.
+    // But since we support them in text mode, we need to return
+    // some sort of metrics.
+    // So if the character is in a script we support but we
+    // don't have metrics for it, just use the metrics for
+    // the Latin capital letter M. This is close enough because
+    // we (currently) only care about the height of the glpyh
+    // not its width.
+    if (supportedCodepoint(ch)) {
+      metrics = fontMetricsData[font][77]; // 77 is the charcode for 'M'
+    }
+  }
+
+  if (metrics) {
+    return {
+      depth: metrics[0],
+      height: metrics[1],
+      italic: metrics[2],
+      skew: metrics[3],
+      width: metrics[4]
+    };
+  }
+}
+var fontMetricsBySizeIndex = {};
+/**
+ * Get the font metrics for a given size.
+ */
+
+function getGlobalMetrics(size) {
+  var sizeIndex;
+
+  if (size >= 5) {
+    sizeIndex = 0;
+  } else if (size >= 3) {
+    sizeIndex = 1;
+  } else {
+    sizeIndex = 2;
+  }
+
+  if (!fontMetricsBySizeIndex[sizeIndex]) {
+    var metrics = fontMetricsBySizeIndex[sizeIndex] = {
+      cssEmPerMu: sigmasAndXis.quad[sizeIndex] / 18
+    };
+
+    for (var key in sigmasAndXis) {
+      if (sigmasAndXis.hasOwnProperty(key)) {
+        metrics[key] = sigmasAndXis[key][sizeIndex];
+      }
+    }
+  }
+
+  return fontMetricsBySizeIndex[sizeIndex];
+}
+// CONCATENATED MODULE: ./src/symbols.js
+/**
+ * This file holds a list of all no-argument functions and single-character
+ * symbols (like 'a' or ';').
+ *
+ * For each of the symbols, there are three properties they can have:
+ * - font (required): the font to be used for this symbol. Either "main" (the
+     normal font), or "ams" (the ams fonts).
+ * - group (required): the ParseNode group type the symbol should have (i.e.
+     "textord", "mathord", etc).
+     See https://github.com/KaTeX/KaTeX/wiki/Examining-TeX#group-types
+ * - replace: the character that this symbol or function should be
+ *   replaced with (i.e. "\phi" has a replace value of "\u03d5", the phi
+ *   character in the main font).
+ *
+ * The outermost map in the table indicates what mode the symbols should be
+ * accepted in (e.g. "math" or "text").
+ */
+// Some of these have a "-token" suffix since these are also used as `ParseNode`
+// types for raw text tokens, and we want to avoid conflicts with higher-level
+// `ParseNode` types. These `ParseNode`s are constructed within `Parser` by
+// looking up the `symbols` map.
+var ATOMS = {
+  "bin": 1,
+  "close": 1,
+  "inner": 1,
+  "open": 1,
+  "punct": 1,
+  "rel": 1
+};
+var NON_ATOMS = {
+  "accent-token": 1,
+  "mathord": 1,
+  "op-token": 1,
+  "spacing": 1,
+  "textord": 1
+};
+var symbols = {
+  "math": {},
+  "text": {}
+};
+/* harmony default export */ var src_symbols = (symbols);
+/** `acceptUnicodeChar = true` is only applicable if `replace` is set. */
+
+function defineSymbol(mode, font, group, replace, name, acceptUnicodeChar) {
+  symbols[mode][name] = {
+    font: font,
+    group: group,
+    replace: replace
+  };
+
+  if (acceptUnicodeChar && replace) {
+    symbols[mode][replace] = symbols[mode][name];
+  }
+} // Some abbreviations for commonly used strings.
+// This helps minify the code, and also spotting typos using jshint.
+// modes:
+
+var symbols_math = "math";
+var symbols_text = "text"; // fonts:
+
+var main = "main";
+var ams = "ams"; // groups:
+
+var symbols_accent = "accent-token";
+var bin = "bin";
+var symbols_close = "close";
+var symbols_inner = "inner";
+var mathord = "mathord";
+var op = "op-token";
+var symbols_open = "open";
+var punct = "punct";
+var rel = "rel";
+var symbols_spacing = "spacing";
+var symbols_textord = "textord"; // Now comes the symbol table
+// Relation Symbols
+
+defineSymbol(symbols_math, main, rel, "\u2261", "\\equiv", true);
+defineSymbol(symbols_math, main, rel, "\u227A", "\\prec", true);
+defineSymbol(symbols_math, main, rel, "\u227B", "\\succ", true);
+defineSymbol(symbols_math, main, rel, "\u223C", "\\sim", true);
+defineSymbol(symbols_math, main, rel, "\u22A5", "\\perp");
+defineSymbol(symbols_math, main, rel, "\u2AAF", "\\preceq", true);
+defineSymbol(symbols_math, main, rel, "\u2AB0", "\\succeq", true);
+defineSymbol(symbols_math, main, rel, "\u2243", "\\simeq", true);
+defineSymbol(symbols_math, main, rel, "\u2223", "\\mid", true);
+defineSymbol(symbols_math, main, rel, "\u226A", "\\ll", true);
+defineSymbol(symbols_math, main, rel, "\u226B", "\\gg", true);
+defineSymbol(symbols_math, main, rel, "\u224D", "\\asymp", true);
+defineSymbol(symbols_math, main, rel, "\u2225", "\\parallel");
+defineSymbol(symbols_math, main, rel, "\u22C8", "\\bowtie", true);
+defineSymbol(symbols_math, main, rel, "\u2323", "\\smile", true);
+defineSymbol(symbols_math, main, rel, "\u2291", "\\sqsubseteq", true);
+defineSymbol(symbols_math, main, rel, "\u2292", "\\sqsupseteq", true);
+defineSymbol(symbols_math, main, rel, "\u2250", "\\doteq", true);
+defineSymbol(symbols_math, main, rel, "\u2322", "\\frown", true);
+defineSymbol(symbols_math, main, rel, "\u220B", "\\ni", true);
+defineSymbol(symbols_math, main, rel, "\u221D", "\\propto", true);
+defineSymbol(symbols_math, main, rel, "\u22A2", "\\vdash", true);
+defineSymbol(symbols_math, main, rel, "\u22A3", "\\dashv", true);
+defineSymbol(symbols_math, main, rel, "\u220B", "\\owns"); // Punctuation
+
+defineSymbol(symbols_math, main, punct, ".", "\\ldotp");
+defineSymbol(symbols_math, main, punct, "\u22C5", "\\cdotp"); // Misc Symbols
+
+defineSymbol(symbols_math, main, symbols_textord, "#", "\\#");
+defineSymbol(symbols_text, main, symbols_textord, "#", "\\#");
+defineSymbol(symbols_math, main, symbols_textord, "&", "\\&");
+defineSymbol(symbols_text, main, symbols_textord, "&", "\\&");
+defineSymbol(symbols_math, main, symbols_textord, "\u2135", "\\aleph", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2200", "\\forall", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u210F", "\\hbar", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2203", "\\exists", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2207", "\\nabla", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u266D", "\\flat", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2113", "\\ell", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u266E", "\\natural", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2663", "\\clubsuit", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2118", "\\wp", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u266F", "\\sharp", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2662", "\\diamondsuit", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u211C", "\\Re", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2661", "\\heartsuit", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2111", "\\Im", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2660", "\\spadesuit", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xA7", "\\S", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xB6", "\\P", true); // Math and Text
+
+defineSymbol(symbols_math, main, symbols_textord, "\u2020", "\\dag");
+defineSymbol(symbols_text, main, symbols_textord, "\u2020", "\\dag");
+defineSymbol(symbols_text, main, symbols_textord, "\u2020", "\\textdagger");
+defineSymbol(symbols_math, main, symbols_textord, "\u2021", "\\ddag");
+defineSymbol(symbols_text, main, symbols_textord, "\u2021", "\\ddag");
+defineSymbol(symbols_text, main, symbols_textord, "\u2021", "\\textdaggerdbl"); // Large Delimiters
+
+defineSymbol(symbols_math, main, symbols_close, "\u23B1", "\\rmoustache", true);
+defineSymbol(symbols_math, main, symbols_open, "\u23B0", "\\lmoustache", true);
+defineSymbol(symbols_math, main, symbols_close, "\u27EF", "\\rgroup", true);
+defineSymbol(symbols_math, main, symbols_open, "\u27EE", "\\lgroup", true); // Binary Operators
+
+defineSymbol(symbols_math, main, bin, "\u2213", "\\mp", true);
+defineSymbol(symbols_math, main, bin, "\u2296", "\\ominus", true);
+defineSymbol(symbols_math, main, bin, "\u228E", "\\uplus", true);
+defineSymbol(symbols_math, main, bin, "\u2293", "\\sqcap", true);
+defineSymbol(symbols_math, main, bin, "\u2217", "\\ast");
+defineSymbol(symbols_math, main, bin, "\u2294", "\\sqcup", true);
+defineSymbol(symbols_math, main, bin, "\u25EF", "\\bigcirc");
+defineSymbol(symbols_math, main, bin, "\u2219", "\\bullet");
+defineSymbol(symbols_math, main, bin, "\u2021", "\\ddagger");
+defineSymbol(symbols_math, main, bin, "\u2240", "\\wr", true);
+defineSymbol(symbols_math, main, bin, "\u2A3F", "\\amalg");
+defineSymbol(symbols_math, main, bin, "&", "\\And"); // from amsmath
+// Arrow Symbols
+
+defineSymbol(symbols_math, main, rel, "\u27F5", "\\longleftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D0", "\\Leftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27F8", "\\Longleftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27F6", "\\longrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D2", "\\Rightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27F9", "\\Longrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2194", "\\leftrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27F7", "\\longleftrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D4", "\\Leftrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u27FA", "\\Longleftrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21A6", "\\mapsto", true);
+defineSymbol(symbols_math, main, rel, "\u27FC", "\\longmapsto", true);
+defineSymbol(symbols_math, main, rel, "\u2197", "\\nearrow", true);
+defineSymbol(symbols_math, main, rel, "\u21A9", "\\hookleftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21AA", "\\hookrightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2198", "\\searrow", true);
+defineSymbol(symbols_math, main, rel, "\u21BC", "\\leftharpoonup", true);
+defineSymbol(symbols_math, main, rel, "\u21C0", "\\rightharpoonup", true);
+defineSymbol(symbols_math, main, rel, "\u2199", "\\swarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21BD", "\\leftharpoondown", true);
+defineSymbol(symbols_math, main, rel, "\u21C1", "\\rightharpoondown", true);
+defineSymbol(symbols_math, main, rel, "\u2196", "\\nwarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21CC", "\\rightleftharpoons", true); // AMS Negated Binary Relations
+
+defineSymbol(symbols_math, ams, rel, "\u226E", "\\nless", true); // Symbol names preceeded by "@" each have a corresponding macro.
+
+defineSymbol(symbols_math, ams, rel, "\uE010", "\\@nleqslant");
+defineSymbol(symbols_math, ams, rel, "\uE011", "\\@nleqq");
+defineSymbol(symbols_math, ams, rel, "\u2A87", "\\lneq", true);
+defineSymbol(symbols_math, ams, rel, "\u2268", "\\lneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE00C", "\\@lvertneqq");
+defineSymbol(symbols_math, ams, rel, "\u22E6", "\\lnsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2A89", "\\lnapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u2280", "\\nprec", true); // unicode-math maps \u22e0 to \npreccurlyeq. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u22E0", "\\npreceq", true);
+defineSymbol(symbols_math, ams, rel, "\u22E8", "\\precnsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB9", "\\precnapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u2241", "\\nsim", true);
+defineSymbol(symbols_math, ams, rel, "\uE006", "\\@nshortmid");
+defineSymbol(symbols_math, ams, rel, "\u2224", "\\nmid", true);
+defineSymbol(symbols_math, ams, rel, "\u22AC", "\\nvdash", true);
+defineSymbol(symbols_math, ams, rel, "\u22AD", "\\nvDash", true);
+defineSymbol(symbols_math, ams, rel, "\u22EA", "\\ntriangleleft");
+defineSymbol(symbols_math, ams, rel, "\u22EC", "\\ntrianglelefteq", true);
+defineSymbol(symbols_math, ams, rel, "\u228A", "\\subsetneq", true);
+defineSymbol(symbols_math, ams, rel, "\uE01A", "\\@varsubsetneq");
+defineSymbol(symbols_math, ams, rel, "\u2ACB", "\\subsetneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE017", "\\@varsubsetneqq");
+defineSymbol(symbols_math, ams, rel, "\u226F", "\\ngtr", true);
+defineSymbol(symbols_math, ams, rel, "\uE00F", "\\@ngeqslant");
+defineSymbol(symbols_math, ams, rel, "\uE00E", "\\@ngeqq");
+defineSymbol(symbols_math, ams, rel, "\u2A88", "\\gneq", true);
+defineSymbol(symbols_math, ams, rel, "\u2269", "\\gneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE00D", "\\@gvertneqq");
+defineSymbol(symbols_math, ams, rel, "\u22E7", "\\gnsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2A8A", "\\gnapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u2281", "\\nsucc", true); // unicode-math maps \u22e1 to \nsucccurlyeq. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u22E1", "\\nsucceq", true);
+defineSymbol(symbols_math, ams, rel, "\u22E9", "\\succnsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2ABA", "\\succnapprox", true); // unicode-math maps \u2246 to \simneqq. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u2246", "\\ncong", true);
+defineSymbol(symbols_math, ams, rel, "\uE007", "\\@nshortparallel");
+defineSymbol(symbols_math, ams, rel, "\u2226", "\\nparallel", true);
+defineSymbol(symbols_math, ams, rel, "\u22AF", "\\nVDash", true);
+defineSymbol(symbols_math, ams, rel, "\u22EB", "\\ntriangleright");
+defineSymbol(symbols_math, ams, rel, "\u22ED", "\\ntrianglerighteq", true);
+defineSymbol(symbols_math, ams, rel, "\uE018", "\\@nsupseteqq");
+defineSymbol(symbols_math, ams, rel, "\u228B", "\\supsetneq", true);
+defineSymbol(symbols_math, ams, rel, "\uE01B", "\\@varsupsetneq");
+defineSymbol(symbols_math, ams, rel, "\u2ACC", "\\supsetneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE019", "\\@varsupsetneqq");
+defineSymbol(symbols_math, ams, rel, "\u22AE", "\\nVdash", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB5", "\\precneqq", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB6", "\\succneqq", true);
+defineSymbol(symbols_math, ams, rel, "\uE016", "\\@nsubseteqq");
+defineSymbol(symbols_math, ams, bin, "\u22B4", "\\unlhd");
+defineSymbol(symbols_math, ams, bin, "\u22B5", "\\unrhd"); // AMS Negated Arrows
+
+defineSymbol(symbols_math, ams, rel, "\u219A", "\\nleftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u219B", "\\nrightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21CD", "\\nLeftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21CF", "\\nRightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21AE", "\\nleftrightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21CE", "\\nLeftrightarrow", true); // AMS Misc
+
+defineSymbol(symbols_math, ams, rel, "\u25B3", "\\vartriangle");
+defineSymbol(symbols_math, ams, symbols_textord, "\u210F", "\\hslash");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25BD", "\\triangledown");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25CA", "\\lozenge");
+defineSymbol(symbols_math, ams, symbols_textord, "\u24C8", "\\circledS");
+defineSymbol(symbols_math, ams, symbols_textord, "\xAE", "\\circledR");
+defineSymbol(symbols_text, ams, symbols_textord, "\xAE", "\\circledR");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2221", "\\measuredangle", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2204", "\\nexists");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2127", "\\mho");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2132", "\\Finv", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2141", "\\Game", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2035", "\\backprime");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25B2", "\\blacktriangle");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25BC", "\\blacktriangledown");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25A0", "\\blacksquare");
+defineSymbol(symbols_math, ams, symbols_textord, "\u29EB", "\\blacklozenge");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2605", "\\bigstar");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2222", "\\sphericalangle", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2201", "\\complement", true); // unicode-math maps U+F0 to \matheth. We map to AMS function \eth
+
+defineSymbol(symbols_math, ams, symbols_textord, "\xF0", "\\eth", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xF0", "\xF0");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2571", "\\diagup");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2572", "\\diagdown");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25A1", "\\square");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25A1", "\\Box");
+defineSymbol(symbols_math, ams, symbols_textord, "\u25CA", "\\Diamond"); // unicode-math maps U+A5 to \mathyen. We map to AMS function \yen
+
+defineSymbol(symbols_math, ams, symbols_textord, "\xA5", "\\yen", true);
+defineSymbol(symbols_text, ams, symbols_textord, "\xA5", "\\yen", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2713", "\\checkmark", true);
+defineSymbol(symbols_text, ams, symbols_textord, "\u2713", "\\checkmark"); // AMS Hebrew
+
+defineSymbol(symbols_math, ams, symbols_textord, "\u2136", "\\beth", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2138", "\\daleth", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2137", "\\gimel", true); // AMS Greek
+
+defineSymbol(symbols_math, ams, symbols_textord, "\u03DD", "\\digamma", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u03F0", "\\varkappa"); // AMS Delimiters
+
+defineSymbol(symbols_math, ams, symbols_open, "\u250C", "\\@ulcorner", true);
+defineSymbol(symbols_math, ams, symbols_close, "\u2510", "\\@urcorner", true);
+defineSymbol(symbols_math, ams, symbols_open, "\u2514", "\\@llcorner", true);
+defineSymbol(symbols_math, ams, symbols_close, "\u2518", "\\@lrcorner", true); // AMS Binary Relations
+
+defineSymbol(symbols_math, ams, rel, "\u2266", "\\leqq", true);
+defineSymbol(symbols_math, ams, rel, "\u2A7D", "\\leqslant", true);
+defineSymbol(symbols_math, ams, rel, "\u2A95", "\\eqslantless", true);
+defineSymbol(symbols_math, ams, rel, "\u2272", "\\lesssim", true);
+defineSymbol(symbols_math, ams, rel, "\u2A85", "\\lessapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u224A", "\\approxeq", true);
+defineSymbol(symbols_math, ams, bin, "\u22D6", "\\lessdot");
+defineSymbol(symbols_math, ams, rel, "\u22D8", "\\lll", true);
+defineSymbol(symbols_math, ams, rel, "\u2276", "\\lessgtr", true);
+defineSymbol(symbols_math, ams, rel, "\u22DA", "\\lesseqgtr", true);
+defineSymbol(symbols_math, ams, rel, "\u2A8B", "\\lesseqqgtr", true);
+defineSymbol(symbols_math, ams, rel, "\u2251", "\\doteqdot");
+defineSymbol(symbols_math, ams, rel, "\u2253", "\\risingdotseq", true);
+defineSymbol(symbols_math, ams, rel, "\u2252", "\\fallingdotseq", true);
+defineSymbol(symbols_math, ams, rel, "\u223D", "\\backsim", true);
+defineSymbol(symbols_math, ams, rel, "\u22CD", "\\backsimeq", true);
+defineSymbol(symbols_math, ams, rel, "\u2AC5", "\\subseteqq", true);
+defineSymbol(symbols_math, ams, rel, "\u22D0", "\\Subset", true);
+defineSymbol(symbols_math, ams, rel, "\u228F", "\\sqsubset", true);
+defineSymbol(symbols_math, ams, rel, "\u227C", "\\preccurlyeq", true);
+defineSymbol(symbols_math, ams, rel, "\u22DE", "\\curlyeqprec", true);
+defineSymbol(symbols_math, ams, rel, "\u227E", "\\precsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB7", "\\precapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u22B2", "\\vartriangleleft");
+defineSymbol(symbols_math, ams, rel, "\u22B4", "\\trianglelefteq");
+defineSymbol(symbols_math, ams, rel, "\u22A8", "\\vDash", true);
+defineSymbol(symbols_math, ams, rel, "\u22AA", "\\Vvdash", true);
+defineSymbol(symbols_math, ams, rel, "\u2323", "\\smallsmile");
+defineSymbol(symbols_math, ams, rel, "\u2322", "\\smallfrown");
+defineSymbol(symbols_math, ams, rel, "\u224F", "\\bumpeq", true);
+defineSymbol(symbols_math, ams, rel, "\u224E", "\\Bumpeq", true);
+defineSymbol(symbols_math, ams, rel, "\u2267", "\\geqq", true);
+defineSymbol(symbols_math, ams, rel, "\u2A7E", "\\geqslant", true);
+defineSymbol(symbols_math, ams, rel, "\u2A96", "\\eqslantgtr", true);
+defineSymbol(symbols_math, ams, rel, "\u2273", "\\gtrsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2A86", "\\gtrapprox", true);
+defineSymbol(symbols_math, ams, bin, "\u22D7", "\\gtrdot");
+defineSymbol(symbols_math, ams, rel, "\u22D9", "\\ggg", true);
+defineSymbol(symbols_math, ams, rel, "\u2277", "\\gtrless", true);
+defineSymbol(symbols_math, ams, rel, "\u22DB", "\\gtreqless", true);
+defineSymbol(symbols_math, ams, rel, "\u2A8C", "\\gtreqqless", true);
+defineSymbol(symbols_math, ams, rel, "\u2256", "\\eqcirc", true);
+defineSymbol(symbols_math, ams, rel, "\u2257", "\\circeq", true);
+defineSymbol(symbols_math, ams, rel, "\u225C", "\\triangleq", true);
+defineSymbol(symbols_math, ams, rel, "\u223C", "\\thicksim");
+defineSymbol(symbols_math, ams, rel, "\u2248", "\\thickapprox");
+defineSymbol(symbols_math, ams, rel, "\u2AC6", "\\supseteqq", true);
+defineSymbol(symbols_math, ams, rel, "\u22D1", "\\Supset", true);
+defineSymbol(symbols_math, ams, rel, "\u2290", "\\sqsupset", true);
+defineSymbol(symbols_math, ams, rel, "\u227D", "\\succcurlyeq", true);
+defineSymbol(symbols_math, ams, rel, "\u22DF", "\\curlyeqsucc", true);
+defineSymbol(symbols_math, ams, rel, "\u227F", "\\succsim", true);
+defineSymbol(symbols_math, ams, rel, "\u2AB8", "\\succapprox", true);
+defineSymbol(symbols_math, ams, rel, "\u22B3", "\\vartriangleright");
+defineSymbol(symbols_math, ams, rel, "\u22B5", "\\trianglerighteq");
+defineSymbol(symbols_math, ams, rel, "\u22A9", "\\Vdash", true);
+defineSymbol(symbols_math, ams, rel, "\u2223", "\\shortmid");
+defineSymbol(symbols_math, ams, rel, "\u2225", "\\shortparallel");
+defineSymbol(symbols_math, ams, rel, "\u226C", "\\between", true);
+defineSymbol(symbols_math, ams, rel, "\u22D4", "\\pitchfork", true);
+defineSymbol(symbols_math, ams, rel, "\u221D", "\\varpropto");
+defineSymbol(symbols_math, ams, rel, "\u25C0", "\\blacktriangleleft"); // unicode-math says that \therefore is a mathord atom.
+// We kept the amssymb atom type, which is rel.
+
+defineSymbol(symbols_math, ams, rel, "\u2234", "\\therefore", true);
+defineSymbol(symbols_math, ams, rel, "\u220D", "\\backepsilon");
+defineSymbol(symbols_math, ams, rel, "\u25B6", "\\blacktriangleright"); // unicode-math says that \because is a mathord atom.
+// We kept the amssymb atom type, which is rel.
+
+defineSymbol(symbols_math, ams, rel, "\u2235", "\\because", true);
+defineSymbol(symbols_math, ams, rel, "\u22D8", "\\llless");
+defineSymbol(symbols_math, ams, rel, "\u22D9", "\\gggtr");
+defineSymbol(symbols_math, ams, bin, "\u22B2", "\\lhd");
+defineSymbol(symbols_math, ams, bin, "\u22B3", "\\rhd");
+defineSymbol(symbols_math, ams, rel, "\u2242", "\\eqsim", true);
+defineSymbol(symbols_math, main, rel, "\u22C8", "\\Join");
+defineSymbol(symbols_math, ams, rel, "\u2251", "\\Doteq", true); // AMS Binary Operators
+
+defineSymbol(symbols_math, ams, bin, "\u2214", "\\dotplus", true);
+defineSymbol(symbols_math, ams, bin, "\u2216", "\\smallsetminus");
+defineSymbol(symbols_math, ams, bin, "\u22D2", "\\Cap", true);
+defineSymbol(symbols_math, ams, bin, "\u22D3", "\\Cup", true);
+defineSymbol(symbols_math, ams, bin, "\u2A5E", "\\doublebarwedge", true);
+defineSymbol(symbols_math, ams, bin, "\u229F", "\\boxminus", true);
+defineSymbol(symbols_math, ams, bin, "\u229E", "\\boxplus", true);
+defineSymbol(symbols_math, ams, bin, "\u22C7", "\\divideontimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22C9", "\\ltimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22CA", "\\rtimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22CB", "\\leftthreetimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22CC", "\\rightthreetimes", true);
+defineSymbol(symbols_math, ams, bin, "\u22CF", "\\curlywedge", true);
+defineSymbol(symbols_math, ams, bin, "\u22CE", "\\curlyvee", true);
+defineSymbol(symbols_math, ams, bin, "\u229D", "\\circleddash", true);
+defineSymbol(symbols_math, ams, bin, "\u229B", "\\circledast", true);
+defineSymbol(symbols_math, ams, bin, "\u22C5", "\\centerdot");
+defineSymbol(symbols_math, ams, bin, "\u22BA", "\\intercal", true);
+defineSymbol(symbols_math, ams, bin, "\u22D2", "\\doublecap");
+defineSymbol(symbols_math, ams, bin, "\u22D3", "\\doublecup");
+defineSymbol(symbols_math, ams, bin, "\u22A0", "\\boxtimes", true); // AMS Arrows
+// Note: unicode-math maps \u21e2 to their own function \rightdasharrow.
+// We'll map it to AMS function \dashrightarrow. It produces the same atom.
+
+defineSymbol(symbols_math, ams, rel, "\u21E2", "\\dashrightarrow", true); // unicode-math maps \u21e0 to \leftdasharrow. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u21E0", "\\dashleftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21C7", "\\leftleftarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21C6", "\\leftrightarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21DA", "\\Lleftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u219E", "\\twoheadleftarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21A2", "\\leftarrowtail", true);
+defineSymbol(symbols_math, ams, rel, "\u21AB", "\\looparrowleft", true);
+defineSymbol(symbols_math, ams, rel, "\u21CB", "\\leftrightharpoons", true);
+defineSymbol(symbols_math, ams, rel, "\u21B6", "\\curvearrowleft", true); // unicode-math maps \u21ba to \acwopencirclearrow. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u21BA", "\\circlearrowleft", true);
+defineSymbol(symbols_math, ams, rel, "\u21B0", "\\Lsh", true);
+defineSymbol(symbols_math, ams, rel, "\u21C8", "\\upuparrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21BF", "\\upharpoonleft", true);
+defineSymbol(symbols_math, ams, rel, "\u21C3", "\\downharpoonleft", true);
+defineSymbol(symbols_math, ams, rel, "\u22B8", "\\multimap", true);
+defineSymbol(symbols_math, ams, rel, "\u21AD", "\\leftrightsquigarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21C9", "\\rightrightarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21C4", "\\rightleftarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21A0", "\\twoheadrightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21A3", "\\rightarrowtail", true);
+defineSymbol(symbols_math, ams, rel, "\u21AC", "\\looparrowright", true);
+defineSymbol(symbols_math, ams, rel, "\u21B7", "\\curvearrowright", true); // unicode-math maps \u21bb to \cwopencirclearrow. We'll use the AMS synonym.
+
+defineSymbol(symbols_math, ams, rel, "\u21BB", "\\circlearrowright", true);
+defineSymbol(symbols_math, ams, rel, "\u21B1", "\\Rsh", true);
+defineSymbol(symbols_math, ams, rel, "\u21CA", "\\downdownarrows", true);
+defineSymbol(symbols_math, ams, rel, "\u21BE", "\\upharpoonright", true);
+defineSymbol(symbols_math, ams, rel, "\u21C2", "\\downharpoonright", true);
+defineSymbol(symbols_math, ams, rel, "\u21DD", "\\rightsquigarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21DD", "\\leadsto");
+defineSymbol(symbols_math, ams, rel, "\u21DB", "\\Rrightarrow", true);
+defineSymbol(symbols_math, ams, rel, "\u21BE", "\\restriction");
+defineSymbol(symbols_math, main, symbols_textord, "\u2018", "`");
+defineSymbol(symbols_math, main, symbols_textord, "$", "\\$");
+defineSymbol(symbols_text, main, symbols_textord, "$", "\\$");
+defineSymbol(symbols_text, main, symbols_textord, "$", "\\textdollar");
+defineSymbol(symbols_math, main, symbols_textord, "%", "\\%");
+defineSymbol(symbols_text, main, symbols_textord, "%", "\\%");
+defineSymbol(symbols_math, main, symbols_textord, "_", "\\_");
+defineSymbol(symbols_text, main, symbols_textord, "_", "\\_");
+defineSymbol(symbols_text, main, symbols_textord, "_", "\\textunderscore");
+defineSymbol(symbols_math, main, symbols_textord, "\u2220", "\\angle", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u221E", "\\infty", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2032", "\\prime");
+defineSymbol(symbols_math, main, symbols_textord, "\u25B3", "\\triangle");
+defineSymbol(symbols_math, main, symbols_textord, "\u0393", "\\Gamma", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u0394", "\\Delta", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u0398", "\\Theta", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u039B", "\\Lambda", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u039E", "\\Xi", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A0", "\\Pi", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A3", "\\Sigma", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A5", "\\Upsilon", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A6", "\\Phi", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A8", "\\Psi", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u03A9", "\\Omega", true);
+defineSymbol(symbols_math, main, symbols_textord, "A", "\u0391");
+defineSymbol(symbols_math, main, symbols_textord, "B", "\u0392");
+defineSymbol(symbols_math, main, symbols_textord, "E", "\u0395");
+defineSymbol(symbols_math, main, symbols_textord, "Z", "\u0396");
+defineSymbol(symbols_math, main, symbols_textord, "H", "\u0397");
+defineSymbol(symbols_math, main, symbols_textord, "I", "\u0399");
+defineSymbol(symbols_math, main, symbols_textord, "K", "\u039A");
+defineSymbol(symbols_math, main, symbols_textord, "M", "\u039C");
+defineSymbol(symbols_math, main, symbols_textord, "N", "\u039D");
+defineSymbol(symbols_math, main, symbols_textord, "O", "\u039F");
+defineSymbol(symbols_math, main, symbols_textord, "P", "\u03A1");
+defineSymbol(symbols_math, main, symbols_textord, "T", "\u03A4");
+defineSymbol(symbols_math, main, symbols_textord, "X", "\u03A7");
+defineSymbol(symbols_math, main, symbols_textord, "\xAC", "\\neg", true);
+defineSymbol(symbols_math, main, symbols_textord, "\xAC", "\\lnot");
+defineSymbol(symbols_math, main, symbols_textord, "\u22A4", "\\top");
+defineSymbol(symbols_math, main, symbols_textord, "\u22A5", "\\bot");
+defineSymbol(symbols_math, main, symbols_textord, "\u2205", "\\emptyset");
+defineSymbol(symbols_math, ams, symbols_textord, "\u2205", "\\varnothing");
+defineSymbol(symbols_math, main, mathord, "\u03B1", "\\alpha", true);
+defineSymbol(symbols_math, main, mathord, "\u03B2", "\\beta", true);
+defineSymbol(symbols_math, main, mathord, "\u03B3", "\\gamma", true);
+defineSymbol(symbols_math, main, mathord, "\u03B4", "\\delta", true);
+defineSymbol(symbols_math, main, mathord, "\u03F5", "\\epsilon", true);
+defineSymbol(symbols_math, main, mathord, "\u03B6", "\\zeta", true);
+defineSymbol(symbols_math, main, mathord, "\u03B7", "\\eta", true);
+defineSymbol(symbols_math, main, mathord, "\u03B8", "\\theta", true);
+defineSymbol(symbols_math, main, mathord, "\u03B9", "\\iota", true);
+defineSymbol(symbols_math, main, mathord, "\u03BA", "\\kappa", true);
+defineSymbol(symbols_math, main, mathord, "\u03BB", "\\lambda", true);
+defineSymbol(symbols_math, main, mathord, "\u03BC", "\\mu", true);
+defineSymbol(symbols_math, main, mathord, "\u03BD", "\\nu", true);
+defineSymbol(symbols_math, main, mathord, "\u03BE", "\\xi", true);
+defineSymbol(symbols_math, main, mathord, "\u03BF", "\\omicron", true);
+defineSymbol(symbols_math, main, mathord, "\u03C0", "\\pi", true);
+defineSymbol(symbols_math, main, mathord, "\u03C1", "\\rho", true);
+defineSymbol(symbols_math, main, mathord, "\u03C3", "\\sigma", true);
+defineSymbol(symbols_math, main, mathord, "\u03C4", "\\tau", true);
+defineSymbol(symbols_math, main, mathord, "\u03C5", "\\upsilon", true);
+defineSymbol(symbols_math, main, mathord, "\u03D5", "\\phi", true);
+defineSymbol(symbols_math, main, mathord, "\u03C7", "\\chi", true);
+defineSymbol(symbols_math, main, mathord, "\u03C8", "\\psi", true);
+defineSymbol(symbols_math, main, mathord, "\u03C9", "\\omega", true);
+defineSymbol(symbols_math, main, mathord, "\u03B5", "\\varepsilon", true);
+defineSymbol(symbols_math, main, mathord, "\u03D1", "\\vartheta", true);
+defineSymbol(symbols_math, main, mathord, "\u03D6", "\\varpi", true);
+defineSymbol(symbols_math, main, mathord, "\u03F1", "\\varrho", true);
+defineSymbol(symbols_math, main, mathord, "\u03C2", "\\varsigma", true);
+defineSymbol(symbols_math, main, mathord, "\u03C6", "\\varphi", true);
+defineSymbol(symbols_math, main, bin, "\u2217", "*");
+defineSymbol(symbols_math, main, bin, "+", "+");
+defineSymbol(symbols_math, main, bin, "\u2212", "-");
+defineSymbol(symbols_math, main, bin, "\u22C5", "\\cdot", true);
+defineSymbol(symbols_math, main, bin, "\u2218", "\\circ");
+defineSymbol(symbols_math, main, bin, "\xF7", "\\div", true);
+defineSymbol(symbols_math, main, bin, "\xB1", "\\pm", true);
+defineSymbol(symbols_math, main, bin, "\xD7", "\\times", true);
+defineSymbol(symbols_math, main, bin, "\u2229", "\\cap", true);
+defineSymbol(symbols_math, main, bin, "\u222A", "\\cup", true);
+defineSymbol(symbols_math, main, bin, "\u2216", "\\setminus");
+defineSymbol(symbols_math, main, bin, "\u2227", "\\land");
+defineSymbol(symbols_math, main, bin, "\u2228", "\\lor");
+defineSymbol(symbols_math, main, bin, "\u2227", "\\wedge", true);
+defineSymbol(symbols_math, main, bin, "\u2228", "\\vee", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u221A", "\\surd");
+defineSymbol(symbols_math, main, symbols_open, "\u27E8", "\\langle", true);
+defineSymbol(symbols_math, main, symbols_open, "\u2223", "\\lvert");
+defineSymbol(symbols_math, main, symbols_open, "\u2225", "\\lVert");
+defineSymbol(symbols_math, main, symbols_close, "?", "?");
+defineSymbol(symbols_math, main, symbols_close, "!", "!");
+defineSymbol(symbols_math, main, symbols_close, "\u27E9", "\\rangle", true);
+defineSymbol(symbols_math, main, symbols_close, "\u2223", "\\rvert");
+defineSymbol(symbols_math, main, symbols_close, "\u2225", "\\rVert");
+defineSymbol(symbols_math, main, rel, "=", "=");
+defineSymbol(symbols_math, main, rel, ":", ":");
+defineSymbol(symbols_math, main, rel, "\u2248", "\\approx", true);
+defineSymbol(symbols_math, main, rel, "\u2245", "\\cong", true);
+defineSymbol(symbols_math, main, rel, "\u2265", "\\ge");
+defineSymbol(symbols_math, main, rel, "\u2265", "\\geq", true);
+defineSymbol(symbols_math, main, rel, "\u2190", "\\gets");
+defineSymbol(symbols_math, main, rel, ">", "\\gt", true);
+defineSymbol(symbols_math, main, rel, "\u2208", "\\in", true);
+defineSymbol(symbols_math, main, rel, "\uE020", "\\@not");
+defineSymbol(symbols_math, main, rel, "\u2282", "\\subset", true);
+defineSymbol(symbols_math, main, rel, "\u2283", "\\supset", true);
+defineSymbol(symbols_math, main, rel, "\u2286", "\\subseteq", true);
+defineSymbol(symbols_math, main, rel, "\u2287", "\\supseteq", true);
+defineSymbol(symbols_math, ams, rel, "\u2288", "\\nsubseteq", true);
+defineSymbol(symbols_math, ams, rel, "\u2289", "\\nsupseteq", true);
+defineSymbol(symbols_math, main, rel, "\u22A8", "\\models");
+defineSymbol(symbols_math, main, rel, "\u2190", "\\leftarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2264", "\\le");
+defineSymbol(symbols_math, main, rel, "\u2264", "\\leq", true);
+defineSymbol(symbols_math, main, rel, "<", "\\lt", true);
+defineSymbol(symbols_math, main, rel, "\u2192", "\\rightarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2192", "\\to");
+defineSymbol(symbols_math, ams, rel, "\u2271", "\\ngeq", true);
+defineSymbol(symbols_math, ams, rel, "\u2270", "\\nleq", true);
+defineSymbol(symbols_math, main, symbols_spacing, "\xA0", "\\ ");
+defineSymbol(symbols_math, main, symbols_spacing, "\xA0", "~");
+defineSymbol(symbols_math, main, symbols_spacing, "\xA0", "\\space"); // Ref: LaTeX Source 2e: \DeclareRobustCommand{\nobreakspace}{%
+
+defineSymbol(symbols_math, main, symbols_spacing, "\xA0", "\\nobreakspace");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", "\\ ");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", " ");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", "~");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", "\\space");
+defineSymbol(symbols_text, main, symbols_spacing, "\xA0", "\\nobreakspace");
+defineSymbol(symbols_math, main, symbols_spacing, null, "\\nobreak");
+defineSymbol(symbols_math, main, symbols_spacing, null, "\\allowbreak");
+defineSymbol(symbols_math, main, punct, ",", ",");
+defineSymbol(symbols_math, main, punct, ";", ";");
+defineSymbol(symbols_math, ams, bin, "\u22BC", "\\barwedge", true);
+defineSymbol(symbols_math, ams, bin, "\u22BB", "\\veebar", true);
+defineSymbol(symbols_math, main, bin, "\u2299", "\\odot", true);
+defineSymbol(symbols_math, main, bin, "\u2295", "\\oplus", true);
+defineSymbol(symbols_math, main, bin, "\u2297", "\\otimes", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u2202", "\\partial", true);
+defineSymbol(symbols_math, main, bin, "\u2298", "\\oslash", true);
+defineSymbol(symbols_math, ams, bin, "\u229A", "\\circledcirc", true);
+defineSymbol(symbols_math, ams, bin, "\u22A1", "\\boxdot", true);
+defineSymbol(symbols_math, main, bin, "\u25B3", "\\bigtriangleup");
+defineSymbol(symbols_math, main, bin, "\u25BD", "\\bigtriangledown");
+defineSymbol(symbols_math, main, bin, "\u2020", "\\dagger");
+defineSymbol(symbols_math, main, bin, "\u22C4", "\\diamond");
+defineSymbol(symbols_math, main, bin, "\u22C6", "\\star");
+defineSymbol(symbols_math, main, bin, "\u25C3", "\\triangleleft");
+defineSymbol(symbols_math, main, bin, "\u25B9", "\\triangleright");
+defineSymbol(symbols_math, main, symbols_open, "{", "\\{");
+defineSymbol(symbols_text, main, symbols_textord, "{", "\\{");
+defineSymbol(symbols_text, main, symbols_textord, "{", "\\textbraceleft");
+defineSymbol(symbols_math, main, symbols_close, "}", "\\}");
+defineSymbol(symbols_text, main, symbols_textord, "}", "\\}");
+defineSymbol(symbols_text, main, symbols_textord, "}", "\\textbraceright");
+defineSymbol(symbols_math, main, symbols_open, "{", "\\lbrace");
+defineSymbol(symbols_math, main, symbols_close, "}", "\\rbrace");
+defineSymbol(symbols_math, main, symbols_open, "[", "\\lbrack", true);
+defineSymbol(symbols_text, main, symbols_textord, "[", "\\lbrack", true);
+defineSymbol(symbols_math, main, symbols_close, "]", "\\rbrack", true);
+defineSymbol(symbols_text, main, symbols_textord, "]", "\\rbrack", true);
+defineSymbol(symbols_math, main, symbols_open, "(", "\\lparen", true);
+defineSymbol(symbols_math, main, symbols_close, ")", "\\rparen", true);
+defineSymbol(symbols_text, main, symbols_textord, "<", "\\textless", true); // in T1 fontenc
+
+defineSymbol(symbols_text, main, symbols_textord, ">", "\\textgreater", true); // in T1 fontenc
+
+defineSymbol(symbols_math, main, symbols_open, "\u230A", "\\lfloor", true);
+defineSymbol(symbols_math, main, symbols_close, "\u230B", "\\rfloor", true);
+defineSymbol(symbols_math, main, symbols_open, "\u2308", "\\lceil", true);
+defineSymbol(symbols_math, main, symbols_close, "\u2309", "\\rceil", true);
+defineSymbol(symbols_math, main, symbols_textord, "\\", "\\backslash");
+defineSymbol(symbols_math, main, symbols_textord, "\u2223", "|");
+defineSymbol(symbols_math, main, symbols_textord, "\u2223", "\\vert");
+defineSymbol(symbols_text, main, symbols_textord, "|", "\\textbar", true); // in T1 fontenc
+
+defineSymbol(symbols_math, main, symbols_textord, "\u2225", "\\|");
+defineSymbol(symbols_math, main, symbols_textord, "\u2225", "\\Vert");
+defineSymbol(symbols_text, main, symbols_textord, "\u2225", "\\textbardbl");
+defineSymbol(symbols_text, main, symbols_textord, "~", "\\textasciitilde");
+defineSymbol(symbols_text, main, symbols_textord, "\\", "\\textbackslash");
+defineSymbol(symbols_text, main, symbols_textord, "^", "\\textasciicircum");
+defineSymbol(symbols_math, main, rel, "\u2191", "\\uparrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D1", "\\Uparrow", true);
+defineSymbol(symbols_math, main, rel, "\u2193", "\\downarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D3", "\\Downarrow", true);
+defineSymbol(symbols_math, main, rel, "\u2195", "\\updownarrow", true);
+defineSymbol(symbols_math, main, rel, "\u21D5", "\\Updownarrow", true);
+defineSymbol(symbols_math, main, op, "\u2210", "\\coprod");
+defineSymbol(symbols_math, main, op, "\u22C1", "\\bigvee");
+defineSymbol(symbols_math, main, op, "\u22C0", "\\bigwedge");
+defineSymbol(symbols_math, main, op, "\u2A04", "\\biguplus");
+defineSymbol(symbols_math, main, op, "\u22C2", "\\bigcap");
+defineSymbol(symbols_math, main, op, "\u22C3", "\\bigcup");
+defineSymbol(symbols_math, main, op, "\u222B", "\\int");
+defineSymbol(symbols_math, main, op, "\u222B", "\\intop");
+defineSymbol(symbols_math, main, op, "\u222C", "\\iint");
+defineSymbol(symbols_math, main, op, "\u222D", "\\iiint");
+defineSymbol(symbols_math, main, op, "\u220F", "\\prod");
+defineSymbol(symbols_math, main, op, "\u2211", "\\sum");
+defineSymbol(symbols_math, main, op, "\u2A02", "\\bigotimes");
+defineSymbol(symbols_math, main, op, "\u2A01", "\\bigoplus");
+defineSymbol(symbols_math, main, op, "\u2A00", "\\bigodot");
+defineSymbol(symbols_math, main, op, "\u222E", "\\oint");
+defineSymbol(symbols_math, main, op, "\u2A06", "\\bigsqcup");
+defineSymbol(symbols_math, main, op, "\u222B", "\\smallint");
+defineSymbol(symbols_text, main, symbols_inner, "\u2026", "\\textellipsis");
+defineSymbol(symbols_math, main, symbols_inner, "\u2026", "\\mathellipsis");
+defineSymbol(symbols_text, main, symbols_inner, "\u2026", "\\ldots", true);
+defineSymbol(symbols_math, main, symbols_inner, "\u2026", "\\ldots", true);
+defineSymbol(symbols_math, main, symbols_inner, "\u22EF", "\\@cdots", true);
+defineSymbol(symbols_math, main, symbols_inner, "\u22F1", "\\ddots", true);
+defineSymbol(symbols_math, main, symbols_textord, "\u22EE", "\\varvdots"); // \vdots is a macro
+
+defineSymbol(symbols_math, main, symbols_accent, "\u02CA", "\\acute");
+defineSymbol(symbols_math, main, symbols_accent, "\u02CB", "\\grave");
+defineSymbol(symbols_math, main, symbols_accent, "\xA8", "\\ddot");
+defineSymbol(symbols_math, main, symbols_accent, "~", "\\tilde");
+defineSymbol(symbols_math, main, symbols_accent, "\u02C9", "\\bar");
+defineSymbol(symbols_math, main, symbols_accent, "\u02D8", "\\breve");
+defineSymbol(symbols_math, main, symbols_accent, "\u02C7", "\\check");
+defineSymbol(symbols_math, main, symbols_accent, "^", "\\hat");
+defineSymbol(symbols_math, main, symbols_accent, "\u20D7", "\\vec");
+defineSymbol(symbols_math, main, symbols_accent, "\u02D9", "\\dot");
+defineSymbol(symbols_math, main, symbols_accent, "\u02DA", "\\mathring"); // \imath and \jmath should be invariant to \mathrm, \mathbf, etc., so use PUA
+
+defineSymbol(symbols_math, main, mathord, "\uE131", "\\@imath");
+defineSymbol(symbols_math, main, mathord, "\uE237", "\\@jmath");
+defineSymbol(symbols_math, main, symbols_textord, "\u0131", "\u0131");
+defineSymbol(symbols_math, main, symbols_textord, "\u0237", "\u0237");
+defineSymbol(symbols_text, main, symbols_textord, "\u0131", "\\i", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u0237", "\\j", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xDF", "\\ss", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xE6", "\\ae", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u0153", "\\oe", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xF8", "\\o", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xC6", "\\AE", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u0152", "\\OE", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xD8", "\\O", true);
+defineSymbol(symbols_text, main, symbols_accent, "\u02CA", "\\'"); // acute
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02CB", "\\`"); // grave
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02C6", "\\^"); // circumflex
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02DC", "\\~"); // tilde
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02C9", "\\="); // macron
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02D8", "\\u"); // breve
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02D9", "\\."); // dot above
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02DA", "\\r"); // ring above
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02C7", "\\v"); // caron
+
+defineSymbol(symbols_text, main, symbols_accent, "\xA8", '\\"'); // diaresis
+
+defineSymbol(symbols_text, main, symbols_accent, "\u02DD", "\\H"); // double acute
+
+defineSymbol(symbols_text, main, symbols_accent, "\u25EF", "\\textcircled"); // \bigcirc glyph
+// These ligatures are detected and created in Parser.js's `formLigatures`.
+
+var ligatures = {
+  "--": true,
+  "---": true,
+  "``": true,
+  "''": true
+};
+defineSymbol(symbols_text, main, symbols_textord, "\u2013", "--", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u2013", "\\textendash");
+defineSymbol(symbols_text, main, symbols_textord, "\u2014", "---", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u2014", "\\textemdash");
+defineSymbol(symbols_text, main, symbols_textord, "\u2018", "`", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u2018", "\\textquoteleft");
+defineSymbol(symbols_text, main, symbols_textord, "\u2019", "'", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u2019", "\\textquoteright");
+defineSymbol(symbols_text, main, symbols_textord, "\u201C", "``", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u201C", "\\textquotedblleft");
+defineSymbol(symbols_text, main, symbols_textord, "\u201D", "''", true);
+defineSymbol(symbols_text, main, symbols_textord, "\u201D", "\\textquotedblright"); //  \degree from gensymb package
+
+defineSymbol(symbols_math, main, symbols_textord, "\xB0", "\\degree", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xB0", "\\degree"); // \textdegree from inputenc package
+
+defineSymbol(symbols_text, main, symbols_textord, "\xB0", "\\textdegree", true); // TODO: In LaTeX, \pounds can generate a different character in text and math
+// mode, but among our fonts, only Main-Regular defines this character "163".
+
+defineSymbol(symbols_math, main, symbols_textord, "\xA3", "\\pounds");
+defineSymbol(symbols_math, main, symbols_textord, "\xA3", "\\mathsterling", true);
+defineSymbol(symbols_text, main, symbols_textord, "\xA3", "\\pounds");
+defineSymbol(symbols_text, main, symbols_textord, "\xA3", "\\textsterling", true);
+defineSymbol(symbols_math, ams, symbols_textord, "\u2720", "\\maltese");
+defineSymbol(symbols_text, ams, symbols_textord, "\u2720", "\\maltese"); // There are lots of symbols which are the same, so we add them in afterwards.
+// All of these are textords in math mode
+
+var mathTextSymbols = "0123456789/@.\"";
+
+for (var symbols_i = 0; symbols_i < mathTextSymbols.length; symbols_i++) {
+  var symbols_ch = mathTextSymbols.charAt(symbols_i);
+  defineSymbol(symbols_math, main, symbols_textord, symbols_ch, symbols_ch);
+} // All of these are textords in text mode
+
+
+var textSymbols = "0123456789!@*()-=+\";:?/.,";
+
+for (var src_symbols_i = 0; src_symbols_i < textSymbols.length; src_symbols_i++) {
+  var _ch = textSymbols.charAt(src_symbols_i);
+
+  defineSymbol(symbols_text, main, symbols_textord, _ch, _ch);
+} // All of these are textords in text mode, and mathords in math mode
+
+
+var letters = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
+
+for (var symbols_i2 = 0; symbols_i2 < letters.length; symbols_i2++) {
+  var _ch2 = letters.charAt(symbols_i2);
+
+  defineSymbol(symbols_math, main, mathord, _ch2, _ch2);
+  defineSymbol(symbols_text, main, symbols_textord, _ch2, _ch2);
+} // Blackboard bold and script letters in Unicode range
+
+
+defineSymbol(symbols_math, ams, symbols_textord, "C", "\u2102"); // blackboard bold
+
+defineSymbol(symbols_text, ams, symbols_textord, "C", "\u2102");
+defineSymbol(symbols_math, ams, symbols_textord, "H", "\u210D");
+defineSymbol(symbols_text, ams, symbols_textord, "H", "\u210D");
+defineSymbol(symbols_math, ams, symbols_textord, "N", "\u2115");
+defineSymbol(symbols_text, ams, symbols_textord, "N", "\u2115");
+defineSymbol(symbols_math, ams, symbols_textord, "P", "\u2119");
+defineSymbol(symbols_text, ams, symbols_textord, "P", "\u2119");
+defineSymbol(symbols_math, ams, symbols_textord, "Q", "\u211A");
+defineSymbol(symbols_text, ams, symbols_textord, "Q", "\u211A");
+defineSymbol(symbols_math, ams, symbols_textord, "R", "\u211D");
+defineSymbol(symbols_text, ams, symbols_textord, "R", "\u211D");
+defineSymbol(symbols_math, ams, symbols_textord, "Z", "\u2124");
+defineSymbol(symbols_text, ams, symbols_textord, "Z", "\u2124");
+defineSymbol(symbols_math, main, mathord, "h", "\u210E"); // italic h, Planck constant
+
+defineSymbol(symbols_text, main, mathord, "h", "\u210E"); // The next loop loads wide (surrogate pair) characters.
+// We support some letters in the Unicode range U+1D400 to U+1D7FF,
+// Mathematical Alphanumeric Symbols.
+// Some editors do not deal well with wide characters. So don't write the
+// string into this file. Instead, create the string from the surrogate pair.
+
+var symbols_wideChar = "";
+
+for (var symbols_i3 = 0; symbols_i3 < letters.length; symbols_i3++) {
+  var _ch3 = letters.charAt(symbols_i3); // The hex numbers in the next line are a surrogate pair.
+  // 0xD835 is the high surrogate for all letters in the range we support.
+  // 0xDC00 is the low surrogate for bold A.
+
+
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDC00 + symbols_i3); // A-Z a-z bold
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDC34 + symbols_i3); // A-Z a-z italic
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDC68 + symbols_i3); // A-Z a-z bold italic
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDD04 + symbols_i3); // A-Z a-z Fractur
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDDA0 + symbols_i3); // A-Z a-z sans-serif
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDDD4 + symbols_i3); // A-Z a-z sans bold
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDE08 + symbols_i3); // A-Z a-z sans italic
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDE70 + symbols_i3); // A-Z a-z monospace
+
+  defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+
+  if (symbols_i3 < 26) {
+    // KaTeX fonts have only capital letters for blackboard bold and script.
+    // See exception for k below.
+    symbols_wideChar = String.fromCharCode(0xD835, 0xDD38 + symbols_i3); // A-Z double struck
+
+    defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+    defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+    symbols_wideChar = String.fromCharCode(0xD835, 0xDC9C + symbols_i3); // A-Z script
+
+    defineSymbol(symbols_math, main, mathord, _ch3, symbols_wideChar);
+    defineSymbol(symbols_text, main, symbols_textord, _ch3, symbols_wideChar);
+  } // TODO: Add bold script when it is supported by a KaTeX font.
+
+} // "k" is the only double struck lower case letter in the KaTeX fonts.
+
+
+symbols_wideChar = String.fromCharCode(0xD835, 0xDD5C); // k double struck
+
+defineSymbol(symbols_math, main, mathord, "k", symbols_wideChar);
+defineSymbol(symbols_text, main, symbols_textord, "k", symbols_wideChar); // Next, some wide character numerals
+
+for (var symbols_i4 = 0; symbols_i4 < 10; symbols_i4++) {
+  var _ch4 = symbols_i4.toString();
+
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDFCE + symbols_i4); // 0-9 bold
+
+  defineSymbol(symbols_math, main, mathord, _ch4, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch4, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDFE2 + symbols_i4); // 0-9 sans serif
+
+  defineSymbol(symbols_math, main, mathord, _ch4, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch4, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDFEC + symbols_i4); // 0-9 bold sans
+
+  defineSymbol(symbols_math, main, mathord, _ch4, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch4, symbols_wideChar);
+  symbols_wideChar = String.fromCharCode(0xD835, 0xDFF6 + symbols_i4); // 0-9 monospace
+
+  defineSymbol(symbols_math, main, mathord, _ch4, symbols_wideChar);
+  defineSymbol(symbols_text, main, symbols_textord, _ch4, symbols_wideChar);
+} // We add these Latin-1 letters as symbols for backwards-compatibility,
+// but they are not actually in the font, nor are they supported by the
+// Unicode accent mechanism, so they fall back to Times font and look ugly.
+// TODO(edemaine): Fix this.
+
+
+var extraLatin = "\xC7\xD0\xDE\xE7\xFE";
+
+for (var _i5 = 0; _i5 < extraLatin.length; _i5++) {
+  var _ch5 = extraLatin.charAt(_i5);
+
+  defineSymbol(symbols_math, main, mathord, _ch5, _ch5);
+  defineSymbol(symbols_text, main, symbols_textord, _ch5, _ch5);
+}
+// CONCATENATED MODULE: ./src/wide-character.js
+/**
+ * This file provides support for Unicode range U+1D400 to U+1D7FF,
+ * Mathematical Alphanumeric Symbols.
+ *
+ * Function wideCharacterFont takes a wide character as input and returns
+ * the font information necessary to render it properly.
+ */
+
+/**
+ * Data below is from https://www.unicode.org/charts/PDF/U1D400.pdf
+ * That document sorts characters into groups by font type, say bold or italic.
+ *
+ * In the arrays below, each subarray consists three elements:
+ *      * The CSS class of that group when in math mode.
+ *      * The CSS class of that group when in text mode.
+ *      * The font name, so that KaTeX can get font metrics.
+ */
+
+var wideLatinLetterData = [["mathbf", "textbf", "Main-Bold"], // A-Z bold upright
+["mathbf", "textbf", "Main-Bold"], // a-z bold upright
+["mathnormal", "textit", "Math-Italic"], // A-Z italic
+["mathnormal", "textit", "Math-Italic"], // a-z italic
+["boldsymbol", "boldsymbol", "Main-BoldItalic"], // A-Z bold italic
+["boldsymbol", "boldsymbol", "Main-BoldItalic"], // a-z bold italic
+// Map fancy A-Z letters to script, not calligraphic.
+// This aligns with unicode-math and math fonts (except Cambria Math).
+["mathscr", "textscr", "Script-Regular"], // A-Z script
+["", "", ""], // a-z script.  No font
+["", "", ""], // A-Z bold script. No font
+["", "", ""], // a-z bold script. No font
+["mathfrak", "textfrak", "Fraktur-Regular"], // A-Z Fraktur
+["mathfrak", "textfrak", "Fraktur-Regular"], // a-z Fraktur
+["mathbb", "textbb", "AMS-Regular"], // A-Z double-struck
+["mathbb", "textbb", "AMS-Regular"], // k double-struck
+["", "", ""], // A-Z bold Fraktur No font metrics
+["", "", ""], // a-z bold Fraktur.   No font.
+["mathsf", "textsf", "SansSerif-Regular"], // A-Z sans-serif
+["mathsf", "textsf", "SansSerif-Regular"], // a-z sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // A-Z bold sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // a-z bold sans-serif
+["mathitsf", "textitsf", "SansSerif-Italic"], // A-Z italic sans-serif
+["mathitsf", "textitsf", "SansSerif-Italic"], // a-z italic sans-serif
+["", "", ""], // A-Z bold italic sans. No font
+["", "", ""], // a-z bold italic sans. No font
+["mathtt", "texttt", "Typewriter-Regular"], // A-Z monospace
+["mathtt", "texttt", "Typewriter-Regular"]];
+var wideNumeralData = [["mathbf", "textbf", "Main-Bold"], // 0-9 bold
+["", "", ""], // 0-9 double-struck. No KaTeX font.
+["mathsf", "textsf", "SansSerif-Regular"], // 0-9 sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // 0-9 bold sans-serif
+["mathtt", "texttt", "Typewriter-Regular"]];
+var wide_character_wideCharacterFont = function wideCharacterFont(wideChar, mode) {
+  // IE doesn't support codePointAt(). So work with the surrogate pair.
+  var H = wideChar.charCodeAt(0); // high surrogate
+
+  var L = wideChar.charCodeAt(1); // low surrogate
+
+  var codePoint = (H - 0xD800) * 0x400 + (L - 0xDC00) + 0x10000;
+  var j = mode === "math" ? 0 : 1; // column index for CSS class.
+
+  if (0x1D400 <= codePoint && codePoint < 0x1D6A4) {
+    // wideLatinLetterData contains exactly 26 chars on each row.
+    // So we can calculate the relevant row. No traverse necessary.
+    var i = Math.floor((codePoint - 0x1D400) / 26);
+    return [wideLatinLetterData[i][2], wideLatinLetterData[i][j]];
+  } else if (0x1D7CE <= codePoint && codePoint <= 0x1D7FF) {
+    // Numerals, ten per row.
+    var _i = Math.floor((codePoint - 0x1D7CE) / 10);
+
+    return [wideNumeralData[_i][2], wideNumeralData[_i][j]];
+  } else if (codePoint === 0x1D6A5 || codePoint === 0x1D6A6) {
+    // dotless i or j
+    return [wideLatinLetterData[0][2], wideLatinLetterData[0][j]];
+  } else if (0x1D6A6 < codePoint && codePoint < 0x1D7CE) {
+    // Greek letters. Not supported, yet.
+    return ["", ""];
+  } else {
+    // We don't support any wide characters outside 1D400–1D7FF.
+    throw new src_ParseError("Unsupported character: " + wideChar);
+  }
+};
+// CONCATENATED MODULE: ./src/Options.js
+/**
+ * This file contains information about the options that the Parser carries
+ * around with it while parsing. Data is held in an `Options` object, and when
+ * recursing, a new `Options` object can be created with the `.with*` and
+ * `.reset` functions.
+ */
+
+var sizeStyleMap = [// Each element contains [textsize, scriptsize, scriptscriptsize].
+// The size mappings are taken from TeX with \normalsize=10pt.
+[1, 1, 1], // size1: [5, 5, 5]              \tiny
+[2, 1, 1], // size2: [6, 5, 5]
+[3, 1, 1], // size3: [7, 5, 5]              \scriptsize
+[4, 2, 1], // size4: [8, 6, 5]              \footnotesize
+[5, 2, 1], // size5: [9, 6, 5]              \small
+[6, 3, 1], // size6: [10, 7, 5]             \normalsize
+[7, 4, 2], // size7: [12, 8, 6]             \large
+[8, 6, 3], // size8: [14.4, 10, 7]          \Large
+[9, 7, 6], // size9: [17.28, 12, 10]        \LARGE
+[10, 8, 7], // size10: [20.74, 14.4, 12]     \huge
+[11, 10, 9]];
+var sizeMultipliers = [// fontMetrics.js:getGlobalMetrics also uses size indexes, so if
+// you change size indexes, change that function.
+0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.44, 1.728, 2.074, 2.488];
+
+var sizeAtStyle = function sizeAtStyle(size, style) {
+  return style.size < 2 ? size : sizeStyleMap[size - 1][style.size - 1];
+}; // In these types, "" (empty string) means "no change".
+
+
+/**
+ * This is the main options class. It contains the current style, size, color,
+ * and font.
+ *
+ * Options objects should not be modified. To create a new Options with
+ * different properties, call a `.having*` method.
+ */
+var Options_Options =
+/*#__PURE__*/
+function () {
+  // A font family applies to a group of fonts (i.e. SansSerif), while a font
+  // represents a specific font (i.e. SansSerif Bold).
+  // See: https://tex.stackexchange.com/questions/22350/difference-between-textrm-and-mathrm
+
+  /**
+   * The base size index.
+   */
+  function Options(data) {
+    this.style = void 0;
+    this.color = void 0;
+    this.size = void 0;
+    this.textSize = void 0;
+    this.phantom = void 0;
+    this.font = void 0;
+    this.fontFamily = void 0;
+    this.fontWeight = void 0;
+    this.fontShape = void 0;
+    this.sizeMultiplier = void 0;
+    this.maxSize = void 0;
+    this.minRuleThickness = void 0;
+    this._fontMetrics = void 0;
+    this.style = data.style;
+    this.color = data.color;
+    this.size = data.size || Options.BASESIZE;
+    this.textSize = data.textSize || this.size;
+    this.phantom = !!data.phantom;
+    this.font = data.font || "";
+    this.fontFamily = data.fontFamily || "";
+    this.fontWeight = data.fontWeight || '';
+    this.fontShape = data.fontShape || '';
+    this.sizeMultiplier = sizeMultipliers[this.size - 1];
+    this.maxSize = data.maxSize;
+    this.minRuleThickness = data.minRuleThickness;
+    this._fontMetrics = undefined;
+  }
+  /**
+   * Returns a new options object with the same properties as "this".  Properties
+   * from "extension" will be copied to the new options object.
+   */
+
+
+  var _proto = Options.prototype;
+
+  _proto.extend = function extend(extension) {
+    var data = {
+      style: this.style,
+      size: this.size,
+      textSize: this.textSize,
+      color: this.color,
+      phantom: this.phantom,
+      font: this.font,
+      fontFamily: this.fontFamily,
+      fontWeight: this.fontWeight,
+      fontShape: this.fontShape,
+      maxSize: this.maxSize,
+      minRuleThickness: this.minRuleThickness
+    };
+
+    for (var key in extension) {
+      if (extension.hasOwnProperty(key)) {
+        data[key] = extension[key];
+      }
+    }
+
+    return new Options(data);
+  }
+  /**
+   * Return an options object with the given style. If `this.style === style`,
+   * returns `this`.
+   */
+  ;
+
+  _proto.havingStyle = function havingStyle(style) {
+    if (this.style === style) {
+      return this;
+    } else {
+      return this.extend({
+        style: style,
+        size: sizeAtStyle(this.textSize, style)
+      });
+    }
+  }
+  /**
+   * Return an options object with a cramped version of the current style. If
+   * the current style is cramped, returns `this`.
+   */
+  ;
+
+  _proto.havingCrampedStyle = function havingCrampedStyle() {
+    return this.havingStyle(this.style.cramp());
+  }
+  /**
+   * Return an options object with the given size and in at least `\textstyle`.
+   * Returns `this` if appropriate.
+   */
+  ;
+
+  _proto.havingSize = function havingSize(size) {
+    if (this.size === size && this.textSize === size) {
+      return this;
+    } else {
+      return this.extend({
+        style: this.style.text(),
+        size: size,
+        textSize: size,
+        sizeMultiplier: sizeMultipliers[size - 1]
+      });
+    }
+  }
+  /**
+   * Like `this.havingSize(BASESIZE).havingStyle(style)`. If `style` is omitted,
+   * changes to at least `\textstyle`.
+   */
+  ;
+
+  _proto.havingBaseStyle = function havingBaseStyle(style) {
+    style = style || this.style.text();
+    var wantSize = sizeAtStyle(Options.BASESIZE, style);
+
+    if (this.size === wantSize && this.textSize === Options.BASESIZE && this.style === style) {
+      return this;
+    } else {
+      return this.extend({
+        style: style,
+        size: wantSize
+      });
+    }
+  }
+  /**
+   * Remove the effect of sizing changes such as \Huge.
+   * Keep the effect of the current style, such as \scriptstyle.
+   */
+  ;
+
+  _proto.havingBaseSizing = function havingBaseSizing() {
+    var size;
+
+    switch (this.style.id) {
+      case 4:
+      case 5:
+        size = 3; // normalsize in scriptstyle
+
+        break;
+
+      case 6:
+      case 7:
+        size = 1; // normalsize in scriptscriptstyle
+
+        break;
+
+      default:
+        size = 6;
+      // normalsize in textstyle or displaystyle
+    }
+
+    return this.extend({
+      style: this.style.text(),
+      size: size
+    });
+  }
+  /**
+   * Create a new options object with the given color.
+   */
+  ;
+
+  _proto.withColor = function withColor(color) {
+    return this.extend({
+      color: color
+    });
+  }
+  /**
+   * Create a new options object with "phantom" set to true.
+   */
+  ;
+
+  _proto.withPhantom = function withPhantom() {
+    return this.extend({
+      phantom: true
+    });
+  }
+  /**
+   * Creates a new options object with the given math font or old text font.
+   * @type {[type]}
+   */
+  ;
+
+  _proto.withFont = function withFont(font) {
+    return this.extend({
+      font: font
+    });
+  }
+  /**
+   * Create a new options objects with the given fontFamily.
+   */
+  ;
+
+  _proto.withTextFontFamily = function withTextFontFamily(fontFamily) {
+    return this.extend({
+      fontFamily: fontFamily,
+      font: ""
+    });
+  }
+  /**
+   * Creates a new options object with the given font weight
+   */
+  ;
+
+  _proto.withTextFontWeight = function withTextFontWeight(fontWeight) {
+    return this.extend({
+      fontWeight: fontWeight,
+      font: ""
+    });
+  }
+  /**
+   * Creates a new options object with the given font weight
+   */
+  ;
+
+  _proto.withTextFontShape = function withTextFontShape(fontShape) {
+    return this.extend({
+      fontShape: fontShape,
+      font: ""
+    });
+  }
+  /**
+   * Return the CSS sizing classes required to switch from enclosing options
+   * `oldOptions` to `this`. Returns an array of classes.
+   */
+  ;
+
+  _proto.sizingClasses = function sizingClasses(oldOptions) {
+    if (oldOptions.size !== this.size) {
+      return ["sizing", "reset-size" + oldOptions.size, "size" + this.size];
+    } else {
+      return [];
+    }
+  }
+  /**
+   * Return the CSS sizing classes required to switch to the base size. Like
+   * `this.havingSize(BASESIZE).sizingClasses(this)`.
+   */
+  ;
+
+  _proto.baseSizingClasses = function baseSizingClasses() {
+    if (this.size !== Options.BASESIZE) {
+      return ["sizing", "reset-size" + this.size, "size" + Options.BASESIZE];
+    } else {
+      return [];
+    }
+  }
+  /**
+   * Return the font metrics for this size.
+   */
+  ;
+
+  _proto.fontMetrics = function fontMetrics() {
+    if (!this._fontMetrics) {
+      this._fontMetrics = getGlobalMetrics(this.size);
+    }
+
+    return this._fontMetrics;
+  }
+  /**
+   * Gets the CSS color of the current options object
+   */
+  ;
+
+  _proto.getColor = function getColor() {
+    if (this.phantom) {
+      return "transparent";
+    } else {
+      return this.color;
+    }
+  };
+
+  return Options;
+}();
+
+Options_Options.BASESIZE = 6;
+/* harmony default export */ var src_Options = (Options_Options);
+// CONCATENATED MODULE: ./src/units.js
+/**
+ * This file does conversion between units.  In particular, it provides
+ * calculateSize to convert other units into ems.
+ */
+
+ // This table gives the number of TeX pts in one of each *absolute* TeX unit.
+// Thus, multiplying a length by this number converts the length from units
+// into pts.  Dividing the result by ptPerEm gives the number of ems
+// *assuming* a font size of ptPerEm (normal size, normal style).
+
+var ptPerUnit = {
+  // https://en.wikibooks.org/wiki/LaTeX/Lengths and
+  // https://tex.stackexchange.com/a/8263
+  "pt": 1,
+  // TeX point
+  "mm": 7227 / 2540,
+  // millimeter
+  "cm": 7227 / 254,
+  // centimeter
+  "in": 72.27,
+  // inch
+  "bp": 803 / 800,
+  // big (PostScript) points
+  "pc": 12,
+  // pica
+  "dd": 1238 / 1157,
+  // didot
+  "cc": 14856 / 1157,
+  // cicero (12 didot)
+  "nd": 685 / 642,
+  // new didot
+  "nc": 1370 / 107,
+  // new cicero (12 new didot)
+  "sp": 1 / 65536,
+  // scaled point (TeX's internal smallest unit)
+  // https://tex.stackexchange.com/a/41371
+  "px": 803 / 800 // \pdfpxdimen defaults to 1 bp in pdfTeX and LuaTeX
+
+}; // Dictionary of relative units, for fast validity testing.
+
+var relativeUnit = {
+  "ex": true,
+  "em": true,
+  "mu": true
+};
+
+/**
+ * Determine whether the specified unit (either a string defining the unit
+ * or a "size" parse node containing a unit field) is valid.
+ */
+var validUnit = function validUnit(unit) {
+  if (typeof unit !== "string") {
+    unit = unit.unit;
+  }
+
+  return unit in ptPerUnit || unit in relativeUnit || unit === "ex";
+};
+/*
+ * Convert a "size" parse node (with numeric "number" and string "unit" fields,
+ * as parsed by functions.js argType "size") into a CSS em value for the
+ * current style/scale.  `options` gives the current options.
+ */
+
+var units_calculateSize = function calculateSize(sizeValue, options) {
+  var scale;
+
+  if (sizeValue.unit in ptPerUnit) {
+    // Absolute units
+    scale = ptPerUnit[sizeValue.unit] // Convert unit to pt
+    / options.fontMetrics().ptPerEm // Convert pt to CSS em
+    / options.sizeMultiplier; // Unscale to make absolute units
+  } else if (sizeValue.unit === "mu") {
+    // `mu` units scale with scriptstyle/scriptscriptstyle.
+    scale = options.fontMetrics().cssEmPerMu;
+  } else {
+    // Other relative units always refer to the *textstyle* font
+    // in the current size.
+    var unitOptions;
+
+    if (options.style.isTight()) {
+      // isTight() means current style is script/scriptscript.
+      unitOptions = options.havingStyle(options.style.text());
+    } else {
+      unitOptions = options;
+    } // TODO: In TeX these units are relative to the quad of the current
+    // *text* font, e.g. cmr10. KaTeX instead uses values from the
+    // comparably-sized *Computer Modern symbol* font. At 10pt, these
+    // match. At 7pt and 5pt, they differ: cmr7=1.138894, cmsy7=1.170641;
+    // cmr5=1.361133, cmsy5=1.472241. Consider $\scriptsize a\kern1emb$.
+    // TeX \showlists shows a kern of 1.13889 * fontsize;
+    // KaTeX shows a kern of 1.171 * fontsize.
+
+
+    if (sizeValue.unit === "ex") {
+      scale = unitOptions.fontMetrics().xHeight;
+    } else if (sizeValue.unit === "em") {
+      scale = unitOptions.fontMetrics().quad;
+    } else {
+      throw new src_ParseError("Invalid unit: '" + sizeValue.unit + "'");
+    }
+
+    if (unitOptions !== options) {
+      scale *= unitOptions.sizeMultiplier / options.sizeMultiplier;
+    }
+  }
+
+  return Math.min(sizeValue.number * scale, options.maxSize);
+};
+// CONCATENATED MODULE: ./src/buildCommon.js
+/* eslint no-console:0 */
+
+/**
+ * This module contains general functions that can be used for building
+ * different kinds of domTree nodes in a consistent manner.
+ */
+
+
+
+
+
+
+
+/**
+ * Looks up the given symbol in fontMetrics, after applying any symbol
+ * replacements defined in symbol.js
+ */
+var buildCommon_lookupSymbol = function lookupSymbol(value, // TODO(#963): Use a union type for this.
+fontName, mode) {
+  // Replace the value with its replaced value from symbol.js
+  if (src_symbols[mode][value] && src_symbols[mode][value].replace) {
+    value = src_symbols[mode][value].replace;
+  }
+
+  return {
+    value: value,
+    metrics: getCharacterMetrics(value, fontName, mode)
+  };
+};
+/**
+ * Makes a symbolNode after translation via the list of symbols in symbols.js.
+ * Correctly pulls out metrics for the character, and optionally takes a list of
+ * classes to be attached to the node.
+ *
+ * TODO: make argument order closer to makeSpan
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ * TODO(#953): Make `options` mandatory and always pass it in.
+ */
+
+
+var buildCommon_makeSymbol = function makeSymbol(value, fontName, mode, options, classes) {
+  var lookup = buildCommon_lookupSymbol(value, fontName, mode);
+  var metrics = lookup.metrics;
+  value = lookup.value;
+  var symbolNode;
+
+  if (metrics) {
+    var italic = metrics.italic;
+
+    if (mode === "text" || options && options.font === "mathit") {
+      italic = 0;
+    }
+
+    symbolNode = new domTree_SymbolNode(value, metrics.height, metrics.depth, italic, metrics.skew, metrics.width, classes);
+  } else {
+    // TODO(emily): Figure out a good way to only print this in development
+    typeof console !== "undefined" && console.warn("No character metrics " + ("for '" + value + "' in style '" + fontName + "' and mode '" + mode + "'"));
+    symbolNode = new domTree_SymbolNode(value, 0, 0, 0, 0, 0, classes);
+  }
+
+  if (options) {
+    symbolNode.maxFontSize = options.sizeMultiplier;
+
+    if (options.style.isTight()) {
+      symbolNode.classes.push("mtight");
+    }
+
+    var color = options.getColor();
+
+    if (color) {
+      symbolNode.style.color = color;
+    }
+  }
+
+  return symbolNode;
+};
+/**
+ * Makes a symbol in Main-Regular or AMS-Regular.
+ * Used for rel, bin, open, close, inner, and punct.
+ */
+
+
+var buildCommon_mathsym = function mathsym(value, mode, options, classes) {
+  if (classes === void 0) {
+    classes = [];
+  }
+
+  // Decide what font to render the symbol in by its entry in the symbols
+  // table.
+  // Have a special case for when the value = \ because the \ is used as a
+  // textord in unsupported command errors but cannot be parsed as a regular
+  // text ordinal and is therefore not present as a symbol in the symbols
+  // table for text, as well as a special case for boldsymbol because it
+  // can be used for bold + and -
+  if (options.font === "boldsymbol" && buildCommon_lookupSymbol(value, "Main-Bold", mode).metrics) {
+    return buildCommon_makeSymbol(value, "Main-Bold", mode, options, classes.concat(["mathbf"]));
+  } else if (value === "\\" || src_symbols[mode][value].font === "main") {
+    return buildCommon_makeSymbol(value, "Main-Regular", mode, options, classes);
+  } else {
+    return buildCommon_makeSymbol(value, "AMS-Regular", mode, options, classes.concat(["amsrm"]));
+  }
+};
+/**
+ * Determines which of the two font names (Main-Bold and Math-BoldItalic) and
+ * corresponding style tags (mathbf or boldsymbol) to use for font "boldsymbol",
+ * depending on the symbol.  Use this function instead of fontMap for font
+ * "boldsymbol".
+ */
+
+
+var boldsymbol = function boldsymbol(value, mode, options, classes, type) {
+  if (type !== "textord" && buildCommon_lookupSymbol(value, "Math-BoldItalic", mode).metrics) {
+    return {
+      fontName: "Math-BoldItalic",
+      fontClass: "boldsymbol"
+    };
+  } else {
+    // Some glyphs do not exist in Math-BoldItalic so we need to use
+    // Main-Bold instead.
+    return {
+      fontName: "Main-Bold",
+      fontClass: "mathbf"
+    };
+  }
+};
+/**
+ * Makes either a mathord or textord in the correct font and color.
+ */
+
+
+var buildCommon_makeOrd = function makeOrd(group, options, type) {
+  var mode = group.mode;
+  var text = group.text;
+  var classes = ["mord"]; // Math mode or Old font (i.e. \rm)
+
+  var isFont = mode === "math" || mode === "text" && options.font;
+  var fontOrFamily = isFont ? options.font : options.fontFamily;
+
+  if (text.charCodeAt(0) === 0xD835) {
+    // surrogate pairs get special treatment
+    var _wideCharacterFont = wide_character_wideCharacterFont(text, mode),
+        wideFontName = _wideCharacterFont[0],
+        wideFontClass = _wideCharacterFont[1];
+
+    return buildCommon_makeSymbol(text, wideFontName, mode, options, classes.concat(wideFontClass));
+  } else if (fontOrFamily) {
+    var fontName;
+    var fontClasses;
+
+    if (fontOrFamily === "boldsymbol") {
+      var fontData = boldsymbol(text, mode, options, classes, type);
+      fontName = fontData.fontName;
+      fontClasses = [fontData.fontClass];
+    } else if (isFont) {
+      fontName = fontMap[fontOrFamily].fontName;
+      fontClasses = [fontOrFamily];
+    } else {
+      fontName = retrieveTextFontName(fontOrFamily, options.fontWeight, options.fontShape);
+      fontClasses = [fontOrFamily, options.fontWeight, options.fontShape];
+    }
+
+    if (buildCommon_lookupSymbol(text, fontName, mode).metrics) {
+      return buildCommon_makeSymbol(text, fontName, mode, options, classes.concat(fontClasses));
+    } else if (ligatures.hasOwnProperty(text) && fontName.substr(0, 10) === "Typewriter") {
+      // Deconstruct ligatures in monospace fonts (\texttt, \tt).
+      var parts = [];
+
+      for (var i = 0; i < text.length; i++) {
+        parts.push(buildCommon_makeSymbol(text[i], fontName, mode, options, classes.concat(fontClasses)));
+      }
+
+      return buildCommon_makeFragment(parts);
+    }
+  } // Makes a symbol in the default font for mathords and textords.
+
+
+  if (type === "mathord") {
+    return buildCommon_makeSymbol(text, "Math-Italic", mode, options, classes.concat(["mathnormal"]));
+  } else if (type === "textord") {
+    var font = src_symbols[mode][text] && src_symbols[mode][text].font;
+
+    if (font === "ams") {
+      var _fontName = retrieveTextFontName("amsrm", options.fontWeight, options.fontShape);
+
+      return buildCommon_makeSymbol(text, _fontName, mode, options, classes.concat("amsrm", options.fontWeight, options.fontShape));
+    } else if (font === "main" || !font) {
+      var _fontName2 = retrieveTextFontName("textrm", options.fontWeight, options.fontShape);
+
+      return buildCommon_makeSymbol(text, _fontName2, mode, options, classes.concat(options.fontWeight, options.fontShape));
+    } else {
+      // fonts added by plugins
+      var _fontName3 = retrieveTextFontName(font, options.fontWeight, options.fontShape); // We add font name as a css class
+
+
+      return buildCommon_makeSymbol(text, _fontName3, mode, options, classes.concat(_fontName3, options.fontWeight, options.fontShape));
+    }
+  } else {
+    throw new Error("unexpected type: " + type + " in makeOrd");
+  }
+};
+/**
+ * Returns true if subsequent symbolNodes have the same classes, skew, maxFont,
+ * and styles.
+ */
+
+
+var buildCommon_canCombine = function canCombine(prev, next) {
+  if (createClass(prev.classes) !== createClass(next.classes) || prev.skew !== next.skew || prev.maxFontSize !== next.maxFontSize) {
+    return false;
+  }
+
+  for (var style in prev.style) {
+    if (prev.style.hasOwnProperty(style) && prev.style[style] !== next.style[style]) {
+      return false;
+    }
+  }
+
+  for (var _style in next.style) {
+    if (next.style.hasOwnProperty(_style) && prev.style[_style] !== next.style[_style]) {
+      return false;
+    }
+  }
+
+  return true;
+};
+/**
+ * Combine consequetive domTree.symbolNodes into a single symbolNode.
+ * Note: this function mutates the argument.
+ */
+
+
+var buildCommon_tryCombineChars = function tryCombineChars(chars) {
+  for (var i = 0; i < chars.length - 1; i++) {
+    var prev = chars[i];
+    var next = chars[i + 1];
+
+    if (prev instanceof domTree_SymbolNode && next instanceof domTree_SymbolNode && buildCommon_canCombine(prev, next)) {
+      prev.text += next.text;
+      prev.height = Math.max(prev.height, next.height);
+      prev.depth = Math.max(prev.depth, next.depth); // Use the last character's italic correction since we use
+      // it to add padding to the right of the span created from
+      // the combined characters.
+
+      prev.italic = next.italic;
+      chars.splice(i + 1, 1);
+      i--;
+    }
+  }
+
+  return chars;
+};
+/**
+ * Calculate the height, depth, and maxFontSize of an element based on its
+ * children.
+ */
+
+
+var sizeElementFromChildren = function sizeElementFromChildren(elem) {
+  var height = 0;
+  var depth = 0;
+  var maxFontSize = 0;
+
+  for (var i = 0; i < elem.children.length; i++) {
+    var child = elem.children[i];
+
+    if (child.height > height) {
+      height = child.height;
+    }
+
+    if (child.depth > depth) {
+      depth = child.depth;
+    }
+
+    if (child.maxFontSize > maxFontSize) {
+      maxFontSize = child.maxFontSize;
+    }
+  }
+
+  elem.height = height;
+  elem.depth = depth;
+  elem.maxFontSize = maxFontSize;
+};
+/**
+ * Makes a span with the given list of classes, list of children, and options.
+ *
+ * TODO(#953): Ensure that `options` is always provided (currently some call
+ * sites don't pass it) and make the type below mandatory.
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ */
+
+
+var buildCommon_makeSpan = function makeSpan(classes, children, options, style) {
+  var span = new domTree_Span(classes, children, options, style);
+  sizeElementFromChildren(span);
+  return span;
+}; // SVG one is simpler -- doesn't require height, depth, max-font setting.
+// This is also a separate method for typesafety.
+
+
+var buildCommon_makeSvgSpan = function makeSvgSpan(classes, children, options, style) {
+  return new domTree_Span(classes, children, options, style);
+};
+
+var makeLineSpan = function makeLineSpan(className, options, thickness) {
+  var line = buildCommon_makeSpan([className], [], options);
+  line.height = Math.max(thickness || options.fontMetrics().defaultRuleThickness, options.minRuleThickness);
+  line.style.borderBottomWidth = line.height + "em";
+  line.maxFontSize = 1.0;
+  return line;
+};
+/**
+ * Makes an anchor with the given href, list of classes, list of children,
+ * and options.
+ */
+
+
+var buildCommon_makeAnchor = function makeAnchor(href, classes, children, options) {
+  var anchor = new domTree_Anchor(href, classes, children, options);
+  sizeElementFromChildren(anchor);
+  return anchor;
+};
+/**
+ * Makes a document fragment with the given list of children.
+ */
+
+
+var buildCommon_makeFragment = function makeFragment(children) {
+  var fragment = new tree_DocumentFragment(children);
+  sizeElementFromChildren(fragment);
+  return fragment;
+};
+/**
+ * Wraps group in a span if it's a document fragment, allowing to apply classes
+ * and styles
+ */
+
+
+var buildCommon_wrapFragment = function wrapFragment(group, options) {
+  if (group instanceof tree_DocumentFragment) {
+    return buildCommon_makeSpan([], [group], options);
+  }
+
+  return group;
+}; // These are exact object types to catch typos in the names of the optional fields.
+
+
+// Computes the updated `children` list and the overall depth.
+//
+// This helper function for makeVList makes it easier to enforce type safety by
+// allowing early exits (returns) in the logic.
+var getVListChildrenAndDepth = function getVListChildrenAndDepth(params) {
+  if (params.positionType === "individualShift") {
+    var oldChildren = params.children;
+    var children = [oldChildren[0]]; // Add in kerns to the list of params.children to get each element to be
+    // shifted to the correct specified shift
+
+    var _depth = -oldChildren[0].shift - oldChildren[0].elem.depth;
+
+    var currPos = _depth;
+
+    for (var i = 1; i < oldChildren.length; i++) {
+      var diff = -oldChildren[i].shift - currPos - oldChildren[i].elem.depth;
+      var size = diff - (oldChildren[i - 1].elem.height + oldChildren[i - 1].elem.depth);
+      currPos = currPos + diff;
+      children.push({
+        type: "kern",
+        size: size
+      });
+      children.push(oldChildren[i]);
+    }
+
+    return {
+      children: children,
+      depth: _depth
+    };
+  }
+
+  var depth;
+
+  if (params.positionType === "top") {
+    // We always start at the bottom, so calculate the bottom by adding up
+    // all the sizes
+    var bottom = params.positionData;
+
+    for (var _i = 0; _i < params.children.length; _i++) {
+      var child = params.children[_i];
+      bottom -= child.type === "kern" ? child.size : child.elem.height + child.elem.depth;
+    }
+
+    depth = bottom;
+  } else if (params.positionType === "bottom") {
+    depth = -params.positionData;
+  } else {
+    var firstChild = params.children[0];
+
+    if (firstChild.type !== "elem") {
+      throw new Error('First child must have type "elem".');
+    }
+
+    if (params.positionType === "shift") {
+      depth = -firstChild.elem.depth - params.positionData;
+    } else if (params.positionType === "firstBaseline") {
+      depth = -firstChild.elem.depth;
+    } else {
+      throw new Error("Invalid positionType " + params.positionType + ".");
+    }
+  }
+
+  return {
+    children: params.children,
+    depth: depth
+  };
+};
+/**
+ * Makes a vertical list by stacking elements and kerns on top of each other.
+ * Allows for many different ways of specifying the positioning method.
+ *
+ * See VListParam documentation above.
+ */
+
+
+var buildCommon_makeVList = function makeVList(params, options) {
+  var _getVListChildrenAndD = getVListChildrenAndDepth(params),
+      children = _getVListChildrenAndD.children,
+      depth = _getVListChildrenAndD.depth; // Create a strut that is taller than any list item. The strut is added to
+  // each item, where it will determine the item's baseline. Since it has
+  // `overflow:hidden`, the strut's top edge will sit on the item's line box's
+  // top edge and the strut's bottom edge will sit on the item's baseline,
+  // with no additional line-height spacing. This allows the item baseline to
+  // be positioned precisely without worrying about font ascent and
+  // line-height.
+
+
+  var pstrutSize = 0;
+
+  for (var i = 0; i < children.length; i++) {
+    var child = children[i];
+
+    if (child.type === "elem") {
+      var elem = child.elem;
+      pstrutSize = Math.max(pstrutSize, elem.maxFontSize, elem.height);
+    }
+  }
+
+  pstrutSize += 2;
+  var pstrut = buildCommon_makeSpan(["pstrut"], []);
+  pstrut.style.height = pstrutSize + "em"; // Create a new list of actual children at the correct offsets
+
+  var realChildren = [];
+  var minPos = depth;
+  var maxPos = depth;
+  var currPos = depth;
+
+  for (var _i2 = 0; _i2 < children.length; _i2++) {
+    var _child = children[_i2];
+
+    if (_child.type === "kern") {
+      currPos += _child.size;
+    } else {
+      var _elem = _child.elem;
+      var classes = _child.wrapperClasses || [];
+      var style = _child.wrapperStyle || {};
+      var childWrap = buildCommon_makeSpan(classes, [pstrut, _elem], undefined, style);
+      childWrap.style.top = -pstrutSize - currPos - _elem.depth + "em";
+
+      if (_child.marginLeft) {
+        childWrap.style.marginLeft = _child.marginLeft;
+      }
+
+      if (_child.marginRight) {
+        childWrap.style.marginRight = _child.marginRight;
+      }
+
+      realChildren.push(childWrap);
+      currPos += _elem.height + _elem.depth;
+    }
+
+    minPos = Math.min(minPos, currPos);
+    maxPos = Math.max(maxPos, currPos);
+  } // The vlist contents go in a table-cell with `vertical-align:bottom`.
+  // This cell's bottom edge will determine the containing table's baseline
+  // without overly expanding the containing line-box.
+
+
+  var vlist = buildCommon_makeSpan(["vlist"], realChildren);
+  vlist.style.height = maxPos + "em"; // A second row is used if necessary to represent the vlist's depth.
+
+  var rows;
+
+  if (minPos < 0) {
+    // We will define depth in an empty span with display: table-cell.
+    // It should render with the height that we define. But Chrome, in
+    // contenteditable mode only, treats that span as if it contains some
+    // text content. And that min-height over-rides our desired height.
+    // So we put another empty span inside the depth strut span.
+    var emptySpan = buildCommon_makeSpan([], []);
+    var depthStrut = buildCommon_makeSpan(["vlist"], [emptySpan]);
+    depthStrut.style.height = -minPos + "em"; // Safari wants the first row to have inline content; otherwise it
+    // puts the bottom of the *second* row on the baseline.
+
+    var topStrut = buildCommon_makeSpan(["vlist-s"], [new domTree_SymbolNode("\u200B")]);
+    rows = [buildCommon_makeSpan(["vlist-r"], [vlist, topStrut]), buildCommon_makeSpan(["vlist-r"], [depthStrut])];
+  } else {
+    rows = [buildCommon_makeSpan(["vlist-r"], [vlist])];
+  }
+
+  var vtable = buildCommon_makeSpan(["vlist-t"], rows);
+
+  if (rows.length === 2) {
+    vtable.classes.push("vlist-t2");
+  }
+
+  vtable.height = maxPos;
+  vtable.depth = -minPos;
+  return vtable;
+}; // Glue is a concept from TeX which is a flexible space between elements in
+// either a vertical or horizontal list. In KaTeX, at least for now, it's
+// static space between elements in a horizontal layout.
+
+
+var buildCommon_makeGlue = function makeGlue(measurement, options) {
+  // Make an empty span for the space
+  var rule = buildCommon_makeSpan(["mspace"], [], options);
+  var size = units_calculateSize(measurement, options);
+  rule.style.marginRight = size + "em";
+  return rule;
+}; // Takes font options, and returns the appropriate fontLookup name
+
+
+var retrieveTextFontName = function retrieveTextFontName(fontFamily, fontWeight, fontShape) {
+  var baseFontName = "";
+
+  switch (fontFamily) {
+    case "amsrm":
+      baseFontName = "AMS";
+      break;
+
+    case "textrm":
+      baseFontName = "Main";
+      break;
+
+    case "textsf":
+      baseFontName = "SansSerif";
+      break;
+
+    case "texttt":
+      baseFontName = "Typewriter";
+      break;
+
+    default:
+      baseFontName = fontFamily;
+    // use fonts added by a plugin
+  }
+
+  var fontStylesName;
+
+  if (fontWeight === "textbf" && fontShape === "textit") {
+    fontStylesName = "BoldItalic";
+  } else if (fontWeight === "textbf") {
+    fontStylesName = "Bold";
+  } else if (fontWeight === "textit") {
+    fontStylesName = "Italic";
+  } else {
+    fontStylesName = "Regular";
+  }
+
+  return baseFontName + "-" + fontStylesName;
+};
+/**
+ * Maps TeX font commands to objects containing:
+ * - variant: string used for "mathvariant" attribute in buildMathML.js
+ * - fontName: the "style" parameter to fontMetrics.getCharacterMetrics
+ */
+// A map between tex font commands an MathML mathvariant attribute values
+
+
+var fontMap = {
+  // styles
+  "mathbf": {
+    variant: "bold",
+    fontName: "Main-Bold"
+  },
+  "mathrm": {
+    variant: "normal",
+    fontName: "Main-Regular"
+  },
+  "textit": {
+    variant: "italic",
+    fontName: "Main-Italic"
+  },
+  "mathit": {
+    variant: "italic",
+    fontName: "Main-Italic"
+  },
+  "mathnormal": {
+    variant: "italic",
+    fontName: "Math-Italic"
+  },
+  // "boldsymbol" is missing because they require the use of multiple fonts:
+  // Math-BoldItalic and Main-Bold.  This is handled by a special case in
+  // makeOrd which ends up calling boldsymbol.
+  // families
+  "mathbb": {
+    variant: "double-struck",
+    fontName: "AMS-Regular"
+  },
+  "mathcal": {
+    variant: "script",
+    fontName: "Caligraphic-Regular"
+  },
+  "mathfrak": {
+    variant: "fraktur",
+    fontName: "Fraktur-Regular"
+  },
+  "mathscr": {
+    variant: "script",
+    fontName: "Script-Regular"
+  },
+  "mathsf": {
+    variant: "sans-serif",
+    fontName: "SansSerif-Regular"
+  },
+  "mathtt": {
+    variant: "monospace",
+    fontName: "Typewriter-Regular"
+  }
+};
+var svgData = {
+  //   path, width, height
+  vec: ["vec", 0.471, 0.714],
+  // values from the font glyph
+  oiintSize1: ["oiintSize1", 0.957, 0.499],
+  // oval to overlay the integrand
+  oiintSize2: ["oiintSize2", 1.472, 0.659],
+  oiiintSize1: ["oiiintSize1", 1.304, 0.499],
+  oiiintSize2: ["oiiintSize2", 1.98, 0.659],
+  leftParenInner: ["leftParenInner", 0.875, 0.3],
+  rightParenInner: ["rightParenInner", 0.875, 0.3]
+};
+
+var buildCommon_staticSvg = function staticSvg(value, options) {
+  // Create a span with inline SVG for the element.
+  var _svgData$value = svgData[value],
+      pathName = _svgData$value[0],
+      width = _svgData$value[1],
+      height = _svgData$value[2];
+  var path = new domTree_PathNode(pathName);
+  var svgNode = new SvgNode([path], {
+    "width": width + "em",
+    "height": height + "em",
+    // Override CSS rule `.katex svg { width: 100% }`
+    "style": "width:" + width + "em",
+    "viewBox": "0 0 " + 1000 * width + " " + 1000 * height,
+    "preserveAspectRatio": "xMinYMin"
+  });
+  var span = buildCommon_makeSvgSpan(["overlay"], [svgNode], options);
+  span.height = height;
+  span.style.height = height + "em";
+  span.style.width = width + "em";
+  return span;
+};
+
+/* harmony default export */ var buildCommon = ({
+  fontMap: fontMap,
+  makeSymbol: buildCommon_makeSymbol,
+  mathsym: buildCommon_mathsym,
+  makeSpan: buildCommon_makeSpan,
+  makeSvgSpan: buildCommon_makeSvgSpan,
+  makeLineSpan: makeLineSpan,
+  makeAnchor: buildCommon_makeAnchor,
+  makeFragment: buildCommon_makeFragment,
+  wrapFragment: buildCommon_wrapFragment,
+  makeVList: buildCommon_makeVList,
+  makeOrd: buildCommon_makeOrd,
+  makeGlue: buildCommon_makeGlue,
+  staticSvg: buildCommon_staticSvg,
+  svgData: svgData,
+  tryCombineChars: buildCommon_tryCombineChars
+});
+// CONCATENATED MODULE: ./src/spacingData.js
+/**
+ * Describes spaces between different classes of atoms.
+ */
+var thinspace = {
+  number: 3,
+  unit: "mu"
+};
+var mediumspace = {
+  number: 4,
+  unit: "mu"
+};
+var thickspace = {
+  number: 5,
+  unit: "mu"
+}; // Making the type below exact with all optional fields doesn't work due to
+// - https://github.com/facebook/flow/issues/4582
+// - https://github.com/facebook/flow/issues/5688
+// However, since *all* fields are optional, $Shape<> works as suggested in 5688
+// above.
+
+// Spacing relationships for display and text styles
+var spacings = {
+  mord: {
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mop: {
+    mord: thinspace,
+    mop: thinspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mbin: {
+    mord: mediumspace,
+    mop: mediumspace,
+    mopen: mediumspace,
+    minner: mediumspace
+  },
+  mrel: {
+    mord: thickspace,
+    mop: thickspace,
+    mopen: thickspace,
+    minner: thickspace
+  },
+  mopen: {},
+  mclose: {
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mpunct: {
+    mord: thinspace,
+    mop: thinspace,
+    mrel: thickspace,
+    mopen: thinspace,
+    mclose: thinspace,
+    mpunct: thinspace,
+    minner: thinspace
+  },
+  minner: {
+    mord: thinspace,
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    mopen: thinspace,
+    mpunct: thinspace,
+    minner: thinspace
+  }
+}; // Spacing relationships for script and scriptscript styles
+
+var tightSpacings = {
+  mord: {
+    mop: thinspace
+  },
+  mop: {
+    mord: thinspace,
+    mop: thinspace
+  },
+  mbin: {},
+  mrel: {},
+  mopen: {},
+  mclose: {
+    mop: thinspace
+  },
+  mpunct: {},
+  minner: {
+    mop: thinspace
+  }
+};
+// CONCATENATED MODULE: ./src/defineFunction.js
+/** Context provided to function handlers for error messages. */
+// Note: reverse the order of the return type union will cause a flow error.
+// See https://github.com/facebook/flow/issues/3663.
+// More general version of `HtmlBuilder` for nodes (e.g. \sum, accent types)
+// whose presence impacts super/subscripting. In this case, ParseNode<"supsub">
+// delegates its HTML building to the HtmlBuilder corresponding to these nodes.
+
+/**
+ * Final function spec for use at parse time.
+ * This is almost identical to `FunctionPropSpec`, except it
+ * 1. includes the function handler, and
+ * 2. requires all arguments except argTypes.
+ * It is generated by `defineFunction()` below.
+ */
+
+/**
+ * All registered functions.
+ * `functions.js` just exports this same dictionary again and makes it public.
+ * `Parser.js` requires this dictionary.
+ */
+var _functions = {};
+/**
+ * All HTML builders. Should be only used in the `define*` and the `build*ML`
+ * functions.
+ */
+
+var _htmlGroupBuilders = {};
+/**
+ * All MathML builders. Should be only used in the `define*` and the `build*ML`
+ * functions.
+ */
+
+var _mathmlGroupBuilders = {};
+function defineFunction(_ref) {
+  var type = _ref.type,
+      names = _ref.names,
+      props = _ref.props,
+      handler = _ref.handler,
+      htmlBuilder = _ref.htmlBuilder,
+      mathmlBuilder = _ref.mathmlBuilder;
+  // Set default values of functions
+  var data = {
+    type: type,
+    numArgs: props.numArgs,
+    argTypes: props.argTypes,
+    greediness: props.greediness === undefined ? 1 : props.greediness,
+    allowedInText: !!props.allowedInText,
+    allowedInMath: props.allowedInMath === undefined ? true : props.allowedInMath,
+    numOptionalArgs: props.numOptionalArgs || 0,
+    infix: !!props.infix,
+    handler: handler
+  };
+
+  for (var i = 0; i < names.length; ++i) {
+    _functions[names[i]] = data;
+  }
+
+  if (type) {
+    if (htmlBuilder) {
+      _htmlGroupBuilders[type] = htmlBuilder;
+    }
+
+    if (mathmlBuilder) {
+      _mathmlGroupBuilders[type] = mathmlBuilder;
+    }
+  }
+}
+/**
+ * Use this to register only the HTML and MathML builders for a function (e.g.
+ * if the function's ParseNode is generated in Parser.js rather than via a
+ * stand-alone handler provided to `defineFunction`).
+ */
+
+function defineFunctionBuilders(_ref2) {
+  var type = _ref2.type,
+      htmlBuilder = _ref2.htmlBuilder,
+      mathmlBuilder = _ref2.mathmlBuilder;
+  defineFunction({
+    type: type,
+    names: [],
+    props: {
+      numArgs: 0
+    },
+    handler: function handler() {
+      throw new Error('Should never be called.');
+    },
+    htmlBuilder: htmlBuilder,
+    mathmlBuilder: mathmlBuilder
+  });
+} // Since the corresponding buildHTML/buildMathML function expects a
+// list of elements, we normalize for different kinds of arguments
+
+var ordargument = function ordargument(arg) {
+  return arg.type === "ordgroup" ? arg.body : [arg];
+};
+// CONCATENATED MODULE: ./src/buildHTML.js
+/**
+ * This file does the main work of building a domTree structure from a parse
+ * tree. The entry point is the `buildHTML` function, which takes a parse tree.
+ * Then, the buildExpression, buildGroup, and various groupBuilders functions
+ * are called, to produce a final HTML tree.
+ */
+
+
+
+
+
+
+
+
+var buildHTML_makeSpan = buildCommon.makeSpan; // Binary atoms (first class `mbin`) change into ordinary atoms (`mord`)
+// depending on their surroundings. See TeXbook pg. 442-446, Rules 5 and 6,
+// and the text before Rule 19.
+
+var binLeftCanceller = ["leftmost", "mbin", "mopen", "mrel", "mop", "mpunct"];
+var binRightCanceller = ["rightmost", "mrel", "mclose", "mpunct"];
+var styleMap = {
+  "display": src_Style.DISPLAY,
+  "text": src_Style.TEXT,
+  "script": src_Style.SCRIPT,
+  "scriptscript": src_Style.SCRIPTSCRIPT
+};
+var DomEnum = {
+  mord: "mord",
+  mop: "mop",
+  mbin: "mbin",
+  mrel: "mrel",
+  mopen: "mopen",
+  mclose: "mclose",
+  mpunct: "mpunct",
+  minner: "minner"
+};
+
+/**
+ * Take a list of nodes, build them in order, and return a list of the built
+ * nodes. documentFragments are flattened into their contents, so the
+ * returned list contains no fragments. `isRealGroup` is true if `expression`
+ * is a real group (no atoms will be added on either side), as opposed to
+ * a partial group (e.g. one created by \color). `surrounding` is an array
+ * consisting type of nodes that will be added to the left and right.
+ */
+var buildHTML_buildExpression = function buildExpression(expression, options, isRealGroup, surrounding) {
+  if (surrounding === void 0) {
+    surrounding = [null, null];
+  }
+
+  // Parse expressions into `groups`.
+  var groups = [];
+
+  for (var i = 0; i < expression.length; i++) {
+    var output = buildHTML_buildGroup(expression[i], options);
+
+    if (output instanceof tree_DocumentFragment) {
+      var children = output.children;
+      groups.push.apply(groups, children);
+    } else {
+      groups.push(output);
+    }
+  } // If `expression` is a partial group, let the parent handle spacings
+  // to avoid processing groups multiple times.
+
+
+  if (!isRealGroup) {
+    return groups;
+  }
+
+  var glueOptions = options;
+
+  if (expression.length === 1) {
+    var node = expression[0];
+
+    if (node.type === "sizing") {
+      glueOptions = options.havingSize(node.size);
+    } else if (node.type === "styling") {
+      glueOptions = options.havingStyle(styleMap[node.style]);
+    }
+  } // Dummy spans for determining spacings between surrounding atoms.
+  // If `expression` has no atoms on the left or right, class "leftmost"
+  // or "rightmost", respectively, is used to indicate it.
+
+
+  var dummyPrev = buildHTML_makeSpan([surrounding[0] || "leftmost"], [], options);
+  var dummyNext = buildHTML_makeSpan([surrounding[1] || "rightmost"], [], options); // TODO: These code assumes that a node's math class is the first element
+  // of its `classes` array. A later cleanup should ensure this, for
+  // instance by changing the signature of `makeSpan`.
+  // Before determining what spaces to insert, perform bin cancellation.
+  // Binary operators change to ordinary symbols in some contexts.
+
+  var isRoot = isRealGroup === "root";
+  traverseNonSpaceNodes(groups, function (node, prev) {
+    var prevType = prev.classes[0];
+    var type = node.classes[0];
+
+    if (prevType === "mbin" && utils.contains(binRightCanceller, type)) {
+      prev.classes[0] = "mord";
+    } else if (type === "mbin" && utils.contains(binLeftCanceller, prevType)) {
+      node.classes[0] = "mord";
+    }
+  }, {
+    node: dummyPrev
+  }, dummyNext, isRoot);
+  traverseNonSpaceNodes(groups, function (node, prev) {
+    var prevType = getTypeOfDomTree(prev);
+    var type = getTypeOfDomTree(node); // 'mtight' indicates that the node is script or scriptscript style.
+
+    var space = prevType && type ? node.hasClass("mtight") ? tightSpacings[prevType][type] : spacings[prevType][type] : null;
+
+    if (space) {
+      // Insert glue (spacing) after the `prev`.
+      return buildCommon.makeGlue(space, glueOptions);
+    }
+  }, {
+    node: dummyPrev
+  }, dummyNext, isRoot);
+  return groups;
+}; // Depth-first traverse non-space `nodes`, calling `callback` with the current and
+// previous node as arguments, optionally returning a node to insert after the
+// previous node. `prev` is an object with the previous node and `insertAfter`
+// function to insert after it. `next` is a node that will be added to the right.
+// Used for bin cancellation and inserting spacings.
+
+var traverseNonSpaceNodes = function traverseNonSpaceNodes(nodes, callback, prev, next, isRoot) {
+  if (next) {
+    // temporarily append the right node, if exists
+    nodes.push(next);
+  }
+
+  var i = 0;
+
+  for (; i < nodes.length; i++) {
+    var node = nodes[i];
+    var partialGroup = buildHTML_checkPartialGroup(node);
+
+    if (partialGroup) {
+      // Recursive DFS
+      // $FlowFixMe: make nodes a $ReadOnlyArray by returning a new array
+      traverseNonSpaceNodes(partialGroup.children, callback, prev, null, isRoot);
+      continue;
+    } // Ignore explicit spaces (e.g., \;, \,) when determining what implicit
+    // spacing should go between atoms of different classes
+
+
+    var nonspace = !node.hasClass("mspace");
+
+    if (nonspace) {
+      var result = callback(node, prev.node);
+
+      if (result) {
+        if (prev.insertAfter) {
+          prev.insertAfter(result);
+        } else {
+          // insert at front
+          nodes.unshift(result);
+          i++;
+        }
+      }
+    }
+
+    if (nonspace) {
+      prev.node = node;
+    } else if (isRoot && node.hasClass("newline")) {
+      prev.node = buildHTML_makeSpan(["leftmost"]); // treat like beginning of line
+    }
+
+    prev.insertAfter = function (index) {
+      return function (n) {
+        nodes.splice(index + 1, 0, n);
+        i++;
+      };
+    }(i);
+  }
+
+  if (next) {
+    nodes.pop();
+  }
+}; // Check if given node is a partial group, i.e., does not affect spacing around.
+
+
+var buildHTML_checkPartialGroup = function checkPartialGroup(node) {
+  if (node instanceof tree_DocumentFragment || node instanceof domTree_Anchor || node instanceof domTree_Span && node.hasClass("enclosing")) {
+    return node;
+  }
+
+  return null;
+}; // Return the outermost node of a domTree.
+
+
+var getOutermostNode = function getOutermostNode(node, side) {
+  var partialGroup = buildHTML_checkPartialGroup(node);
+
+  if (partialGroup) {
+    var children = partialGroup.children;
+
+    if (children.length) {
+      if (side === "right") {
+        return getOutermostNode(children[children.length - 1], "right");
+      } else if (side === "left") {
+        return getOutermostNode(children[0], "left");
+      }
+    }
+  }
+
+  return node;
+}; // Return math atom class (mclass) of a domTree.
+// If `side` is given, it will get the type of the outermost node at given side.
+
+
+var getTypeOfDomTree = function getTypeOfDomTree(node, side) {
+  if (!node) {
+    return null;
+  }
+
+  if (side) {
+    node = getOutermostNode(node, side);
+  } // This makes a lot of assumptions as to where the type of atom
+  // appears.  We should do a better job of enforcing this.
+
+
+  return DomEnum[node.classes[0]] || null;
+};
+var makeNullDelimiter = function makeNullDelimiter(options, classes) {
+  var moreClasses = ["nulldelimiter"].concat(options.baseSizingClasses());
+  return buildHTML_makeSpan(classes.concat(moreClasses));
+};
+/**
+ * buildGroup is the function that takes a group and calls the correct groupType
+ * function for it. It also handles the interaction of size and style changes
+ * between parents and children.
+ */
+
+var buildHTML_buildGroup = function buildGroup(group, options, baseOptions) {
+  if (!group) {
+    return buildHTML_makeSpan();
+  }
+
+  if (_htmlGroupBuilders[group.type]) {
+    // Call the groupBuilders function
+    var groupNode = _htmlGroupBuilders[group.type](group, options); // If the size changed between the parent and the current group, account
+    // for that size difference.
+
+    if (baseOptions && options.size !== baseOptions.size) {
+      groupNode = buildHTML_makeSpan(options.sizingClasses(baseOptions), [groupNode], options);
+      var multiplier = options.sizeMultiplier / baseOptions.sizeMultiplier;
+      groupNode.height *= multiplier;
+      groupNode.depth *= multiplier;
+    }
+
+    return groupNode;
+  } else {
+    throw new src_ParseError("Got group of unknown type: '" + group.type + "'");
+  }
+};
+/**
+ * Combine an array of HTML DOM nodes (e.g., the output of `buildExpression`)
+ * into an unbreakable HTML node of class .base, with proper struts to
+ * guarantee correct vertical extent.  `buildHTML` calls this repeatedly to
+ * make up the entire expression as a sequence of unbreakable units.
+ */
+
+function buildHTMLUnbreakable(children, options) {
+  // Compute height and depth of this chunk.
+  var body = buildHTML_makeSpan(["base"], children, options); // Add strut, which ensures that the top of the HTML element falls at
+  // the height of the expression, and the bottom of the HTML element
+  // falls at the depth of the expression.
+
+  var strut = buildHTML_makeSpan(["strut"]);
+  strut.style.height = body.height + body.depth + "em";
+  strut.style.verticalAlign = -body.depth + "em";
+  body.children.unshift(strut);
+  return body;
+}
+/**
+ * Take an entire parse tree, and build it into an appropriate set of HTML
+ * nodes.
+ */
+
+
+function buildHTML(tree, options) {
+  // Strip off outer tag wrapper for processing below.
+  var tag = null;
+
+  if (tree.length === 1 && tree[0].type === "tag") {
+    tag = tree[0].tag;
+    tree = tree[0].body;
+  } // Build the expression contained in the tree
+
+
+  var expression = buildHTML_buildExpression(tree, options, "root");
+  var children = []; // Create one base node for each chunk between potential line breaks.
+  // The TeXBook [p.173] says "A formula will be broken only after a
+  // relation symbol like $=$ or $<$ or $\rightarrow$, or after a binary
+  // operation symbol like $+$ or $-$ or $\times$, where the relation or
+  // binary operation is on the ``outer level'' of the formula (i.e., not
+  // enclosed in {...} and not part of an \over construction)."
+
+  var parts = [];
+
+  for (var i = 0; i < expression.length; i++) {
+    parts.push(expression[i]);
+
+    if (expression[i].hasClass("mbin") || expression[i].hasClass("mrel") || expression[i].hasClass("allowbreak")) {
+      // Put any post-operator glue on same line as operator.
+      // Watch for \nobreak along the way, and stop at \newline.
+      var nobreak = false;
+
+      while (i < expression.length - 1 && expression[i + 1].hasClass("mspace") && !expression[i + 1].hasClass("newline")) {
+        i++;
+        parts.push(expression[i]);
+
+        if (expression[i].hasClass("nobreak")) {
+          nobreak = true;
+        }
+      } // Don't allow break if \nobreak among the post-operator glue.
+
+
+      if (!nobreak) {
+        children.push(buildHTMLUnbreakable(parts, options));
+        parts = [];
+      }
+    } else if (expression[i].hasClass("newline")) {
+      // Write the line except the newline
+      parts.pop();
+
+      if (parts.length > 0) {
+        children.push(buildHTMLUnbreakable(parts, options));
+        parts = [];
+      } // Put the newline at the top level
+
+
+      children.push(expression[i]);
+    }
+  }
+
+  if (parts.length > 0) {
+    children.push(buildHTMLUnbreakable(parts, options));
+  } // Now, if there was a tag, build it too and append it as a final child.
+
+
+  var tagChild;
+
+  if (tag) {
+    tagChild = buildHTMLUnbreakable(buildHTML_buildExpression(tag, options, true));
+    tagChild.classes = ["tag"];
+    children.push(tagChild);
+  }
+
+  var htmlNode = buildHTML_makeSpan(["katex-html"], children);
+  htmlNode.setAttribute("aria-hidden", "true"); // Adjust the strut of the tag to be the maximum height of all children
+  // (the height of the enclosing htmlNode) for proper vertical alignment.
+
+  if (tagChild) {
+    var strut = tagChild.children[0];
+    strut.style.height = htmlNode.height + htmlNode.depth + "em";
+    strut.style.verticalAlign = -htmlNode.depth + "em";
+  }
+
+  return htmlNode;
+}
+// CONCATENATED MODULE: ./src/mathMLTree.js
+/**
+ * These objects store data about MathML nodes. This is the MathML equivalent
+ * of the types in domTree.js. Since MathML handles its own rendering, and
+ * since we're mainly using MathML to improve accessibility, we don't manage
+ * any of the styling state that the plain DOM nodes do.
+ *
+ * The `toNode` and `toMarkup` functions work simlarly to how they do in
+ * domTree.js, creating namespaced DOM nodes and HTML text markup respectively.
+ */
+
+
+function newDocumentFragment(children) {
+  return new tree_DocumentFragment(children);
+}
+/**
+ * This node represents a general purpose MathML node of any type. The
+ * constructor requires the type of node to create (for example, `"mo"` or
+ * `"mspace"`, corresponding to `<mo>` and `<mspace>` tags).
+ */
+
+var mathMLTree_MathNode =
+/*#__PURE__*/
+function () {
+  function MathNode(type, children) {
+    this.type = void 0;
+    this.attributes = void 0;
+    this.children = void 0;
+    this.type = type;
+    this.attributes = {};
+    this.children = children || [];
+  }
+  /**
+   * Sets an attribute on a MathML node. MathML depends on attributes to convey a
+   * semantic content, so this is used heavily.
+   */
+
+
+  var _proto = MathNode.prototype;
+
+  _proto.setAttribute = function setAttribute(name, value) {
+    this.attributes[name] = value;
+  }
+  /**
+   * Gets an attribute on a MathML node.
+   */
+  ;
+
+  _proto.getAttribute = function getAttribute(name) {
+    return this.attributes[name];
+  }
+  /**
+   * Converts the math node into a MathML-namespaced DOM element.
+   */
+  ;
+
+  _proto.toNode = function toNode() {
+    var node = document.createElementNS("http://www.w3.org/1998/Math/MathML", this.type);
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    for (var i = 0; i < this.children.length; i++) {
+      node.appendChild(this.children[i].toNode());
+    }
+
+    return node;
+  }
+  /**
+   * Converts the math node into an HTML markup string.
+   */
+  ;
+
+  _proto.toMarkup = function toMarkup() {
+    var markup = "<" + this.type; // Add the attributes
+
+    for (var attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += " " + attr + "=\"";
+        markup += utils.escape(this.attributes[attr]);
+        markup += "\"";
+      }
+    }
+
+    markup += ">";
+
+    for (var i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    markup += "</" + this.type + ">";
+    return markup;
+  }
+  /**
+   * Converts the math node into a string, similar to innerText, but escaped.
+   */
+  ;
+
+  _proto.toText = function toText() {
+    return this.children.map(function (child) {
+      return child.toText();
+    }).join("");
+  };
+
+  return MathNode;
+}();
+/**
+ * This node represents a piece of text.
+ */
+
+var mathMLTree_TextNode =
+/*#__PURE__*/
+function () {
+  function TextNode(text) {
+    this.text = void 0;
+    this.text = text;
+  }
+  /**
+   * Converts the text node into a DOM text node.
+   */
+
+
+  var _proto2 = TextNode.prototype;
+
+  _proto2.toNode = function toNode() {
+    return document.createTextNode(this.text);
+  }
+  /**
+   * Converts the text node into escaped HTML markup
+   * (representing the text itself).
+   */
+  ;
+
+  _proto2.toMarkup = function toMarkup() {
+    return utils.escape(this.toText());
+  }
+  /**
+   * Converts the text node into a string
+   * (representing the text iteself).
+   */
+  ;
+
+  _proto2.toText = function toText() {
+    return this.text;
+  };
+
+  return TextNode;
+}();
+/**
+ * This node represents a space, but may render as <mspace.../> or as text,
+ * depending on the width.
+ */
+
+var SpaceNode =
+/*#__PURE__*/
+function () {
+  /**
+   * Create a Space node with width given in CSS ems.
+   */
+  function SpaceNode(width) {
+    this.width = void 0;
+    this.character = void 0;
+    this.width = width; // See https://www.w3.org/TR/2000/WD-MathML2-20000328/chapter6.html
+    // for a table of space-like characters.  We use Unicode
+    // representations instead of &LongNames; as it's not clear how to
+    // make the latter via document.createTextNode.
+
+    if (width >= 0.05555 && width <= 0.05556) {
+      this.character = "\u200A"; // &VeryThinSpace;
+    } else if (width >= 0.1666 && width <= 0.1667) {
+      this.character = "\u2009"; // &ThinSpace;
+    } else if (width >= 0.2222 && width <= 0.2223) {
+      this.character = "\u2005"; // &MediumSpace;
+    } else if (width >= 0.2777 && width <= 0.2778) {
+      this.character = "\u2005\u200A"; // &ThickSpace;
+    } else if (width >= -0.05556 && width <= -0.05555) {
+      this.character = "\u200A\u2063"; // &NegativeVeryThinSpace;
+    } else if (width >= -0.1667 && width <= -0.1666) {
+      this.character = "\u2009\u2063"; // &NegativeThinSpace;
+    } else if (width >= -0.2223 && width <= -0.2222) {
+      this.character = "\u205F\u2063"; // &NegativeMediumSpace;
+    } else if (width >= -0.2778 && width <= -0.2777) {
+      this.character = "\u2005\u2063"; // &NegativeThickSpace;
+    } else {
+      this.character = null;
+    }
+  }
+  /**
+   * Converts the math node into a MathML-namespaced DOM element.
+   */
+
+
+  var _proto3 = SpaceNode.prototype;
+
+  _proto3.toNode = function toNode() {
+    if (this.character) {
+      return document.createTextNode(this.character);
+    } else {
+      var node = document.createElementNS("http://www.w3.org/1998/Math/MathML", "mspace");
+      node.setAttribute("width", this.width + "em");
+      return node;
+    }
+  }
+  /**
+   * Converts the math node into an HTML markup string.
+   */
+  ;
+
+  _proto3.toMarkup = function toMarkup() {
+    if (this.character) {
+      return "<mtext>" + this.character + "</mtext>";
+    } else {
+      return "<mspace width=\"" + this.width + "em\"/>";
+    }
+  }
+  /**
+   * Converts the math node into a string, similar to innerText.
+   */
+  ;
+
+  _proto3.toText = function toText() {
+    if (this.character) {
+      return this.character;
+    } else {
+      return " ";
+    }
+  };
+
+  return SpaceNode;
+}();
+
+/* harmony default export */ var mathMLTree = ({
+  MathNode: mathMLTree_MathNode,
+  TextNode: mathMLTree_TextNode,
+  SpaceNode: SpaceNode,
+  newDocumentFragment: newDocumentFragment
+});
+// CONCATENATED MODULE: ./src/buildMathML.js
+/**
+ * This file converts a parse tree into a cooresponding MathML tree. The main
+ * entry point is the `buildMathML` function, which takes a parse tree from the
+ * parser.
+ */
+
+
+
+
+
+
+
+
+
+/**
+ * Takes a symbol and converts it into a MathML text node after performing
+ * optional replacement from symbols.js.
+ */
+var buildMathML_makeText = function makeText(text, mode, options) {
+  if (src_symbols[mode][text] && src_symbols[mode][text].replace && text.charCodeAt(0) !== 0xD835 && !(ligatures.hasOwnProperty(text) && options && (options.fontFamily && options.fontFamily.substr(4, 2) === "tt" || options.font && options.font.substr(4, 2) === "tt"))) {
+    text = src_symbols[mode][text].replace;
+  }
+
+  return new mathMLTree.TextNode(text);
+};
+/**
+ * Wrap the given array of nodes in an <mrow> node if needed, i.e.,
+ * unless the array has length 1.  Always returns a single node.
+ */
+
+var buildMathML_makeRow = function makeRow(body) {
+  if (body.length === 1) {
+    return body[0];
+  } else {
+    return new mathMLTree.MathNode("mrow", body);
+  }
+};
+/**
+ * Returns the math variant as a string or null if none is required.
+ */
+
+var buildMathML_getVariant = function getVariant(group, options) {
+  // Handle \text... font specifiers as best we can.
+  // MathML has a limited list of allowable mathvariant specifiers; see
+  // https://www.w3.org/TR/MathML3/chapter3.html#presm.commatt
+  if (options.fontFamily === "texttt") {
+    return "monospace";
+  } else if (options.fontFamily === "textsf") {
+    if (options.fontShape === "textit" && options.fontWeight === "textbf") {
+      return "sans-serif-bold-italic";
+    } else if (options.fontShape === "textit") {
+      return "sans-serif-italic";
+    } else if (options.fontWeight === "textbf") {
+      return "bold-sans-serif";
+    } else {
+      return "sans-serif";
+    }
+  } else if (options.fontShape === "textit" && options.fontWeight === "textbf") {
+    return "bold-italic";
+  } else if (options.fontShape === "textit") {
+    return "italic";
+  } else if (options.fontWeight === "textbf") {
+    return "bold";
+  }
+
+  var font = options.font;
+
+  if (!font || font === "mathnormal") {
+    return null;
+  }
+
+  var mode = group.mode;
+
+  if (font === "mathit") {
+    return "italic";
+  } else if (font === "boldsymbol") {
+    return group.type === "textord" ? "bold" : "bold-italic";
+  } else if (font === "mathbf") {
+    return "bold";
+  } else if (font === "mathbb") {
+    return "double-struck";
+  } else if (font === "mathfrak") {
+    return "fraktur";
+  } else if (font === "mathscr" || font === "mathcal") {
+    // MathML makes no distinction between script and caligrahpic
+    return "script";
+  } else if (font === "mathsf") {
+    return "sans-serif";
+  } else if (font === "mathtt") {
+    return "monospace";
+  }
+
+  var text = group.text;
+
+  if (utils.contains(["\\imath", "\\jmath"], text)) {
+    return null;
+  }
+
+  if (src_symbols[mode][text] && src_symbols[mode][text].replace) {
+    text = src_symbols[mode][text].replace;
+  }
+
+  var fontName = buildCommon.fontMap[font].fontName;
+
+  if (getCharacterMetrics(text, fontName, mode)) {
+    return buildCommon.fontMap[font].variant;
+  }
+
+  return null;
+};
+/**
+ * Takes a list of nodes, builds them, and returns a list of the generated
+ * MathML nodes.  Also combine consecutive <mtext> outputs into a single
+ * <mtext> tag.
+ */
+
+var buildMathML_buildExpression = function buildExpression(expression, options, isOrdgroup) {
+  if (expression.length === 1) {
+    var group = buildMathML_buildGroup(expression[0], options);
+
+    if (isOrdgroup && group instanceof mathMLTree_MathNode && group.type === "mo") {
+      // When TeX writers want to suppress spacing on an operator,
+      // they often put the operator by itself inside braces.
+      group.setAttribute("lspace", "0em");
+      group.setAttribute("rspace", "0em");
+    }
+
+    return [group];
+  }
+
+  var groups = [];
+  var lastGroup;
+
+  for (var i = 0; i < expression.length; i++) {
+    var _group = buildMathML_buildGroup(expression[i], options);
+
+    if (_group instanceof mathMLTree_MathNode && lastGroup instanceof mathMLTree_MathNode) {
+      // Concatenate adjacent <mtext>s
+      if (_group.type === 'mtext' && lastGroup.type === 'mtext' && _group.getAttribute('mathvariant') === lastGroup.getAttribute('mathvariant')) {
+        var _lastGroup$children;
+
+        (_lastGroup$children = lastGroup.children).push.apply(_lastGroup$children, _group.children);
+
+        continue; // Concatenate adjacent <mn>s
+      } else if (_group.type === 'mn' && lastGroup.type === 'mn') {
+        var _lastGroup$children2;
+
+        (_lastGroup$children2 = lastGroup.children).push.apply(_lastGroup$children2, _group.children);
+
+        continue; // Concatenate <mn>...</mn> followed by <mi>.</mi>
+      } else if (_group.type === 'mi' && _group.children.length === 1 && lastGroup.type === 'mn') {
+        var child = _group.children[0];
+
+        if (child instanceof mathMLTree_TextNode && child.text === '.') {
+          var _lastGroup$children3;
+
+          (_lastGroup$children3 = lastGroup.children).push.apply(_lastGroup$children3, _group.children);
+
+          continue;
+        }
+      } else if (lastGroup.type === 'mi' && lastGroup.children.length === 1) {
+        var lastChild = lastGroup.children[0];
+
+        if (lastChild instanceof mathMLTree_TextNode && lastChild.text === "\u0338" && (_group.type === 'mo' || _group.type === 'mi' || _group.type === 'mn')) {
+          var _child = _group.children[0];
+
+          if (_child instanceof mathMLTree_TextNode && _child.text.length > 0) {
+            // Overlay with combining character long solidus
+            _child.text = _child.text.slice(0, 1) + "\u0338" + _child.text.slice(1);
+            groups.pop();
+          }
+        }
+      }
+    }
+
+    groups.push(_group);
+    lastGroup = _group;
+  }
+
+  return groups;
+};
+/**
+ * Equivalent to buildExpression, but wraps the elements in an <mrow>
+ * if there's more than one.  Returns a single node instead of an array.
+ */
+
+var buildExpressionRow = function buildExpressionRow(expression, options, isOrdgroup) {
+  return buildMathML_makeRow(buildMathML_buildExpression(expression, options, isOrdgroup));
+};
+/**
+ * Takes a group from the parser and calls the appropriate groupBuilders function
+ * on it to produce a MathML node.
+ */
+
+var buildMathML_buildGroup = function buildGroup(group, options) {
+  if (!group) {
+    return new mathMLTree.MathNode("mrow");
+  }
+
+  if (_mathmlGroupBuilders[group.type]) {
+    // Call the groupBuilders function
+    var result = _mathmlGroupBuilders[group.type](group, options);
+    return result;
+  } else {
+    throw new src_ParseError("Got group of unknown type: '" + group.type + "'");
+  }
+};
+/**
+ * Takes a full parse tree and settings and builds a MathML representation of
+ * it. In particular, we put the elements from building the parse tree into a
+ * <semantics> tag so we can also include that TeX source as an annotation.
+ *
+ * Note that we actually return a domTree element with a `<math>` inside it so
+ * we can do appropriate styling.
+ */
+
+function buildMathML(tree, texExpression, options, isDisplayMode, forMathmlOnly) {
+  var expression = buildMathML_buildExpression(tree, options); // Wrap up the expression in an mrow so it is presented in the semantics
+  // tag correctly, unless it's a single <mrow> or <mtable>.
+
+  var wrapper;
+
+  if (expression.length === 1 && expression[0] instanceof mathMLTree_MathNode && utils.contains(["mrow", "mtable"], expression[0].type)) {
+    wrapper = expression[0];
+  } else {
+    wrapper = new mathMLTree.MathNode("mrow", expression);
+  } // Build a TeX annotation of the source
+
+
+  var annotation = new mathMLTree.MathNode("annotation", [new mathMLTree.TextNode(texExpression)]);
+  annotation.setAttribute("encoding", "application/x-tex");
+  var semantics = new mathMLTree.MathNode("semantics", [wrapper, annotation]);
+  var math = new mathMLTree.MathNode("math", [semantics]);
+  math.setAttribute("xmlns", "http://www.w3.org/1998/Math/MathML");
+
+  if (isDisplayMode) {
+    math.setAttribute("display", "block");
+  } // You can't style <math> nodes, so we wrap the node in a span.
+  // NOTE: The span class is not typed to have <math> nodes as children, and
+  // we don't want to make the children type more generic since the children
+  // of span are expected to have more fields in `buildHtml` contexts.
+
+
+  var wrapperClass = forMathmlOnly ? "katex" : "katex-mathml"; // $FlowFixMe
+
+  return buildCommon.makeSpan([wrapperClass], [math]);
+}
+// CONCATENATED MODULE: ./src/buildTree.js
+
+
+
+
+
+
+
+var buildTree_optionsFromSettings = function optionsFromSettings(settings) {
+  return new src_Options({
+    style: settings.displayMode ? src_Style.DISPLAY : src_Style.TEXT,
+    maxSize: settings.maxSize,
+    minRuleThickness: settings.minRuleThickness
+  });
+};
+
+var buildTree_displayWrap = function displayWrap(node, settings) {
+  if (settings.displayMode) {
+    var classes = ["katex-display"];
+
+    if (settings.leqno) {
+      classes.push("leqno");
+    }
+
+    if (settings.fleqn) {
+      classes.push("fleqn");
+    }
+
+    node = buildCommon.makeSpan(classes, [node]);
+  }
+
+  return node;
+};
+
+var buildTree_buildTree = function buildTree(tree, expression, settings) {
+  var options = buildTree_optionsFromSettings(settings);
+  var katexNode;
+
+  if (settings.output === "mathml") {
+    return buildMathML(tree, expression, options, settings.displayMode, true);
+  } else if (settings.output === "html") {
+    var htmlNode = buildHTML(tree, options);
+    katexNode = buildCommon.makeSpan(["katex"], [htmlNode]);
+  } else {
+    var mathMLNode = buildMathML(tree, expression, options, settings.displayMode, false);
+
+    var _htmlNode = buildHTML(tree, options);
+
+    katexNode = buildCommon.makeSpan(["katex"], [mathMLNode, _htmlNode]);
+  }
+
+  return buildTree_displayWrap(katexNode, settings);
+};
+var buildTree_buildHTMLTree = function buildHTMLTree(tree, expression, settings) {
+  var options = buildTree_optionsFromSettings(settings);
+  var htmlNode = buildHTML(tree, options);
+  var katexNode = buildCommon.makeSpan(["katex"], [htmlNode]);
+  return buildTree_displayWrap(katexNode, settings);
+};
+/* harmony default export */ var src_buildTree = (buildTree_buildTree);
+// CONCATENATED MODULE: ./src/stretchy.js
+/**
+ * This file provides support to buildMathML.js and buildHTML.js
+ * for stretchy wide elements rendered from SVG files
+ * and other CSS trickery.
+ */
+
+
+
+
+var stretchyCodePoint = {
+  widehat: "^",
+  widecheck: "ˇ",
+  widetilde: "~",
+  utilde: "~",
+  overleftarrow: "\u2190",
+  underleftarrow: "\u2190",
+  xleftarrow: "\u2190",
+  overrightarrow: "\u2192",
+  underrightarrow: "\u2192",
+  xrightarrow: "\u2192",
+  underbrace: "\u23DF",
+  overbrace: "\u23DE",
+  overgroup: "\u23E0",
+  undergroup: "\u23E1",
+  overleftrightarrow: "\u2194",
+  underleftrightarrow: "\u2194",
+  xleftrightarrow: "\u2194",
+  Overrightarrow: "\u21D2",
+  xRightarrow: "\u21D2",
+  overleftharpoon: "\u21BC",
+  xleftharpoonup: "\u21BC",
+  overrightharpoon: "\u21C0",
+  xrightharpoonup: "\u21C0",
+  xLeftarrow: "\u21D0",
+  xLeftrightarrow: "\u21D4",
+  xhookleftarrow: "\u21A9",
+  xhookrightarrow: "\u21AA",
+  xmapsto: "\u21A6",
+  xrightharpoondown: "\u21C1",
+  xleftharpoondown: "\u21BD",
+  xrightleftharpoons: "\u21CC",
+  xleftrightharpoons: "\u21CB",
+  xtwoheadleftarrow: "\u219E",
+  xtwoheadrightarrow: "\u21A0",
+  xlongequal: "=",
+  xtofrom: "\u21C4",
+  xrightleftarrows: "\u21C4",
+  xrightequilibrium: "\u21CC",
+  // Not a perfect match.
+  xleftequilibrium: "\u21CB" // None better available.
+
+};
+
+var stretchy_mathMLnode = function mathMLnode(label) {
+  var node = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(stretchyCodePoint[label.substr(1)])]);
+  node.setAttribute("stretchy", "true");
+  return node;
+}; // Many of the KaTeX SVG images have been adapted from glyphs in KaTeX fonts.
+// Copyright (c) 2009-2010, Design Science, Inc. (<www.mathjax.org>)
+// Copyright (c) 2014-2017 Khan Academy (<www.khanacademy.org>)
+// Licensed under the SIL Open Font License, Version 1.1.
+// See \nhttp://scripts.sil.org/OFL
+// Very Long SVGs
+//    Many of the KaTeX stretchy wide elements use a long SVG image and an
+//    overflow: hidden tactic to achieve a stretchy image while avoiding
+//    distortion of arrowheads or brace corners.
+//    The SVG typically contains a very long (400 em) arrow.
+//    The SVG is in a container span that has overflow: hidden, so the span
+//    acts like a window that exposes only part of the  SVG.
+//    The SVG always has a longer, thinner aspect ratio than the container span.
+//    After the SVG fills 100% of the height of the container span,
+//    there is a long arrow shaft left over. That left-over shaft is not shown.
+//    Instead, it is sliced off because the span's CSS has overflow: hidden.
+//    Thus, the reader sees an arrow that matches the subject matter width
+//    without distortion.
+//    Some functions, such as \cancel, need to vary their aspect ratio. These
+//    functions do not get the overflow SVG treatment.
+// Second Brush Stroke
+//    Low resolution monitors struggle to display images in fine detail.
+//    So browsers apply anti-aliasing. A long straight arrow shaft therefore
+//    will sometimes appear as if it has a blurred edge.
+//    To mitigate this, these SVG files contain a second "brush-stroke" on the
+//    arrow shafts. That is, a second long thin rectangular SVG path has been
+//    written directly on top of each arrow shaft. This reinforcement causes
+//    some of the screen pixels to display as black instead of the anti-aliased
+//    gray pixel that a  single path would generate. So we get arrow shafts
+//    whose edges appear to be sharper.
+// In the katexImagesData object just below, the dimensions all
+// correspond to path geometry inside the relevant SVG.
+// For example, \overrightarrow uses the same arrowhead as glyph U+2192
+// from the KaTeX Main font. The scaling factor is 1000.
+// That is, inside the font, that arrowhead is 522 units tall, which
+// corresponds to 0.522 em inside the document.
+
+
+var katexImagesData = {
+  //   path(s), minWidth, height, align
+  overrightarrow: [["rightarrow"], 0.888, 522, "xMaxYMin"],
+  overleftarrow: [["leftarrow"], 0.888, 522, "xMinYMin"],
+  underrightarrow: [["rightarrow"], 0.888, 522, "xMaxYMin"],
+  underleftarrow: [["leftarrow"], 0.888, 522, "xMinYMin"],
+  xrightarrow: [["rightarrow"], 1.469, 522, "xMaxYMin"],
+  xleftarrow: [["leftarrow"], 1.469, 522, "xMinYMin"],
+  Overrightarrow: [["doublerightarrow"], 0.888, 560, "xMaxYMin"],
+  xRightarrow: [["doublerightarrow"], 1.526, 560, "xMaxYMin"],
+  xLeftarrow: [["doubleleftarrow"], 1.526, 560, "xMinYMin"],
+  overleftharpoon: [["leftharpoon"], 0.888, 522, "xMinYMin"],
+  xleftharpoonup: [["leftharpoon"], 0.888, 522, "xMinYMin"],
+  xleftharpoondown: [["leftharpoondown"], 0.888, 522, "xMinYMin"],
+  overrightharpoon: [["rightharpoon"], 0.888, 522, "xMaxYMin"],
+  xrightharpoonup: [["rightharpoon"], 0.888, 522, "xMaxYMin"],
+  xrightharpoondown: [["rightharpoondown"], 0.888, 522, "xMaxYMin"],
+  xlongequal: [["longequal"], 0.888, 334, "xMinYMin"],
+  xtwoheadleftarrow: [["twoheadleftarrow"], 0.888, 334, "xMinYMin"],
+  xtwoheadrightarrow: [["twoheadrightarrow"], 0.888, 334, "xMaxYMin"],
+  overleftrightarrow: [["leftarrow", "rightarrow"], 0.888, 522],
+  overbrace: [["leftbrace", "midbrace", "rightbrace"], 1.6, 548],
+  underbrace: [["leftbraceunder", "midbraceunder", "rightbraceunder"], 1.6, 548],
+  underleftrightarrow: [["leftarrow", "rightarrow"], 0.888, 522],
+  xleftrightarrow: [["leftarrow", "rightarrow"], 1.75, 522],
+  xLeftrightarrow: [["doubleleftarrow", "doublerightarrow"], 1.75, 560],
+  xrightleftharpoons: [["leftharpoondownplus", "rightharpoonplus"], 1.75, 716],
+  xleftrightharpoons: [["leftharpoonplus", "rightharpoondownplus"], 1.75, 716],
+  xhookleftarrow: [["leftarrow", "righthook"], 1.08, 522],
+  xhookrightarrow: [["lefthook", "rightarrow"], 1.08, 522],
+  overlinesegment: [["leftlinesegment", "rightlinesegment"], 0.888, 522],
+  underlinesegment: [["leftlinesegment", "rightlinesegment"], 0.888, 522],
+  overgroup: [["leftgroup", "rightgroup"], 0.888, 342],
+  undergroup: [["leftgroupunder", "rightgroupunder"], 0.888, 342],
+  xmapsto: [["leftmapsto", "rightarrow"], 1.5, 522],
+  xtofrom: [["leftToFrom", "rightToFrom"], 1.75, 528],
+  // The next three arrows are from the mhchem package.
+  // In mhchem.sty, min-length is 2.0em. But these arrows might appear in the
+  // document as \xrightarrow or \xrightleftharpoons. Those have
+  // min-length = 1.75em, so we set min-length on these next three to match.
+  xrightleftarrows: [["baraboveleftarrow", "rightarrowabovebar"], 1.75, 901],
+  xrightequilibrium: [["baraboveshortleftharpoon", "rightharpoonaboveshortbar"], 1.75, 716],
+  xleftequilibrium: [["shortbaraboveleftharpoon", "shortrightharpoonabovebar"], 1.75, 716]
+};
+
+var groupLength = function groupLength(arg) {
+  if (arg.type === "ordgroup") {
+    return arg.body.length;
+  } else {
+    return 1;
+  }
+};
+
+var stretchy_svgSpan = function svgSpan(group, options) {
+  // Create a span with inline SVG for the element.
+  function buildSvgSpan_() {
+    var viewBoxWidth = 400000; // default
+
+    var label = group.label.substr(1);
+
+    if (utils.contains(["widehat", "widecheck", "widetilde", "utilde"], label)) {
+      // Each type in the `if` statement corresponds to one of the ParseNode
+      // types below. This narrowing is required to access `grp.base`.
+      var grp = group; // There are four SVG images available for each function.
+      // Choose a taller image when there are more characters.
+
+      var numChars = groupLength(grp.base);
+      var viewBoxHeight;
+      var pathName;
+
+      var _height;
+
+      if (numChars > 5) {
+        if (label === "widehat" || label === "widecheck") {
+          viewBoxHeight = 420;
+          viewBoxWidth = 2364;
+          _height = 0.42;
+          pathName = label + "4";
+        } else {
+          viewBoxHeight = 312;
+          viewBoxWidth = 2340;
+          _height = 0.34;
+          pathName = "tilde4";
+        }
+      } else {
+        var imgIndex = [1, 1, 2, 2, 3, 3][numChars];
+
+        if (label === "widehat" || label === "widecheck") {
+          viewBoxWidth = [0, 1062, 2364, 2364, 2364][imgIndex];
+          viewBoxHeight = [0, 239, 300, 360, 420][imgIndex];
+          _height = [0, 0.24, 0.3, 0.3, 0.36, 0.42][imgIndex];
+          pathName = label + imgIndex;
+        } else {
+          viewBoxWidth = [0, 600, 1033, 2339, 2340][imgIndex];
+          viewBoxHeight = [0, 260, 286, 306, 312][imgIndex];
+          _height = [0, 0.26, 0.286, 0.3, 0.306, 0.34][imgIndex];
+          pathName = "tilde" + imgIndex;
+        }
+      }
+
+      var path = new domTree_PathNode(pathName);
+      var svgNode = new SvgNode([path], {
+        "width": "100%",
+        "height": _height + "em",
+        "viewBox": "0 0 " + viewBoxWidth + " " + viewBoxHeight,
+        "preserveAspectRatio": "none"
+      });
+      return {
+        span: buildCommon.makeSvgSpan([], [svgNode], options),
+        minWidth: 0,
+        height: _height
+      };
+    } else {
+      var spans = [];
+      var data = katexImagesData[label];
+      var paths = data[0],
+          _minWidth = data[1],
+          _viewBoxHeight = data[2];
+
+      var _height2 = _viewBoxHeight / 1000;
+
+      var numSvgChildren = paths.length;
+      var widthClasses;
+      var aligns;
+
+      if (numSvgChildren === 1) {
+        // $FlowFixMe: All these cases must be of the 4-tuple type.
+        var align1 = data[3];
+        widthClasses = ["hide-tail"];
+        aligns = [align1];
+      } else if (numSvgChildren === 2) {
+        widthClasses = ["halfarrow-left", "halfarrow-right"];
+        aligns = ["xMinYMin", "xMaxYMin"];
+      } else if (numSvgChildren === 3) {
+        widthClasses = ["brace-left", "brace-center", "brace-right"];
+        aligns = ["xMinYMin", "xMidYMin", "xMaxYMin"];
+      } else {
+        throw new Error("Correct katexImagesData or update code here to support\n                    " + numSvgChildren + " children.");
+      }
+
+      for (var i = 0; i < numSvgChildren; i++) {
+        var _path = new domTree_PathNode(paths[i]);
+
+        var _svgNode = new SvgNode([_path], {
+          "width": "400em",
+          "height": _height2 + "em",
+          "viewBox": "0 0 " + viewBoxWidth + " " + _viewBoxHeight,
+          "preserveAspectRatio": aligns[i] + " slice"
+        });
+
+        var _span = buildCommon.makeSvgSpan([widthClasses[i]], [_svgNode], options);
+
+        if (numSvgChildren === 1) {
+          return {
+            span: _span,
+            minWidth: _minWidth,
+            height: _height2
+          };
+        } else {
+          _span.style.height = _height2 + "em";
+          spans.push(_span);
+        }
+      }
+
+      return {
+        span: buildCommon.makeSpan(["stretchy"], spans, options),
+        minWidth: _minWidth,
+        height: _height2
+      };
+    }
+  } // buildSvgSpan_()
+
+
+  var _buildSvgSpan_ = buildSvgSpan_(),
+      span = _buildSvgSpan_.span,
+      minWidth = _buildSvgSpan_.minWidth,
+      height = _buildSvgSpan_.height; // Note that we are returning span.depth = 0.
+  // Any adjustments relative to the baseline must be done in buildHTML.
+
+
+  span.height = height;
+  span.style.height = height + "em";
+
+  if (minWidth > 0) {
+    span.style.minWidth = minWidth + "em";
+  }
+
+  return span;
+};
+
+var stretchy_encloseSpan = function encloseSpan(inner, label, pad, options) {
+  // Return an image span for \cancel, \bcancel, \xcancel, or \fbox
+  var img;
+  var totalHeight = inner.height + inner.depth + 2 * pad;
+
+  if (/fbox|color/.test(label)) {
+    img = buildCommon.makeSpan(["stretchy", label], [], options);
+
+    if (label === "fbox") {
+      var color = options.color && options.getColor();
+
+      if (color) {
+        img.style.borderColor = color;
+      }
+    }
+  } else {
+    // \cancel, \bcancel, or \xcancel
+    // Since \cancel's SVG is inline and it omits the viewBox attribute,
+    // its stroke-width will not vary with span area.
+    var lines = [];
+
+    if (/^[bx]cancel$/.test(label)) {
+      lines.push(new LineNode({
+        "x1": "0",
+        "y1": "0",
+        "x2": "100%",
+        "y2": "100%",
+        "stroke-width": "0.046em"
+      }));
+    }
+
+    if (/^x?cancel$/.test(label)) {
+      lines.push(new LineNode({
+        "x1": "0",
+        "y1": "100%",
+        "x2": "100%",
+        "y2": "0",
+        "stroke-width": "0.046em"
+      }));
+    }
+
+    var svgNode = new SvgNode(lines, {
+      "width": "100%",
+      "height": totalHeight + "em"
+    });
+    img = buildCommon.makeSvgSpan([], [svgNode], options);
+  }
+
+  img.height = totalHeight;
+  img.style.height = totalHeight + "em";
+  return img;
+};
+
+/* harmony default export */ var stretchy = ({
+  encloseSpan: stretchy_encloseSpan,
+  mathMLnode: stretchy_mathMLnode,
+  svgSpan: stretchy_svgSpan
+});
+// CONCATENATED MODULE: ./src/parseNode.js
+
+
+/**
+ * Asserts that the node is of the given type and returns it with stricter
+ * typing. Throws if the node's type does not match.
+ */
+function assertNodeType(node, type) {
+  if (!node || node.type !== type) {
+    throw new Error("Expected node of type " + type + ", but got " + (node ? "node of type " + node.type : String(node)));
+  }
+
+  return node;
+}
+/**
+ * Returns the node more strictly typed iff it is of the given type. Otherwise,
+ * returns null.
+ */
+
+function assertSymbolNodeType(node) {
+  var typedNode = checkSymbolNodeType(node);
+
+  if (!typedNode) {
+    throw new Error("Expected node of symbol group type, but got " + (node ? "node of type " + node.type : String(node)));
+  }
+
+  return typedNode;
+}
+/**
+ * Returns the node more strictly typed iff it is of the given type. Otherwise,
+ * returns null.
+ */
+
+function checkSymbolNodeType(node) {
+  if (node && (node.type === "atom" || NON_ATOMS.hasOwnProperty(node.type))) {
+    // $FlowFixMe
+    return node;
+  }
+
+  return null;
+}
+// CONCATENATED MODULE: ./src/functions/accent.js
+
+
+
+
+
+
+
+
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only "accent", but
+var accent_htmlBuilder = function htmlBuilder(grp, options) {
+  // Accents are handled in the TeXbook pg. 443, rule 12.
+  var base;
+  var group;
+  var supSubGroup;
+
+  if (grp && grp.type === "supsub") {
+    // If our base is a character box, and we have superscripts and
+    // subscripts, the supsub will defer to us. In particular, we want
+    // to attach the superscripts and subscripts to the inner body (so
+    // that the position of the superscripts and subscripts won't be
+    // affected by the height of the accent). We accomplish this by
+    // sticking the base of the accent into the base of the supsub, and
+    // rendering that, while keeping track of where the accent is.
+    // The real accent group is the base of the supsub group
+    group = assertNodeType(grp.base, "accent"); // The character box is the base of the accent group
+
+    base = group.base; // Stick the character box into the base of the supsub group
+
+    grp.base = base; // Rerender the supsub group with its new base, and store that
+    // result.
+
+    supSubGroup = assertSpan(buildHTML_buildGroup(grp, options)); // reset original base
+
+    grp.base = group;
+  } else {
+    group = assertNodeType(grp, "accent");
+    base = group.base;
+  } // Build the base group
+
+
+  var body = buildHTML_buildGroup(base, options.havingCrampedStyle()); // Does the accent need to shift for the skew of a character?
+
+  var mustShift = group.isShifty && utils.isCharacterBox(base); // Calculate the skew of the accent. This is based on the line "If the
+  // nucleus is not a single character, let s = 0; otherwise set s to the
+  // kern amount for the nucleus followed by the \skewchar of its font."
+  // Note that our skew metrics are just the kern between each character
+  // and the skewchar.
+
+  var skew = 0;
+
+  if (mustShift) {
+    // If the base is a character box, then we want the skew of the
+    // innermost character. To do that, we find the innermost character:
+    var baseChar = utils.getBaseElem(base); // Then, we render its group to get the symbol inside it
+
+    var baseGroup = buildHTML_buildGroup(baseChar, options.havingCrampedStyle()); // Finally, we pull the skew off of the symbol.
+
+    skew = assertSymbolDomNode(baseGroup).skew; // Note that we now throw away baseGroup, because the layers we
+    // removed with getBaseElem might contain things like \color which
+    // we can't get rid of.
+    // TODO(emily): Find a better way to get the skew
+  } // calculate the amount of space between the body and the accent
+
+
+  var clearance = Math.min(body.height, options.fontMetrics().xHeight); // Build the accent
+
+  var accentBody;
+
+  if (!group.isStretchy) {
+    var accent;
+    var width;
+
+    if (group.label === "\\vec") {
+      // Before version 0.9, \vec used the combining font glyph U+20D7.
+      // But browsers, especially Safari, are not consistent in how they
+      // render combining characters when not preceded by a character.
+      // So now we use an SVG.
+      // If Safari reforms, we should consider reverting to the glyph.
+      accent = buildCommon.staticSvg("vec", options);
+      width = buildCommon.svgData.vec[1];
+    } else {
+      accent = buildCommon.makeOrd({
+        mode: group.mode,
+        text: group.label
+      }, options, "textord");
+      accent = assertSymbolDomNode(accent); // Remove the italic correction of the accent, because it only serves to
+      // shift the accent over to a place we don't want.
+
+      accent.italic = 0;
+      width = accent.width;
+    }
+
+    accentBody = buildCommon.makeSpan(["accent-body"], [accent]); // "Full" accents expand the width of the resulting symbol to be
+    // at least the width of the accent, and overlap directly onto the
+    // character without any vertical offset.
+
+    var accentFull = group.label === "\\textcircled";
+
+    if (accentFull) {
+      accentBody.classes.push('accent-full');
+      clearance = body.height;
+    } // Shift the accent over by the skew.
+
+
+    var left = skew; // CSS defines `.katex .accent .accent-body:not(.accent-full) { width: 0 }`
+    // so that the accent doesn't contribute to the bounding box.
+    // We need to shift the character by its width (effectively half
+    // its width) to compensate.
+
+    if (!accentFull) {
+      left -= width / 2;
+    }
+
+    accentBody.style.left = left + "em"; // \textcircled uses the \bigcirc glyph, so it needs some
+    // vertical adjustment to match LaTeX.
+
+    if (group.label === "\\textcircled") {
+      accentBody.style.top = ".2em";
+    }
+
+    accentBody = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "kern",
+        size: -clearance
+      }, {
+        type: "elem",
+        elem: accentBody
+      }]
+    }, options);
+  } else {
+    accentBody = stretchy.svgSpan(group, options);
+    accentBody = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "elem",
+        elem: accentBody,
+        wrapperClasses: ["svg-align"],
+        wrapperStyle: skew > 0 ? {
+          width: "calc(100% - " + 2 * skew + "em)",
+          marginLeft: 2 * skew + "em"
+        } : undefined
+      }]
+    }, options);
+  }
+
+  var accentWrap = buildCommon.makeSpan(["mord", "accent"], [accentBody], options);
+
+  if (supSubGroup) {
+    // Here, we replace the "base" child of the supsub with our newly
+    // generated accent.
+    supSubGroup.children[0] = accentWrap; // Since we don't rerun the height calculation after replacing the
+    // accent, we manually recalculate height.
+
+    supSubGroup.height = Math.max(accentWrap.height, supSubGroup.height); // Accents should always be ords, even when their innards are not.
+
+    supSubGroup.classes[0] = "mord";
+    return supSubGroup;
+  } else {
+    return accentWrap;
+  }
+};
+
+var accent_mathmlBuilder = function mathmlBuilder(group, options) {
+  var accentNode = group.isStretchy ? stretchy.mathMLnode(group.label) : new mathMLTree.MathNode("mo", [buildMathML_makeText(group.label, group.mode)]);
+  var node = new mathMLTree.MathNode("mover", [buildMathML_buildGroup(group.base, options), accentNode]);
+  node.setAttribute("accent", "true");
+  return node;
+};
+
+var NON_STRETCHY_ACCENT_REGEX = new RegExp(["\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve", "\\check", "\\hat", "\\vec", "\\dot", "\\mathring"].map(function (accent) {
+  return "\\" + accent;
+}).join("|")); // Accents
+
+defineFunction({
+  type: "accent",
+  names: ["\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve", "\\check", "\\hat", "\\vec", "\\dot", "\\mathring", "\\widecheck", "\\widehat", "\\widetilde", "\\overrightarrow", "\\overleftarrow", "\\Overrightarrow", "\\overleftrightarrow", "\\overgroup", "\\overlinesegment", "\\overleftharpoon", "\\overrightharpoon"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    var base = args[0];
+    var isStretchy = !NON_STRETCHY_ACCENT_REGEX.test(context.funcName);
+    var isShifty = !isStretchy || context.funcName === "\\widehat" || context.funcName === "\\widetilde" || context.funcName === "\\widecheck";
+    return {
+      type: "accent",
+      mode: context.parser.mode,
+      label: context.funcName,
+      isStretchy: isStretchy,
+      isShifty: isShifty,
+      base: base
+    };
+  },
+  htmlBuilder: accent_htmlBuilder,
+  mathmlBuilder: accent_mathmlBuilder
+}); // Text-mode accents
+
+defineFunction({
+  type: "accent",
+  names: ["\\'", "\\`", "\\^", "\\~", "\\=", "\\u", "\\.", '\\"', "\\r", "\\H", "\\v", "\\textcircled"],
+  props: {
+    numArgs: 1,
+    allowedInText: true,
+    allowedInMath: false
+  },
+  handler: function handler(context, args) {
+    var base = args[0];
+    return {
+      type: "accent",
+      mode: context.parser.mode,
+      label: context.funcName,
+      isStretchy: false,
+      isShifty: true,
+      base: base
+    };
+  },
+  htmlBuilder: accent_htmlBuilder,
+  mathmlBuilder: accent_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/accentunder.js
+// Horizontal overlap functions
+
+
+
+
+
+
+defineFunction({
+  type: "accentUnder",
+  names: ["\\underleftarrow", "\\underrightarrow", "\\underleftrightarrow", "\\undergroup", "\\underlinesegment", "\\utilde"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var base = args[0];
+    return {
+      type: "accentUnder",
+      mode: parser.mode,
+      label: funcName,
+      base: base
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Treat under accents much like underlines.
+    var innerGroup = buildHTML_buildGroup(group.base, options);
+    var accentBody = stretchy.svgSpan(group, options);
+    var kern = group.label === "\\utilde" ? 0.12 : 0; // Generate the vlist, with the appropriate kerns
+
+    var vlist = buildCommon.makeVList({
+      positionType: "top",
+      positionData: innerGroup.height,
+      children: [{
+        type: "elem",
+        elem: accentBody,
+        wrapperClasses: ["svg-align"]
+      }, {
+        type: "kern",
+        size: kern
+      }, {
+        type: "elem",
+        elem: innerGroup
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "accentunder"], [vlist], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var accentNode = stretchy.mathMLnode(group.label);
+    var node = new mathMLTree.MathNode("munder", [buildMathML_buildGroup(group.base, options), accentNode]);
+    node.setAttribute("accentunder", "true");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/arrow.js
+
+
+
+
+
+
+
+// Helper function
+var arrow_paddedNode = function paddedNode(group) {
+  var node = new mathMLTree.MathNode("mpadded", group ? [group] : []);
+  node.setAttribute("width", "+0.6em");
+  node.setAttribute("lspace", "0.3em");
+  return node;
+}; // Stretchy arrows with an optional argument
+
+
+defineFunction({
+  type: "xArrow",
+  names: ["\\xleftarrow", "\\xrightarrow", "\\xLeftarrow", "\\xRightarrow", "\\xleftrightarrow", "\\xLeftrightarrow", "\\xhookleftarrow", "\\xhookrightarrow", "\\xmapsto", "\\xrightharpoondown", "\\xrightharpoonup", "\\xleftharpoondown", "\\xleftharpoonup", "\\xrightleftharpoons", "\\xleftrightharpoons", "\\xlongequal", "\\xtwoheadrightarrow", "\\xtwoheadleftarrow", "\\xtofrom", // The next 3 functions are here to support the mhchem extension.
+  // Direct use of these functions is discouraged and may break someday.
+  "\\xrightleftarrows", "\\xrightequilibrium", "\\xleftequilibrium"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    return {
+      type: "xArrow",
+      mode: parser.mode,
+      label: funcName,
+      body: args[0],
+      below: optArgs[0]
+    };
+  },
+  // Flow is unable to correctly infer the type of `group`, even though it's
+  // unamibiguously determined from the passed-in `type` above.
+  htmlBuilder: function htmlBuilder(group, options) {
+    var style = options.style; // Build the argument groups in the appropriate style.
+    // Ref: amsmath.dtx:   \hbox{$\scriptstyle\mkern#3mu{#6}\mkern#4mu$}%
+    // Some groups can return document fragments.  Handle those by wrapping
+    // them in a span.
+
+    var newOptions = options.havingStyle(style.sup());
+    var upperGroup = buildCommon.wrapFragment(buildHTML_buildGroup(group.body, newOptions, options), options);
+    upperGroup.classes.push("x-arrow-pad");
+    var lowerGroup;
+
+    if (group.below) {
+      // Build the lower group
+      newOptions = options.havingStyle(style.sub());
+      lowerGroup = buildCommon.wrapFragment(buildHTML_buildGroup(group.below, newOptions, options), options);
+      lowerGroup.classes.push("x-arrow-pad");
+    }
+
+    var arrowBody = stretchy.svgSpan(group, options); // Re shift: Note that stretchy.svgSpan returned arrowBody.depth = 0.
+    // The point we want on the math axis is at 0.5 * arrowBody.height.
+
+    var arrowShift = -options.fontMetrics().axisHeight + 0.5 * arrowBody.height; // 2 mu kern. Ref: amsmath.dtx: #7\if0#2\else\mkern#2mu\fi
+
+    var upperShift = -options.fontMetrics().axisHeight - 0.5 * arrowBody.height - 0.111; // 0.111 em = 2 mu
+
+    if (upperGroup.depth > 0.25 || group.label === "\\xleftequilibrium") {
+      upperShift -= upperGroup.depth; // shift up if depth encroaches
+    } // Generate the vlist
+
+
+    var vlist;
+
+    if (lowerGroup) {
+      var lowerShift = -options.fontMetrics().axisHeight + lowerGroup.height + 0.5 * arrowBody.height + 0.111;
+      vlist = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: upperGroup,
+          shift: upperShift
+        }, {
+          type: "elem",
+          elem: arrowBody,
+          shift: arrowShift
+        }, {
+          type: "elem",
+          elem: lowerGroup,
+          shift: lowerShift
+        }]
+      }, options);
+    } else {
+      vlist = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: upperGroup,
+          shift: upperShift
+        }, {
+          type: "elem",
+          elem: arrowBody,
+          shift: arrowShift
+        }]
+      }, options);
+    } // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+
+    vlist.children[0].children[0].children[1].classes.push("svg-align");
+    return buildCommon.makeSpan(["mrel", "x-arrow"], [vlist], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var arrowNode = stretchy.mathMLnode(group.label);
+    var node;
+
+    if (group.body) {
+      var upperNode = arrow_paddedNode(buildMathML_buildGroup(group.body, options));
+
+      if (group.below) {
+        var lowerNode = arrow_paddedNode(buildMathML_buildGroup(group.below, options));
+        node = new mathMLTree.MathNode("munderover", [arrowNode, lowerNode, upperNode]);
+      } else {
+        node = new mathMLTree.MathNode("mover", [arrowNode, upperNode]);
+      }
+    } else if (group.below) {
+      var _lowerNode = arrow_paddedNode(buildMathML_buildGroup(group.below, options));
+
+      node = new mathMLTree.MathNode("munder", [arrowNode, _lowerNode]);
+    } else {
+      // This should never happen.
+      // Parser.js throws an error if there is no argument.
+      node = arrow_paddedNode();
+      node = new mathMLTree.MathNode("mover", [arrowNode, node]);
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/char.js
+
+
+ // \@char is an internal function that takes a grouped decimal argument like
+// {123} and converts into symbol with code 123.  It is used by the *macro*
+// \char defined in macros.js.
+
+defineFunction({
+  type: "textord",
+  names: ["\\@char"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var arg = assertNodeType(args[0], "ordgroup");
+    var group = arg.body;
+    var number = "";
+
+    for (var i = 0; i < group.length; i++) {
+      var node = assertNodeType(group[i], "textord");
+      number += node.text;
+    }
+
+    var code = parseInt(number);
+
+    if (isNaN(code)) {
+      throw new src_ParseError("\\@char has non-numeric argument " + number);
+    }
+
+    return {
+      type: "textord",
+      mode: parser.mode,
+      text: String.fromCharCode(code)
+    };
+  }
+});
+// CONCATENATED MODULE: ./src/functions/color.js
+
+
+
+
+
+
+
+var color_htmlBuilder = function htmlBuilder(group, options) {
+  var elements = buildHTML_buildExpression(group.body, options.withColor(group.color), false); // \color isn't supposed to affect the type of the elements it contains.
+  // To accomplish this, we wrap the results in a fragment, so the inner
+  // elements will be able to directly interact with their neighbors. For
+  // example, `\color{red}{2 +} 3` has the same spacing as `2 + 3`
+
+  return buildCommon.makeFragment(elements);
+};
+
+var color_mathmlBuilder = function mathmlBuilder(group, options) {
+  var inner = buildMathML_buildExpression(group.body, options.withColor(group.color));
+  var node = new mathMLTree.MathNode("mstyle", inner);
+  node.setAttribute("mathcolor", group.color);
+  return node;
+};
+
+defineFunction({
+  type: "color",
+  names: ["\\textcolor"],
+  props: {
+    numArgs: 2,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "original"]
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var color = assertNodeType(args[0], "color-token").color;
+    var body = args[1];
+    return {
+      type: "color",
+      mode: parser.mode,
+      color: color,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: color_htmlBuilder,
+  mathmlBuilder: color_mathmlBuilder
+});
+defineFunction({
+  type: "color",
+  names: ["\\color"],
+  props: {
+    numArgs: 1,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color"]
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser,
+        breakOnTokenText = _ref2.breakOnTokenText;
+    var color = assertNodeType(args[0], "color-token").color; // Set macro \current@color in current namespace to store the current
+    // color, mimicking the behavior of color.sty.
+    // This is currently used just to correctly color a \right
+    // that follows a \color command.
+
+    parser.gullet.macros.set("\\current@color", color); // Parse out the implicit body that should be colored.
+
+    var body = parser.parseExpression(true, breakOnTokenText);
+    return {
+      type: "color",
+      mode: parser.mode,
+      color: color,
+      body: body
+    };
+  },
+  htmlBuilder: color_htmlBuilder,
+  mathmlBuilder: color_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/cr.js
+// Row breaks within tabular environments, and line breaks at top level
+
+
+
+
+
+ // \\ is a macro mapping to either \cr or \newline.  Because they have the
+// same signature, we implement them as one megafunction, with newRow
+// indicating whether we're in the \cr case, and newLine indicating whether
+// to break the line in the \newline case.
+
+defineFunction({
+  type: "cr",
+  names: ["\\cr", "\\newline"],
+  props: {
+    numArgs: 0,
+    numOptionalArgs: 1,
+    argTypes: ["size"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var size = optArgs[0];
+    var newRow = funcName === "\\cr";
+    var newLine = false;
+
+    if (!newRow) {
+      if (parser.settings.displayMode && parser.settings.useStrictBehavior("newLineInDisplayMode", "In LaTeX, \\\\ or \\newline " + "does nothing in display mode")) {
+        newLine = false;
+      } else {
+        newLine = true;
+      }
+    }
+
+    return {
+      type: "cr",
+      mode: parser.mode,
+      newLine: newLine,
+      newRow: newRow,
+      size: size && assertNodeType(size, "size").value
+    };
+  },
+  // The following builders are called only at the top level,
+  // not within tabular/array environments.
+  htmlBuilder: function htmlBuilder(group, options) {
+    if (group.newRow) {
+      throw new src_ParseError("\\cr valid only within a tabular/array environment");
+    }
+
+    var span = buildCommon.makeSpan(["mspace"], [], options);
+
+    if (group.newLine) {
+      span.classes.push("newline");
+
+      if (group.size) {
+        span.style.marginTop = units_calculateSize(group.size, options) + "em";
+      }
+    }
+
+    return span;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mspace");
+
+    if (group.newLine) {
+      node.setAttribute("linebreak", "newline");
+
+      if (group.size) {
+        node.setAttribute("height", units_calculateSize(group.size, options) + "em");
+      }
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/def.js
+
+
+
+var globalMap = {
+  "\\global": "\\global",
+  "\\long": "\\\\globallong",
+  "\\\\globallong": "\\\\globallong",
+  "\\def": "\\gdef",
+  "\\gdef": "\\gdef",
+  "\\edef": "\\xdef",
+  "\\xdef": "\\xdef",
+  "\\let": "\\\\globallet",
+  "\\futurelet": "\\\\globalfuture"
+};
+
+var def_checkControlSequence = function checkControlSequence(tok) {
+  var name = tok.text;
+
+  if (/^(?:[\\{}$&#^_]|EOF)$/.test(name)) {
+    throw new src_ParseError("Expected a control sequence", tok);
+  }
+
+  return name;
+};
+
+var getRHS = function getRHS(parser) {
+  var tok = parser.gullet.popToken();
+
+  if (tok.text === "=") {
+    // consume optional equals
+    tok = parser.gullet.popToken();
+
+    if (tok.text === " ") {
+      // consume one optional space
+      tok = parser.gullet.popToken();
+    }
+  }
+
+  return tok;
+};
+
+var letCommand = function letCommand(parser, name, tok, global) {
+  var macro = parser.gullet.macros.get(tok.text);
+
+  if (macro == null) {
+    // don't expand it later even if a macro with the same name is defined
+    // e.g., \let\foo=\frac \def\frac{\relax} \frac12
+    tok.noexpand = true;
+    macro = {
+      tokens: [tok],
+      numArgs: 0,
+      // reproduce the same behavior in expansion
+      unexpandable: !parser.gullet.isExpandable(tok.text)
+    };
+  }
+
+  parser.gullet.macros.set(name, macro, global);
+}; // <assignment> -> <non-macro assignment>|<macro assignment>
+// <non-macro assignment> -> <simple assignment>|\global<non-macro assignment>
+// <macro assignment> -> <definition>|<prefix><macro assignment>
+// <prefix> -> \global|\long|\outer
+
+
+defineFunction({
+  type: "internal",
+  names: ["\\global", "\\long", "\\\\globallong"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    parser.consumeSpaces();
+    var token = parser.fetch();
+
+    if (globalMap[token.text]) {
+      // KaTeX doesn't have \par, so ignore \long
+      if (funcName === "\\global" || funcName === "\\\\globallong") {
+        token.text = globalMap[token.text];
+      }
+
+      return assertNodeType(parser.parseFunction(), "internal");
+    }
+
+    throw new src_ParseError("Invalid token after macro prefix", token);
+  }
+}); // Basic support for macro definitions: \def, \gdef, \edef, \xdef
+// <definition> -> <def><control sequence><definition text>
+// <def> -> \def|\gdef|\edef|\xdef
+// <definition text> -> <parameter text><left brace><balanced text><right brace>
+
+defineFunction({
+  type: "internal",
+  names: ["\\def", "\\gdef", "\\edef", "\\xdef"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref2) {
+    var parser = _ref2.parser,
+        funcName = _ref2.funcName;
+    var arg = parser.gullet.consumeArgs(1)[0];
+
+    if (arg.length !== 1) {
+      throw new src_ParseError("\\gdef's first argument must be a macro name");
+    }
+
+    var name = arg[0].text; // Count argument specifiers, and check they are in the order #1 #2 ...
+
+    var numArgs = 0;
+    arg = parser.gullet.consumeArgs(1)[0];
+
+    while (arg.length === 1 && arg[0].text === "#") {
+      arg = parser.gullet.consumeArgs(1)[0];
+
+      if (arg.length !== 1) {
+        throw new src_ParseError("Invalid argument number length \"" + arg.length + "\"");
+      }
+
+      if (!/^[1-9]$/.test(arg[0].text)) {
+        throw new src_ParseError("Invalid argument number \"" + arg[0].text + "\"");
+      }
+
+      numArgs++;
+
+      if (parseInt(arg[0].text) !== numArgs) {
+        throw new src_ParseError("Argument number \"" + arg[0].text + "\" out of order");
+      }
+
+      arg = parser.gullet.consumeArgs(1)[0];
+    }
+
+    if (funcName === "\\edef" || funcName === "\\xdef") {
+      arg = parser.gullet.expandTokens(arg);
+      arg.reverse(); // to fit in with stack order
+    } // Final arg is the expansion of the macro
+
+
+    parser.gullet.macros.set(name, {
+      tokens: arg,
+      numArgs: numArgs
+    }, funcName === globalMap[funcName]);
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+}); // <simple assignment> -> <let assignment>
+// <let assignment> -> \futurelet<control sequence><token><token>
+//     | \let<control sequence><equals><one optional space><token>
+// <equals> -> <optional spaces>|<optional spaces>=
+
+defineFunction({
+  type: "internal",
+  names: ["\\let", "\\\\globallet"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref3) {
+    var parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    var name = def_checkControlSequence(parser.gullet.popToken());
+    parser.gullet.consumeSpaces();
+    var tok = getRHS(parser);
+    letCommand(parser, name, tok, funcName === "\\\\globallet");
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+}); // ref: https://www.tug.org/TUGboat/tb09-3/tb22bechtolsheim.pdf
+
+defineFunction({
+  type: "internal",
+  names: ["\\futurelet", "\\\\globalfuture"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref4) {
+    var parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    var name = def_checkControlSequence(parser.gullet.popToken());
+    var middle = parser.gullet.popToken();
+    var tok = parser.gullet.popToken();
+    letCommand(parser, name, tok, funcName === "\\\\globalfuture");
+    parser.gullet.pushToken(tok);
+    parser.gullet.pushToken(middle);
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+});
+// CONCATENATED MODULE: ./src/delimiter.js
+/**
+ * This file deals with creating delimiters of various sizes. The TeXbook
+ * discusses these routines on page 441-442, in the "Another subroutine sets box
+ * x to a specified variable delimiter" paragraph.
+ *
+ * There are three main routines here. `makeSmallDelim` makes a delimiter in the
+ * normal font, but in either text, script, or scriptscript style.
+ * `makeLargeDelim` makes a delimiter in textstyle, but in one of the Size1,
+ * Size2, Size3, or Size4 fonts. `makeStackedDelim` makes a delimiter out of
+ * smaller pieces that are stacked on top of one another.
+ *
+ * The functions take a parameter `center`, which determines if the delimiter
+ * should be centered around the axis.
+ *
+ * Then, there are three exposed functions. `sizedDelim` makes a delimiter in
+ * one of the given sizes. This is used for things like `\bigl`.
+ * `customSizedDelim` makes a delimiter with a given total height+depth. It is
+ * called in places like `\sqrt`. `leftRightDelim` makes an appropriate
+ * delimiter which surrounds an expression of a given height an depth. It is
+ * used in `\left` and `\right`.
+ */
+
+
+
+
+
+
+
+
+
+/**
+ * Get the metrics for a given symbol and font, after transformation (i.e.
+ * after following replacement from symbols.js)
+ */
+var delimiter_getMetrics = function getMetrics(symbol, font, mode) {
+  var replace = src_symbols.math[symbol] && src_symbols.math[symbol].replace;
+  var metrics = getCharacterMetrics(replace || symbol, font, mode);
+
+  if (!metrics) {
+    throw new Error("Unsupported symbol " + symbol + " and font size " + font + ".");
+  }
+
+  return metrics;
+};
+/**
+ * Puts a delimiter span in a given style, and adds appropriate height, depth,
+ * and maxFontSizes.
+ */
+
+
+var delimiter_styleWrap = function styleWrap(delim, toStyle, options, classes) {
+  var newOptions = options.havingBaseStyle(toStyle);
+  var span = buildCommon.makeSpan(classes.concat(newOptions.sizingClasses(options)), [delim], options);
+  var delimSizeMultiplier = newOptions.sizeMultiplier / options.sizeMultiplier;
+  span.height *= delimSizeMultiplier;
+  span.depth *= delimSizeMultiplier;
+  span.maxFontSize = newOptions.sizeMultiplier;
+  return span;
+};
+
+var centerSpan = function centerSpan(span, options, style) {
+  var newOptions = options.havingBaseStyle(style);
+  var shift = (1 - options.sizeMultiplier / newOptions.sizeMultiplier) * options.fontMetrics().axisHeight;
+  span.classes.push("delimcenter");
+  span.style.top = shift + "em";
+  span.height -= shift;
+  span.depth += shift;
+};
+/**
+ * Makes a small delimiter. This is a delimiter that comes in the Main-Regular
+ * font, but is restyled to either be in textstyle, scriptstyle, or
+ * scriptscriptstyle.
+ */
+
+
+var delimiter_makeSmallDelim = function makeSmallDelim(delim, style, center, options, mode, classes) {
+  var text = buildCommon.makeSymbol(delim, "Main-Regular", mode, options);
+  var span = delimiter_styleWrap(text, style, options, classes);
+
+  if (center) {
+    centerSpan(span, options, style);
+  }
+
+  return span;
+};
+/**
+ * Builds a symbol in the given font size (note size is an integer)
+ */
+
+
+var delimiter_mathrmSize = function mathrmSize(value, size, mode, options) {
+  return buildCommon.makeSymbol(value, "Size" + size + "-Regular", mode, options);
+};
+/**
+ * Makes a large delimiter. This is a delimiter that comes in the Size1, Size2,
+ * Size3, or Size4 fonts. It is always rendered in textstyle.
+ */
+
+
+var delimiter_makeLargeDelim = function makeLargeDelim(delim, size, center, options, mode, classes) {
+  var inner = delimiter_mathrmSize(delim, size, mode, options);
+  var span = delimiter_styleWrap(buildCommon.makeSpan(["delimsizing", "size" + size], [inner], options), src_Style.TEXT, options, classes);
+
+  if (center) {
+    centerSpan(span, options, src_Style.TEXT);
+  }
+
+  return span;
+};
+/**
+ * Make an inner span with the given offset and in the given font. This is used
+ * in `makeStackedDelim` to make the stacking pieces for the delimiter.
+ */
+
+
+var delimiter_makeInner = function makeInner(symbol, font, mode) {
+  var sizeClass; // Apply the correct CSS class to choose the right font.
+
+  if (font === "Size1-Regular") {
+    sizeClass = "delim-size1";
+  } else
+    /* if (font === "Size4-Regular") */
+    {
+      sizeClass = "delim-size4";
+    }
+
+  var inner = buildCommon.makeSpan(["delimsizinginner", sizeClass], [buildCommon.makeSpan([], [buildCommon.makeSymbol(symbol, font, mode)])]); // Since this will be passed into `makeVList` in the end, wrap the element
+  // in the appropriate tag that VList uses.
+
+  return {
+    type: "elem",
+    elem: inner
+  };
+}; // Helper for makeStackedDelim
+
+
+var lap = {
+  type: "kern",
+  size: -0.005
+};
+/**
+ * Make a stacked delimiter out of a given delimiter, with the total height at
+ * least `heightTotal`. This routine is mentioned on page 442 of the TeXbook.
+ */
+
+var delimiter_makeStackedDelim = function makeStackedDelim(delim, heightTotal, center, options, mode, classes) {
+  // There are four parts, the top, an optional middle, a repeated part, and a
+  // bottom.
+  var top;
+  var middle;
+  var repeat;
+  var bottom;
+  top = repeat = bottom = delim;
+  middle = null; // Also keep track of what font the delimiters are in
+
+  var font = "Size1-Regular"; // We set the parts and font based on the symbol. Note that we use
+  // '\u23d0' instead of '|' and '\u2016' instead of '\\|' for the
+  // repeats of the arrows
+
+  if (delim === "\\uparrow") {
+    repeat = bottom = "\u23D0";
+  } else if (delim === "\\Uparrow") {
+    repeat = bottom = "\u2016";
+  } else if (delim === "\\downarrow") {
+    top = repeat = "\u23D0";
+  } else if (delim === "\\Downarrow") {
+    top = repeat = "\u2016";
+  } else if (delim === "\\updownarrow") {
+    top = "\\uparrow";
+    repeat = "\u23D0";
+    bottom = "\\downarrow";
+  } else if (delim === "\\Updownarrow") {
+    top = "\\Uparrow";
+    repeat = "\u2016";
+    bottom = "\\Downarrow";
+  } else if (delim === "[" || delim === "\\lbrack") {
+    top = "\u23A1";
+    repeat = "\u23A2";
+    bottom = "\u23A3";
+    font = "Size4-Regular";
+  } else if (delim === "]" || delim === "\\rbrack") {
+    top = "\u23A4";
+    repeat = "\u23A5";
+    bottom = "\u23A6";
+    font = "Size4-Regular";
+  } else if (delim === "\\lfloor" || delim === "\u230A") {
+    repeat = top = "\u23A2";
+    bottom = "\u23A3";
+    font = "Size4-Regular";
+  } else if (delim === "\\lceil" || delim === "\u2308") {
+    top = "\u23A1";
+    repeat = bottom = "\u23A2";
+    font = "Size4-Regular";
+  } else if (delim === "\\rfloor" || delim === "\u230B") {
+    repeat = top = "\u23A5";
+    bottom = "\u23A6";
+    font = "Size4-Regular";
+  } else if (delim === "\\rceil" || delim === "\u2309") {
+    top = "\u23A4";
+    repeat = bottom = "\u23A5";
+    font = "Size4-Regular";
+  } else if (delim === "(" || delim === "\\lparen") {
+    top = "\u239B";
+    repeat = "\u239C";
+    bottom = "\u239D";
+    font = "Size4-Regular";
+  } else if (delim === ")" || delim === "\\rparen") {
+    top = "\u239E";
+    repeat = "\u239F";
+    bottom = "\u23A0";
+    font = "Size4-Regular";
+  } else if (delim === "\\{" || delim === "\\lbrace") {
+    top = "\u23A7";
+    middle = "\u23A8";
+    bottom = "\u23A9";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\}" || delim === "\\rbrace") {
+    top = "\u23AB";
+    middle = "\u23AC";
+    bottom = "\u23AD";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\lgroup" || delim === "\u27EE") {
+    top = "\u23A7";
+    bottom = "\u23A9";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\rgroup" || delim === "\u27EF") {
+    top = "\u23AB";
+    bottom = "\u23AD";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\lmoustache" || delim === "\u23B0") {
+    top = "\u23A7";
+    bottom = "\u23AD";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } else if (delim === "\\rmoustache" || delim === "\u23B1") {
+    top = "\u23AB";
+    bottom = "\u23A9";
+    repeat = "\u23AA";
+    font = "Size4-Regular";
+  } // Get the metrics of the four sections
+
+
+  var topMetrics = delimiter_getMetrics(top, font, mode);
+  var topHeightTotal = topMetrics.height + topMetrics.depth;
+  var repeatMetrics = delimiter_getMetrics(repeat, font, mode);
+  var repeatHeightTotal = repeatMetrics.height + repeatMetrics.depth;
+  var bottomMetrics = delimiter_getMetrics(bottom, font, mode);
+  var bottomHeightTotal = bottomMetrics.height + bottomMetrics.depth;
+  var middleHeightTotal = 0;
+  var middleFactor = 1;
+
+  if (middle !== null) {
+    var middleMetrics = delimiter_getMetrics(middle, font, mode);
+    middleHeightTotal = middleMetrics.height + middleMetrics.depth;
+    middleFactor = 2; // repeat symmetrically above and below middle
+  } // Calcuate the minimal height that the delimiter can have.
+  // It is at least the size of the top, bottom, and optional middle combined.
+
+
+  var minHeight = topHeightTotal + bottomHeightTotal + middleHeightTotal; // Compute the number of copies of the repeat symbol we will need
+
+  var repeatCount = Math.max(0, Math.ceil((heightTotal - minHeight) / (middleFactor * repeatHeightTotal))); // Compute the total height of the delimiter including all the symbols
+
+  var realHeightTotal = minHeight + repeatCount * middleFactor * repeatHeightTotal; // The center of the delimiter is placed at the center of the axis. Note
+  // that in this context, "center" means that the delimiter should be
+  // centered around the axis in the current style, while normally it is
+  // centered around the axis in textstyle.
+
+  var axisHeight = options.fontMetrics().axisHeight;
+
+  if (center) {
+    axisHeight *= options.sizeMultiplier;
+  } // Calculate the depth
+
+
+  var depth = realHeightTotal / 2 - axisHeight; // This function differs from the TeX procedure in one way.
+  // We shift each repeat element downwards by 0.005em, to prevent a gap
+  // due to browser floating point rounding error.
+  // Then, at the last element-to element joint, we add one extra repeat
+  // element to cover the gap created by the shifts.
+  // Find the shift needed to align the upper end of the extra element at a point
+  // 0.005em above the lower end of the top element.
+
+  var shiftOfExtraElement = (repeatCount + 1) * 0.005 - repeatHeightTotal; // Now, we start building the pieces that will go into the vlist
+  // Keep a list of the inner pieces
+
+  var inners = []; // Add the bottom symbol
+
+  inners.push(delimiter_makeInner(bottom, font, mode));
+
+  if (middle === null) {
+    // Add that many symbols
+    for (var i = 0; i < repeatCount; i++) {
+      inners.push(lap); // overlap
+
+      inners.push(delimiter_makeInner(repeat, font, mode));
+    }
+  } else {
+    // When there is a middle bit, we need the middle part and two repeated
+    // sections
+    for (var _i = 0; _i < repeatCount; _i++) {
+      inners.push(lap);
+      inners.push(delimiter_makeInner(repeat, font, mode));
+    } // Insert one extra repeat element.
+
+
+    inners.push({
+      type: "kern",
+      size: shiftOfExtraElement
+    });
+    inners.push(delimiter_makeInner(repeat, font, mode));
+    inners.push(lap); // Now insert the middle of the brace.
+
+    inners.push(delimiter_makeInner(middle, font, mode));
+
+    for (var _i2 = 0; _i2 < repeatCount; _i2++) {
+      inners.push(lap);
+      inners.push(delimiter_makeInner(repeat, font, mode));
+    }
+  } // To cover the gap create by the overlaps, insert one more repeat element,
+  // at a position that juts 0.005 above the bottom of the top element.
+
+
+  if ((repeat === "\u239C" || repeat === "\u239F") && repeatCount === 0) {
+    // Parentheses need a short repeat element in order to avoid an overrun.
+    // We'll make a 0.3em tall element from a SVG.
+    var overlap = buildCommon.svgData.leftParenInner[2] / 2;
+    inners.push({
+      type: "kern",
+      size: -overlap
+    });
+    var pathName = repeat === "\u239C" ? "leftParenInner" : "rightParenInner";
+    var innerSpan = buildCommon.staticSvg(pathName, options);
+    inners.push({
+      type: "elem",
+      elem: innerSpan
+    });
+    inners.push({
+      type: "kern",
+      size: -overlap
+    });
+  } else {
+    inners.push({
+      type: "kern",
+      size: shiftOfExtraElement
+    });
+    inners.push(delimiter_makeInner(repeat, font, mode));
+    inners.push(lap);
+  } // Add the top symbol
+
+
+  inners.push(delimiter_makeInner(top, font, mode)); // Finally, build the vlist
+
+  var newOptions = options.havingBaseStyle(src_Style.TEXT);
+  var inner = buildCommon.makeVList({
+    positionType: "bottom",
+    positionData: depth,
+    children: inners
+  }, newOptions);
+  return delimiter_styleWrap(buildCommon.makeSpan(["delimsizing", "mult"], [inner], newOptions), src_Style.TEXT, options, classes);
+}; // All surds have 0.08em padding above the viniculum inside the SVG.
+// That keeps browser span height rounding error from pinching the line.
+
+
+var vbPad = 80; // padding above the surd, measured inside the viewBox.
+
+var emPad = 0.08; // padding, in ems, measured in the document.
+
+var delimiter_sqrtSvg = function sqrtSvg(sqrtName, height, viewBoxHeight, extraViniculum, options) {
+  var path = sqrtPath(sqrtName, extraViniculum, viewBoxHeight);
+  var pathNode = new domTree_PathNode(sqrtName, path);
+  var svg = new SvgNode([pathNode], {
+    // Note: 1000:1 ratio of viewBox to document em width.
+    "width": "400em",
+    "height": height + "em",
+    "viewBox": "0 0 400000 " + viewBoxHeight,
+    "preserveAspectRatio": "xMinYMin slice"
+  });
+  return buildCommon.makeSvgSpan(["hide-tail"], [svg], options);
+};
+/**
+ * Make a sqrt image of the given height,
+ */
+
+
+var makeSqrtImage = function makeSqrtImage(height, options) {
+  // Define a newOptions that removes the effect of size changes such as \Huge.
+  // We don't pick different a height surd for \Huge. For it, we scale up.
+  var newOptions = options.havingBaseSizing(); // Pick the desired surd glyph from a sequence of surds.
+
+  var delim = traverseSequence("\\surd", height * newOptions.sizeMultiplier, stackLargeDelimiterSequence, newOptions);
+  var sizeMultiplier = newOptions.sizeMultiplier; // default
+  // The standard sqrt SVGs each have a 0.04em thick viniculum.
+  // If Settings.minRuleThickness is larger than that, we add extraViniculum.
+
+  var extraViniculum = Math.max(0, options.minRuleThickness - options.fontMetrics().sqrtRuleThickness); // Create a span containing an SVG image of a sqrt symbol.
+
+  var span;
+  var spanHeight = 0;
+  var texHeight = 0;
+  var viewBoxHeight = 0;
+  var advanceWidth; // We create viewBoxes with 80 units of "padding" above each surd.
+  // Then browser rounding error on the parent span height will not
+  // encroach on the ink of the viniculum. But that padding is not
+  // included in the TeX-like `height` used for calculation of
+  // vertical alignment. So texHeight = span.height < span.style.height.
+
+  if (delim.type === "small") {
+    // Get an SVG that is derived from glyph U+221A in font KaTeX-Main.
+    // 1000 unit normal glyph height.
+    viewBoxHeight = 1000 + 1000 * extraViniculum + vbPad;
+
+    if (height < 1.0) {
+      sizeMultiplier = 1.0; // mimic a \textfont radical
+    } else if (height < 1.4) {
+      sizeMultiplier = 0.7; // mimic a \scriptfont radical
+    }
+
+    spanHeight = (1.0 + extraViniculum + emPad) / sizeMultiplier;
+    texHeight = (1.00 + extraViniculum) / sizeMultiplier;
+    span = delimiter_sqrtSvg("sqrtMain", spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "0.853em";
+    advanceWidth = 0.833 / sizeMultiplier; // from the font.
+  } else if (delim.type === "large") {
+    // These SVGs come from fonts: KaTeX_Size1, _Size2, etc.
+    viewBoxHeight = (1000 + vbPad) * sizeToMaxHeight[delim.size];
+    texHeight = (sizeToMaxHeight[delim.size] + extraViniculum) / sizeMultiplier;
+    spanHeight = (sizeToMaxHeight[delim.size] + extraViniculum + emPad) / sizeMultiplier;
+    span = delimiter_sqrtSvg("sqrtSize" + delim.size, spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "1.02em";
+    advanceWidth = 1.0 / sizeMultiplier; // 1.0 from the font.
+  } else {
+    // Tall sqrt. In TeX, this would be stacked using multiple glyphs.
+    // We'll use a single SVG to accomplish the same thing.
+    spanHeight = height + extraViniculum + emPad;
+    texHeight = height + extraViniculum;
+    viewBoxHeight = Math.floor(1000 * height + extraViniculum) + vbPad;
+    span = delimiter_sqrtSvg("sqrtTall", spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "0.742em";
+    advanceWidth = 1.056;
+  }
+
+  span.height = texHeight;
+  span.style.height = spanHeight + "em";
+  return {
+    span: span,
+    advanceWidth: advanceWidth,
+    // Calculate the actual line width.
+    // This actually should depend on the chosen font -- e.g. \boldmath
+    // should use the thicker surd symbols from e.g. KaTeX_Main-Bold, and
+    // have thicker rules.
+    ruleWidth: (options.fontMetrics().sqrtRuleThickness + extraViniculum) * sizeMultiplier
+  };
+}; // There are three kinds of delimiters, delimiters that stack when they become
+// too large
+
+
+var stackLargeDelimiters = ["(", "\\lparen", ")", "\\rparen", "[", "\\lbrack", "]", "\\rbrack", "\\{", "\\lbrace", "\\}", "\\rbrace", "\\lfloor", "\\rfloor", "\u230A", "\u230B", "\\lceil", "\\rceil", "\u2308", "\u2309", "\\surd"]; // delimiters that always stack
+
+var stackAlwaysDelimiters = ["\\uparrow", "\\downarrow", "\\updownarrow", "\\Uparrow", "\\Downarrow", "\\Updownarrow", "|", "\\|", "\\vert", "\\Vert", "\\lvert", "\\rvert", "\\lVert", "\\rVert", "\\lgroup", "\\rgroup", "\u27EE", "\u27EF", "\\lmoustache", "\\rmoustache", "\u23B0", "\u23B1"]; // and delimiters that never stack
+
+var stackNeverDelimiters = ["<", ">", "\\langle", "\\rangle", "/", "\\backslash", "\\lt", "\\gt"]; // Metrics of the different sizes. Found by looking at TeX's output of
+// $\bigl| // \Bigl| \biggl| \Biggl| \showlists$
+// Used to create stacked delimiters of appropriate sizes in makeSizedDelim.
+
+var sizeToMaxHeight = [0, 1.2, 1.8, 2.4, 3.0];
+/**
+ * Used to create a delimiter of a specific size, where `size` is 1, 2, 3, or 4.
+ */
+
+var delimiter_makeSizedDelim = function makeSizedDelim(delim, size, options, mode, classes) {
+  // < and > turn into \langle and \rangle in delimiters
+  if (delim === "<" || delim === "\\lt" || delim === "\u27E8") {
+    delim = "\\langle";
+  } else if (delim === ">" || delim === "\\gt" || delim === "\u27E9") {
+    delim = "\\rangle";
+  } // Sized delimiters are never centered.
+
+
+  if (utils.contains(stackLargeDelimiters, delim) || utils.contains(stackNeverDelimiters, delim)) {
+    return delimiter_makeLargeDelim(delim, size, false, options, mode, classes);
+  } else if (utils.contains(stackAlwaysDelimiters, delim)) {
+    return delimiter_makeStackedDelim(delim, sizeToMaxHeight[size], false, options, mode, classes);
+  } else {
+    throw new src_ParseError("Illegal delimiter: '" + delim + "'");
+  }
+};
+/**
+ * There are three different sequences of delimiter sizes that the delimiters
+ * follow depending on the kind of delimiter. This is used when creating custom
+ * sized delimiters to decide whether to create a small, large, or stacked
+ * delimiter.
+ *
+ * In real TeX, these sequences aren't explicitly defined, but are instead
+ * defined inside the font metrics. Since there are only three sequences that
+ * are possible for the delimiters that TeX defines, it is easier to just encode
+ * them explicitly here.
+ */
+
+
+// Delimiters that never stack try small delimiters and large delimiters only
+var stackNeverDelimiterSequence = [{
+  type: "small",
+  style: src_Style.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: src_Style.SCRIPT
+}, {
+  type: "small",
+  style: src_Style.TEXT
+}, {
+  type: "large",
+  size: 1
+}, {
+  type: "large",
+  size: 2
+}, {
+  type: "large",
+  size: 3
+}, {
+  type: "large",
+  size: 4
+}]; // Delimiters that always stack try the small delimiters first, then stack
+
+var stackAlwaysDelimiterSequence = [{
+  type: "small",
+  style: src_Style.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: src_Style.SCRIPT
+}, {
+  type: "small",
+  style: src_Style.TEXT
+}, {
+  type: "stack"
+}]; // Delimiters that stack when large try the small and then large delimiters, and
+// stack afterwards
+
+var stackLargeDelimiterSequence = [{
+  type: "small",
+  style: src_Style.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: src_Style.SCRIPT
+}, {
+  type: "small",
+  style: src_Style.TEXT
+}, {
+  type: "large",
+  size: 1
+}, {
+  type: "large",
+  size: 2
+}, {
+  type: "large",
+  size: 3
+}, {
+  type: "large",
+  size: 4
+}, {
+  type: "stack"
+}];
+/**
+ * Get the font used in a delimiter based on what kind of delimiter it is.
+ * TODO(#963) Use more specific font family return type once that is introduced.
+ */
+
+var delimTypeToFont = function delimTypeToFont(type) {
+  if (type.type === "small") {
+    return "Main-Regular";
+  } else if (type.type === "large") {
+    return "Size" + type.size + "-Regular";
+  } else if (type.type === "stack") {
+    return "Size4-Regular";
+  } else {
+    throw new Error("Add support for delim type '" + type.type + "' here.");
+  }
+};
+/**
+ * Traverse a sequence of types of delimiters to decide what kind of delimiter
+ * should be used to create a delimiter of the given height+depth.
+ */
+
+
+var traverseSequence = function traverseSequence(delim, height, sequence, options) {
+  // Here, we choose the index we should start at in the sequences. In smaller
+  // sizes (which correspond to larger numbers in style.size) we start earlier
+  // in the sequence. Thus, scriptscript starts at index 3-3=0, script starts
+  // at index 3-2=1, text starts at 3-1=2, and display starts at min(2,3-0)=2
+  var start = Math.min(2, 3 - options.style.size);
+
+  for (var i = start; i < sequence.length; i++) {
+    if (sequence[i].type === "stack") {
+      // This is always the last delimiter, so we just break the loop now.
+      break;
+    }
+
+    var metrics = delimiter_getMetrics(delim, delimTypeToFont(sequence[i]), "math");
+    var heightDepth = metrics.height + metrics.depth; // Small delimiters are scaled down versions of the same font, so we
+    // account for the style change size.
+
+    if (sequence[i].type === "small") {
+      var newOptions = options.havingBaseStyle(sequence[i].style);
+      heightDepth *= newOptions.sizeMultiplier;
+    } // Check if the delimiter at this size works for the given height.
+
+
+    if (heightDepth > height) {
+      return sequence[i];
+    }
+  } // If we reached the end of the sequence, return the last sequence element.
+
+
+  return sequence[sequence.length - 1];
+};
+/**
+ * Make a delimiter of a given height+depth, with optional centering. Here, we
+ * traverse the sequences, and create a delimiter that the sequence tells us to.
+ */
+
+
+var delimiter_makeCustomSizedDelim = function makeCustomSizedDelim(delim, height, center, options, mode, classes) {
+  if (delim === "<" || delim === "\\lt" || delim === "\u27E8") {
+    delim = "\\langle";
+  } else if (delim === ">" || delim === "\\gt" || delim === "\u27E9") {
+    delim = "\\rangle";
+  } // Decide what sequence to use
+
+
+  var sequence;
+
+  if (utils.contains(stackNeverDelimiters, delim)) {
+    sequence = stackNeverDelimiterSequence;
+  } else if (utils.contains(stackLargeDelimiters, delim)) {
+    sequence = stackLargeDelimiterSequence;
+  } else {
+    sequence = stackAlwaysDelimiterSequence;
+  } // Look through the sequence
+
+
+  var delimType = traverseSequence(delim, height, sequence, options); // Get the delimiter from font glyphs.
+  // Depending on the sequence element we decided on, call the
+  // appropriate function.
+
+  if (delimType.type === "small") {
+    return delimiter_makeSmallDelim(delim, delimType.style, center, options, mode, classes);
+  } else if (delimType.type === "large") {
+    return delimiter_makeLargeDelim(delim, delimType.size, center, options, mode, classes);
+  } else
+    /* if (delimType.type === "stack") */
+    {
+      return delimiter_makeStackedDelim(delim, height, center, options, mode, classes);
+    }
+};
+/**
+ * Make a delimiter for use with `\left` and `\right`, given a height and depth
+ * of an expression that the delimiters surround.
+ */
+
+
+var makeLeftRightDelim = function makeLeftRightDelim(delim, height, depth, options, mode, classes) {
+  // We always center \left/\right delimiters, so the axis is always shifted
+  var axisHeight = options.fontMetrics().axisHeight * options.sizeMultiplier; // Taken from TeX source, tex.web, function make_left_right
+
+  var delimiterFactor = 901;
+  var delimiterExtend = 5.0 / options.fontMetrics().ptPerEm;
+  var maxDistFromAxis = Math.max(height - axisHeight, depth + axisHeight);
+  var totalHeight = Math.max( // In real TeX, calculations are done using integral values which are
+  // 65536 per pt, or 655360 per em. So, the division here truncates in
+  // TeX but doesn't here, producing different results. If we wanted to
+  // exactly match TeX's calculation, we could do
+  //   Math.floor(655360 * maxDistFromAxis / 500) *
+  //    delimiterFactor / 655360
+  // (To see the difference, compare
+  //    x^{x^{\left(\rule{0.1em}{0.68em}\right)}}
+  // in TeX and KaTeX)
+  maxDistFromAxis / 500 * delimiterFactor, 2 * maxDistFromAxis - delimiterExtend); // Finally, we defer to `makeCustomSizedDelim` with our calculated total
+  // height
+
+  return delimiter_makeCustomSizedDelim(delim, totalHeight, true, options, mode, classes);
+};
+
+/* harmony default export */ var delimiter = ({
+  sqrtImage: makeSqrtImage,
+  sizedDelim: delimiter_makeSizedDelim,
+  customSizedDelim: delimiter_makeCustomSizedDelim,
+  leftRightDelim: makeLeftRightDelim
+});
+// CONCATENATED MODULE: ./src/functions/delimsizing.js
+
+
+
+
+
+
+
+
+
+// Extra data needed for the delimiter handler down below
+var delimiterSizes = {
+  "\\bigl": {
+    mclass: "mopen",
+    size: 1
+  },
+  "\\Bigl": {
+    mclass: "mopen",
+    size: 2
+  },
+  "\\biggl": {
+    mclass: "mopen",
+    size: 3
+  },
+  "\\Biggl": {
+    mclass: "mopen",
+    size: 4
+  },
+  "\\bigr": {
+    mclass: "mclose",
+    size: 1
+  },
+  "\\Bigr": {
+    mclass: "mclose",
+    size: 2
+  },
+  "\\biggr": {
+    mclass: "mclose",
+    size: 3
+  },
+  "\\Biggr": {
+    mclass: "mclose",
+    size: 4
+  },
+  "\\bigm": {
+    mclass: "mrel",
+    size: 1
+  },
+  "\\Bigm": {
+    mclass: "mrel",
+    size: 2
+  },
+  "\\biggm": {
+    mclass: "mrel",
+    size: 3
+  },
+  "\\Biggm": {
+    mclass: "mrel",
+    size: 4
+  },
+  "\\big": {
+    mclass: "mord",
+    size: 1
+  },
+  "\\Big": {
+    mclass: "mord",
+    size: 2
+  },
+  "\\bigg": {
+    mclass: "mord",
+    size: 3
+  },
+  "\\Bigg": {
+    mclass: "mord",
+    size: 4
+  }
+};
+var delimiters = ["(", "\\lparen", ")", "\\rparen", "[", "\\lbrack", "]", "\\rbrack", "\\{", "\\lbrace", "\\}", "\\rbrace", "\\lfloor", "\\rfloor", "\u230A", "\u230B", "\\lceil", "\\rceil", "\u2308", "\u2309", "<", ">", "\\langle", "\u27E8", "\\rangle", "\u27E9", "\\lt", "\\gt", "\\lvert", "\\rvert", "\\lVert", "\\rVert", "\\lgroup", "\\rgroup", "\u27EE", "\u27EF", "\\lmoustache", "\\rmoustache", "\u23B0", "\u23B1", "/", "\\backslash", "|", "\\vert", "\\|", "\\Vert", "\\uparrow", "\\Uparrow", "\\downarrow", "\\Downarrow", "\\updownarrow", "\\Updownarrow", "."];
+
+// Delimiter functions
+function checkDelimiter(delim, context) {
+  var symDelim = checkSymbolNodeType(delim);
+
+  if (symDelim && utils.contains(delimiters, symDelim.text)) {
+    return symDelim;
+  } else if (symDelim) {
+    throw new src_ParseError("Invalid delimiter '" + symDelim.text + "' after '" + context.funcName + "'", delim);
+  } else {
+    throw new src_ParseError("Invalid delimiter type '" + delim.type + "'", delim);
+  }
+}
+
+defineFunction({
+  type: "delimsizing",
+  names: ["\\bigl", "\\Bigl", "\\biggl", "\\Biggl", "\\bigr", "\\Bigr", "\\biggr", "\\Biggr", "\\bigm", "\\Bigm", "\\biggm", "\\Biggm", "\\big", "\\Big", "\\bigg", "\\Bigg"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    var delim = checkDelimiter(args[0], context);
+    return {
+      type: "delimsizing",
+      mode: context.parser.mode,
+      size: delimiterSizes[context.funcName].size,
+      mclass: delimiterSizes[context.funcName].mclass,
+      delim: delim.text
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    if (group.delim === ".") {
+      // Empty delimiters still count as elements, even though they don't
+      // show anything.
+      return buildCommon.makeSpan([group.mclass]);
+    } // Use delimiter.sizedDelim to generate the delimiter.
+
+
+    return delimiter.sizedDelim(group.delim, group.size, options, group.mode, [group.mclass]);
+  },
+  mathmlBuilder: function mathmlBuilder(group) {
+    var children = [];
+
+    if (group.delim !== ".") {
+      children.push(buildMathML_makeText(group.delim, group.mode));
+    }
+
+    var node = new mathMLTree.MathNode("mo", children);
+
+    if (group.mclass === "mopen" || group.mclass === "mclose") {
+      // Only some of the delimsizing functions act as fences, and they
+      // return "mopen" or "mclose" mclass.
+      node.setAttribute("fence", "true");
+    } else {
+      // Explicitly disable fencing if it's not a fence, to override the
+      // defaults.
+      node.setAttribute("fence", "false");
+    }
+
+    return node;
+  }
+});
+
+function assertParsed(group) {
+  if (!group.body) {
+    throw new Error("Bug: The leftright ParseNode wasn't fully parsed.");
+  }
+}
+
+defineFunction({
+  type: "leftright-right",
+  names: ["\\right"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    // \left case below triggers parsing of \right in
+    //   `const right = parser.parseFunction();`
+    // uses this return value.
+    var color = context.parser.gullet.macros.get("\\current@color");
+
+    if (color && typeof color !== "string") {
+      throw new src_ParseError("\\current@color set to non-string in \\right");
+    }
+
+    return {
+      type: "leftright-right",
+      mode: context.parser.mode,
+      delim: checkDelimiter(args[0], context).text,
+      color: color // undefined if not set via \color
+
+    };
+  }
+});
+defineFunction({
+  type: "leftright",
+  names: ["\\left"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    var delim = checkDelimiter(args[0], context);
+    var parser = context.parser; // Parse out the implicit body
+
+    ++parser.leftrightDepth; // parseExpression stops before '\\right'
+
+    var body = parser.parseExpression(false);
+    --parser.leftrightDepth; // Check the next token
+
+    parser.expect("\\right", false);
+    var right = assertNodeType(parser.parseFunction(), "leftright-right");
+    return {
+      type: "leftright",
+      mode: parser.mode,
+      body: body,
+      left: delim.text,
+      right: right.delim,
+      rightColor: right.color
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    assertParsed(group); // Build the inner expression
+
+    var inner = buildHTML_buildExpression(group.body, options, true, ["mopen", "mclose"]);
+    var innerHeight = 0;
+    var innerDepth = 0;
+    var hadMiddle = false; // Calculate its height and depth
+
+    for (var i = 0; i < inner.length; i++) {
+      // Property `isMiddle` not defined on `span`. See comment in
+      // "middle"'s htmlBuilder.
+      // $FlowFixMe
+      if (inner[i].isMiddle) {
+        hadMiddle = true;
+      } else {
+        innerHeight = Math.max(inner[i].height, innerHeight);
+        innerDepth = Math.max(inner[i].depth, innerDepth);
+      }
+    } // The size of delimiters is the same, regardless of what style we are
+    // in. Thus, to correctly calculate the size of delimiter we need around
+    // a group, we scale down the inner size based on the size.
+
+
+    innerHeight *= options.sizeMultiplier;
+    innerDepth *= options.sizeMultiplier;
+    var leftDelim;
+
+    if (group.left === ".") {
+      // Empty delimiters in \left and \right make null delimiter spaces.
+      leftDelim = makeNullDelimiter(options, ["mopen"]);
+    } else {
+      // Otherwise, use leftRightDelim to generate the correct sized
+      // delimiter.
+      leftDelim = delimiter.leftRightDelim(group.left, innerHeight, innerDepth, options, group.mode, ["mopen"]);
+    } // Add it to the beginning of the expression
+
+
+    inner.unshift(leftDelim); // Handle middle delimiters
+
+    if (hadMiddle) {
+      for (var _i = 1; _i < inner.length; _i++) {
+        var middleDelim = inner[_i]; // Property `isMiddle` not defined on `span`. See comment in
+        // "middle"'s htmlBuilder.
+        // $FlowFixMe
+
+        var isMiddle = middleDelim.isMiddle;
+
+        if (isMiddle) {
+          // Apply the options that were active when \middle was called
+          inner[_i] = delimiter.leftRightDelim(isMiddle.delim, innerHeight, innerDepth, isMiddle.options, group.mode, []);
+        }
+      }
+    }
+
+    var rightDelim; // Same for the right delimiter, but using color specified by \color
+
+    if (group.right === ".") {
+      rightDelim = makeNullDelimiter(options, ["mclose"]);
+    } else {
+      var colorOptions = group.rightColor ? options.withColor(group.rightColor) : options;
+      rightDelim = delimiter.leftRightDelim(group.right, innerHeight, innerDepth, colorOptions, group.mode, ["mclose"]);
+    } // Add it to the end of the expression.
+
+
+    inner.push(rightDelim);
+    return buildCommon.makeSpan(["minner"], inner, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    assertParsed(group);
+    var inner = buildMathML_buildExpression(group.body, options);
+
+    if (group.left !== ".") {
+      var leftNode = new mathMLTree.MathNode("mo", [buildMathML_makeText(group.left, group.mode)]);
+      leftNode.setAttribute("fence", "true");
+      inner.unshift(leftNode);
+    }
+
+    if (group.right !== ".") {
+      var rightNode = new mathMLTree.MathNode("mo", [buildMathML_makeText(group.right, group.mode)]);
+      rightNode.setAttribute("fence", "true");
+
+      if (group.rightColor) {
+        rightNode.setAttribute("mathcolor", group.rightColor);
+      }
+
+      inner.push(rightNode);
+    }
+
+    return buildMathML_makeRow(inner);
+  }
+});
+defineFunction({
+  type: "middle",
+  names: ["\\middle"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    var delim = checkDelimiter(args[0], context);
+
+    if (!context.parser.leftrightDepth) {
+      throw new src_ParseError("\\middle without preceding \\left", delim);
+    }
+
+    return {
+      type: "middle",
+      mode: context.parser.mode,
+      delim: delim.text
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var middleDelim;
+
+    if (group.delim === ".") {
+      middleDelim = makeNullDelimiter(options, []);
+    } else {
+      middleDelim = delimiter.sizedDelim(group.delim, 1, options, group.mode, []);
+      var isMiddle = {
+        delim: group.delim,
+        options: options
+      }; // Property `isMiddle` not defined on `span`. It is only used in
+      // this file above.
+      // TODO: Fix this violation of the `span` type and possibly rename
+      // things since `isMiddle` sounds like a boolean, but is a struct.
+      // $FlowFixMe
+
+      middleDelim.isMiddle = isMiddle;
+    }
+
+    return middleDelim;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    // A Firefox \middle will strech a character vertically only if it
+    // is in the fence part of the operator dictionary at:
+    // https://www.w3.org/TR/MathML3/appendixc.html.
+    // So we need to avoid U+2223 and use plain "|" instead.
+    var textNode = group.delim === "\\vert" || group.delim === "|" ? buildMathML_makeText("|", "text") : buildMathML_makeText(group.delim, group.mode);
+    var middleNode = new mathMLTree.MathNode("mo", [textNode]);
+    middleNode.setAttribute("fence", "true"); // MathML gives 5/18em spacing to each <mo> element.
+    // \middle should get delimiter spacing instead.
+
+    middleNode.setAttribute("lspace", "0.05em");
+    middleNode.setAttribute("rspace", "0.05em");
+    return middleNode;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/enclose.js
+
+
+
+
+
+
+
+
+
+var enclose_htmlBuilder = function htmlBuilder(group, options) {
+  // \cancel, \bcancel, \xcancel, \sout, \fbox, \colorbox, \fcolorbox
+  // Some groups can return document fragments.  Handle those by wrapping
+  // them in a span.
+  var inner = buildCommon.wrapFragment(buildHTML_buildGroup(group.body, options), options);
+  var label = group.label.substr(1);
+  var scale = options.sizeMultiplier;
+  var img;
+  var imgShift = 0; // In the LaTeX cancel package, line geometry is slightly different
+  // depending on whether the subject is wider than it is tall, or vice versa.
+  // We don't know the width of a group, so as a proxy, we test if
+  // the subject is a single character. This captures most of the
+  // subjects that should get the "tall" treatment.
+
+  var isSingleChar = utils.isCharacterBox(group.body);
+
+  if (label === "sout") {
+    img = buildCommon.makeSpan(["stretchy", "sout"]);
+    img.height = options.fontMetrics().defaultRuleThickness / scale;
+    imgShift = -0.5 * options.fontMetrics().xHeight;
+  } else {
+    // Add horizontal padding
+    if (/cancel/.test(label)) {
+      if (!isSingleChar) {
+        inner.classes.push("cancel-pad");
+      }
+    } else {
+      inner.classes.push("boxpad");
+    } // Add vertical padding
+
+
+    var vertPad = 0;
+    var ruleThickness = 0; // ref: cancel package: \advance\totalheight2\p@ % "+2"
+
+    if (/box/.test(label)) {
+      ruleThickness = Math.max(options.fontMetrics().fboxrule, // default
+      options.minRuleThickness // User override.
+      );
+      vertPad = options.fontMetrics().fboxsep + (label === "colorbox" ? 0 : ruleThickness);
+    } else {
+      vertPad = isSingleChar ? 0.2 : 0;
+    }
+
+    img = stretchy.encloseSpan(inner, label, vertPad, options);
+
+    if (/fbox|boxed|fcolorbox/.test(label)) {
+      img.style.borderStyle = "solid";
+      img.style.borderWidth = ruleThickness + "em";
+    }
+
+    imgShift = inner.depth + vertPad;
+
+    if (group.backgroundColor) {
+      img.style.backgroundColor = group.backgroundColor;
+
+      if (group.borderColor) {
+        img.style.borderColor = group.borderColor;
+      }
+    }
+  }
+
+  var vlist;
+
+  if (group.backgroundColor) {
+    vlist = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [// Put the color background behind inner;
+      {
+        type: "elem",
+        elem: img,
+        shift: imgShift
+      }, {
+        type: "elem",
+        elem: inner,
+        shift: 0
+      }]
+    }, options);
+  } else {
+    vlist = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [// Write the \cancel stroke on top of inner.
+      {
+        type: "elem",
+        elem: inner,
+        shift: 0
+      }, {
+        type: "elem",
+        elem: img,
+        shift: imgShift,
+        wrapperClasses: /cancel/.test(label) ? ["svg-align"] : []
+      }]
+    }, options);
+  }
+
+  if (/cancel/.test(label)) {
+    // The cancel package documentation says that cancel lines add their height
+    // to the expression, but tests show that isn't how it actually works.
+    vlist.height = inner.height;
+    vlist.depth = inner.depth;
+  }
+
+  if (/cancel/.test(label) && !isSingleChar) {
+    // cancel does not create horiz space for its line extension.
+    return buildCommon.makeSpan(["mord", "cancel-lap"], [vlist], options);
+  } else {
+    return buildCommon.makeSpan(["mord"], [vlist], options);
+  }
+};
+
+var enclose_mathmlBuilder = function mathmlBuilder(group, options) {
+  var fboxsep = 0;
+  var node = new mathMLTree.MathNode(group.label.indexOf("colorbox") > -1 ? "mpadded" : "menclose", [buildMathML_buildGroup(group.body, options)]);
+
+  switch (group.label) {
+    case "\\cancel":
+      node.setAttribute("notation", "updiagonalstrike");
+      break;
+
+    case "\\bcancel":
+      node.setAttribute("notation", "downdiagonalstrike");
+      break;
+
+    case "\\sout":
+      node.setAttribute("notation", "horizontalstrike");
+      break;
+
+    case "\\fbox":
+      node.setAttribute("notation", "box");
+      break;
+
+    case "\\fcolorbox":
+    case "\\colorbox":
+      // <menclose> doesn't have a good notation option. So use <mpadded>
+      // instead. Set some attributes that come included with <menclose>.
+      fboxsep = options.fontMetrics().fboxsep * options.fontMetrics().ptPerEm;
+      node.setAttribute("width", "+" + 2 * fboxsep + "pt");
+      node.setAttribute("height", "+" + 2 * fboxsep + "pt");
+      node.setAttribute("lspace", fboxsep + "pt"); //
+
+      node.setAttribute("voffset", fboxsep + "pt");
+
+      if (group.label === "\\fcolorbox") {
+        var thk = Math.max(options.fontMetrics().fboxrule, // default
+        options.minRuleThickness // user override
+        );
+        node.setAttribute("style", "border: " + thk + "em solid " + String(group.borderColor));
+      }
+
+      break;
+
+    case "\\xcancel":
+      node.setAttribute("notation", "updiagonalstrike downdiagonalstrike");
+      break;
+  }
+
+  if (group.backgroundColor) {
+    node.setAttribute("mathbackground", group.backgroundColor);
+  }
+
+  return node;
+};
+
+defineFunction({
+  type: "enclose",
+  names: ["\\colorbox"],
+  props: {
+    numArgs: 2,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "text"]
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var color = assertNodeType(args[0], "color-token").color;
+    var body = args[1];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      backgroundColor: color,
+      body: body
+    };
+  },
+  htmlBuilder: enclose_htmlBuilder,
+  mathmlBuilder: enclose_mathmlBuilder
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\fcolorbox"],
+  props: {
+    numArgs: 3,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "color", "text"]
+  },
+  handler: function handler(_ref2, args, optArgs) {
+    var parser = _ref2.parser,
+        funcName = _ref2.funcName;
+    var borderColor = assertNodeType(args[0], "color-token").color;
+    var backgroundColor = assertNodeType(args[1], "color-token").color;
+    var body = args[2];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      backgroundColor: backgroundColor,
+      borderColor: borderColor,
+      body: body
+    };
+  },
+  htmlBuilder: enclose_htmlBuilder,
+  mathmlBuilder: enclose_mathmlBuilder
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\fbox"],
+  props: {
+    numArgs: 1,
+    argTypes: ["hbox"],
+    allowedInText: true
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser;
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: "\\fbox",
+      body: args[0]
+    };
+  }
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\cancel", "\\bcancel", "\\xcancel", "\\sout"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref4, args, optArgs) {
+    var parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    var body = args[0];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      body: body
+    };
+  },
+  htmlBuilder: enclose_htmlBuilder,
+  mathmlBuilder: enclose_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/defineEnvironment.js
+
+
+/**
+ * All registered environments.
+ * `environments.js` exports this same dictionary again and makes it public.
+ * `Parser.js` requires this dictionary via `environments.js`.
+ */
+var _environments = {};
+function defineEnvironment(_ref) {
+  var type = _ref.type,
+      names = _ref.names,
+      props = _ref.props,
+      handler = _ref.handler,
+      htmlBuilder = _ref.htmlBuilder,
+      mathmlBuilder = _ref.mathmlBuilder;
+  // Set default values of environments.
+  var data = {
+    type: type,
+    numArgs: props.numArgs || 0,
+    greediness: 1,
+    allowedInText: false,
+    numOptionalArgs: 0,
+    handler: handler
+  };
+
+  for (var i = 0; i < names.length; ++i) {
+    // TODO: The value type of _environments should be a type union of all
+    // possible `EnvSpec<>` possibilities instead of `EnvSpec<*>`, which is
+    // an existential type.
+    // $FlowFixMe
+    _environments[names[i]] = data;
+  }
+
+  if (htmlBuilder) {
+    _htmlGroupBuilders[type] = htmlBuilder;
+  }
+
+  if (mathmlBuilder) {
+    _mathmlGroupBuilders[type] = mathmlBuilder;
+  }
+}
+// CONCATENATED MODULE: ./src/environments/array.js
+
+
+
+
+
+
+
+
+
+
+
+
+
+function getHLines(parser) {
+  // Return an array. The array length = number of hlines.
+  // Each element in the array tells if the line is dashed.
+  var hlineInfo = [];
+  parser.consumeSpaces();
+  var nxt = parser.fetch().text;
+
+  while (nxt === "\\hline" || nxt === "\\hdashline") {
+    parser.consume();
+    hlineInfo.push(nxt === "\\hdashline");
+    parser.consumeSpaces();
+    nxt = parser.fetch().text;
+  }
+
+  return hlineInfo;
+}
+/**
+ * Parse the body of the environment, with rows delimited by \\ and
+ * columns delimited by &, and create a nested list in row-major order
+ * with one group per cell.  If given an optional argument style
+ * ("text", "display", etc.), then each cell is cast into that style.
+ */
+
+
+function parseArray(parser, _ref, style) {
+  var hskipBeforeAndAfter = _ref.hskipBeforeAndAfter,
+      addJot = _ref.addJot,
+      cols = _ref.cols,
+      arraystretch = _ref.arraystretch,
+      colSeparationType = _ref.colSeparationType;
+  // Parse body of array with \\ temporarily mapped to \cr
+  parser.gullet.beginGroup();
+  parser.gullet.macros.set("\\\\", "\\cr"); // Get current arraystretch if it's not set by the environment
+
+  if (!arraystretch) {
+    var stretch = parser.gullet.expandMacroAsText("\\arraystretch");
+
+    if (stretch == null) {
+      // Default \arraystretch from lttab.dtx
+      arraystretch = 1;
+    } else {
+      arraystretch = parseFloat(stretch);
+
+      if (!arraystretch || arraystretch < 0) {
+        throw new src_ParseError("Invalid \\arraystretch: " + stretch);
+      }
+    }
+  } // Start group for first cell
+
+
+  parser.gullet.beginGroup();
+  var row = [];
+  var body = [row];
+  var rowGaps = [];
+  var hLinesBeforeRow = []; // Test for \hline at the top of the array.
+
+  hLinesBeforeRow.push(getHLines(parser));
+
+  while (true) {
+    // eslint-disable-line no-constant-condition
+    // Parse each cell in its own group (namespace)
+    var cell = parser.parseExpression(false, "\\cr");
+    parser.gullet.endGroup();
+    parser.gullet.beginGroup();
+    cell = {
+      type: "ordgroup",
+      mode: parser.mode,
+      body: cell
+    };
+
+    if (style) {
+      cell = {
+        type: "styling",
+        mode: parser.mode,
+        style: style,
+        body: [cell]
+      };
+    }
+
+    row.push(cell);
+    var next = parser.fetch().text;
+
+    if (next === "&") {
+      parser.consume();
+    } else if (next === "\\end") {
+      // Arrays terminate newlines with `\crcr` which consumes a `\cr` if
+      // the last line is empty.
+      // NOTE: Currently, `cell` is the last item added into `row`.
+      if (row.length === 1 && cell.type === "styling" && cell.body[0].body.length === 0) {
+        body.pop();
+      }
+
+      if (hLinesBeforeRow.length < body.length + 1) {
+        hLinesBeforeRow.push([]);
+      }
+
+      break;
+    } else if (next === "\\cr") {
+      var cr = assertNodeType(parser.parseFunction(), "cr");
+      rowGaps.push(cr.size); // check for \hline(s) following the row separator
+
+      hLinesBeforeRow.push(getHLines(parser));
+      row = [];
+      body.push(row);
+    } else {
+      throw new src_ParseError("Expected & or \\\\ or \\cr or \\end", parser.nextToken);
+    }
+  } // End cell group
+
+
+  parser.gullet.endGroup(); // End array group defining \\
+
+  parser.gullet.endGroup();
+  return {
+    type: "array",
+    mode: parser.mode,
+    addJot: addJot,
+    arraystretch: arraystretch,
+    body: body,
+    cols: cols,
+    rowGaps: rowGaps,
+    hskipBeforeAndAfter: hskipBeforeAndAfter,
+    hLinesBeforeRow: hLinesBeforeRow,
+    colSeparationType: colSeparationType
+  };
+} // Decides on a style for cells in an array according to whether the given
+// environment name starts with the letter 'd'.
+
+
+function dCellStyle(envName) {
+  if (envName.substr(0, 1) === "d") {
+    return "display";
+  } else {
+    return "text";
+  }
+}
+
+var array_htmlBuilder = function htmlBuilder(group, options) {
+  var r;
+  var c;
+  var nr = group.body.length;
+  var hLinesBeforeRow = group.hLinesBeforeRow;
+  var nc = 0;
+  var body = new Array(nr);
+  var hlines = [];
+  var ruleThickness = Math.max( // From LaTeX \showthe\arrayrulewidth. Equals 0.04 em.
+  options.fontMetrics().arrayRuleWidth, options.minRuleThickness // User override.
+  ); // Horizontal spacing
+
+  var pt = 1 / options.fontMetrics().ptPerEm;
+  var arraycolsep = 5 * pt; // default value, i.e. \arraycolsep in article.cls
+
+  if (group.colSeparationType && group.colSeparationType === "small") {
+    // We're in a {smallmatrix}. Default column space is \thickspace,
+    // i.e. 5/18em = 0.2778em, per amsmath.dtx for {smallmatrix}.
+    // But that needs adjustment because LaTeX applies \scriptstyle to the
+    // entire array, including the colspace, but this function applies
+    // \scriptstyle only inside each element.
+    var localMultiplier = options.havingStyle(src_Style.SCRIPT).sizeMultiplier;
+    arraycolsep = 0.2778 * (localMultiplier / options.sizeMultiplier);
+  } // Vertical spacing
+
+
+  var baselineskip = 12 * pt; // see size10.clo
+  // Default \jot from ltmath.dtx
+  // TODO(edemaine): allow overriding \jot via \setlength (#687)
+
+  var jot = 3 * pt;
+  var arrayskip = group.arraystretch * baselineskip;
+  var arstrutHeight = 0.7 * arrayskip; // \strutbox in ltfsstrc.dtx and
+
+  var arstrutDepth = 0.3 * arrayskip; // \@arstrutbox in lttab.dtx
+
+  var totalHeight = 0; // Set a position for \hline(s) at the top of the array, if any.
+
+  function setHLinePos(hlinesInGap) {
+    for (var i = 0; i < hlinesInGap.length; ++i) {
+      if (i > 0) {
+        totalHeight += 0.25;
+      }
+
+      hlines.push({
+        pos: totalHeight,
+        isDashed: hlinesInGap[i]
+      });
+    }
+  }
+
+  setHLinePos(hLinesBeforeRow[0]);
+
+  for (r = 0; r < group.body.length; ++r) {
+    var inrow = group.body[r];
+    var height = arstrutHeight; // \@array adds an \@arstrut
+
+    var depth = arstrutDepth; // to each tow (via the template)
+
+    if (nc < inrow.length) {
+      nc = inrow.length;
+    }
+
+    var outrow = new Array(inrow.length);
+
+    for (c = 0; c < inrow.length; ++c) {
+      var elt = buildHTML_buildGroup(inrow[c], options);
+
+      if (depth < elt.depth) {
+        depth = elt.depth;
+      }
+
+      if (height < elt.height) {
+        height = elt.height;
+      }
+
+      outrow[c] = elt;
+    }
+
+    var rowGap = group.rowGaps[r];
+    var gap = 0;
+
+    if (rowGap) {
+      gap = units_calculateSize(rowGap, options);
+
+      if (gap > 0) {
+        // \@argarraycr
+        gap += arstrutDepth;
+
+        if (depth < gap) {
+          depth = gap; // \@xargarraycr
+        }
+
+        gap = 0;
+      }
+    } // In AMS multiline environments such as aligned and gathered, rows
+    // correspond to lines that have additional \jot added to the
+    // \baselineskip via \openup.
+
+
+    if (group.addJot) {
+      depth += jot;
+    }
+
+    outrow.height = height;
+    outrow.depth = depth;
+    totalHeight += height;
+    outrow.pos = totalHeight;
+    totalHeight += depth + gap; // \@yargarraycr
+
+    body[r] = outrow; // Set a position for \hline(s), if any.
+
+    setHLinePos(hLinesBeforeRow[r + 1]);
+  }
+
+  var offset = totalHeight / 2 + options.fontMetrics().axisHeight;
+  var colDescriptions = group.cols || [];
+  var cols = [];
+  var colSep;
+  var colDescrNum;
+
+  for (c = 0, colDescrNum = 0; // Continue while either there are more columns or more column
+  // descriptions, so trailing separators don't get lost.
+  c < nc || colDescrNum < colDescriptions.length; ++c, ++colDescrNum) {
+    var colDescr = colDescriptions[colDescrNum] || {};
+    var firstSeparator = true;
+
+    while (colDescr.type === "separator") {
+      // If there is more than one separator in a row, add a space
+      // between them.
+      if (!firstSeparator) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = options.fontMetrics().doubleRuleSep + "em";
+        cols.push(colSep);
+      }
+
+      if (colDescr.separator === "|" || colDescr.separator === ":") {
+        var lineType = colDescr.separator === "|" ? "solid" : "dashed";
+        var separator = buildCommon.makeSpan(["vertical-separator"], [], options);
+        separator.style.height = totalHeight + "em";
+        separator.style.borderRightWidth = ruleThickness + "em";
+        separator.style.borderRightStyle = lineType;
+        separator.style.margin = "0 -" + ruleThickness / 2 + "em";
+        separator.style.verticalAlign = -(totalHeight - offset) + "em";
+        cols.push(separator);
+      } else {
+        throw new src_ParseError("Invalid separator type: " + colDescr.separator);
+      }
+
+      colDescrNum++;
+      colDescr = colDescriptions[colDescrNum] || {};
+      firstSeparator = false;
+    }
+
+    if (c >= nc) {
+      continue;
+    }
+
+    var sepwidth = void 0;
+
+    if (c > 0 || group.hskipBeforeAndAfter) {
+      sepwidth = utils.deflt(colDescr.pregap, arraycolsep);
+
+      if (sepwidth !== 0) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = sepwidth + "em";
+        cols.push(colSep);
+      }
+    }
+
+    var col = [];
+
+    for (r = 0; r < nr; ++r) {
+      var row = body[r];
+      var elem = row[c];
+
+      if (!elem) {
+        continue;
+      }
+
+      var shift = row.pos - offset;
+      elem.depth = row.depth;
+      elem.height = row.height;
+      col.push({
+        type: "elem",
+        elem: elem,
+        shift: shift
+      });
+    }
+
+    col = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: col
+    }, options);
+    col = buildCommon.makeSpan(["col-align-" + (colDescr.align || "c")], [col]);
+    cols.push(col);
+
+    if (c < nc - 1 || group.hskipBeforeAndAfter) {
+      sepwidth = utils.deflt(colDescr.postgap, arraycolsep);
+
+      if (sepwidth !== 0) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = sepwidth + "em";
+        cols.push(colSep);
+      }
+    }
+  }
+
+  body = buildCommon.makeSpan(["mtable"], cols); // Add \hline(s), if any.
+
+  if (hlines.length > 0) {
+    var line = buildCommon.makeLineSpan("hline", options, ruleThickness);
+    var dashes = buildCommon.makeLineSpan("hdashline", options, ruleThickness);
+    var vListElems = [{
+      type: "elem",
+      elem: body,
+      shift: 0
+    }];
+
+    while (hlines.length > 0) {
+      var hline = hlines.pop();
+      var lineShift = hline.pos - offset;
+
+      if (hline.isDashed) {
+        vListElems.push({
+          type: "elem",
+          elem: dashes,
+          shift: lineShift
+        });
+      } else {
+        vListElems.push({
+          type: "elem",
+          elem: line,
+          shift: lineShift
+        });
+      }
+    }
+
+    body = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: vListElems
+    }, options);
+  }
+
+  return buildCommon.makeSpan(["mord"], [body], options);
+};
+
+var alignMap = {
+  c: "center ",
+  l: "left ",
+  r: "right "
+};
+
+var array_mathmlBuilder = function mathmlBuilder(group, options) {
+  var table = new mathMLTree.MathNode("mtable", group.body.map(function (row) {
+    return new mathMLTree.MathNode("mtr", row.map(function (cell) {
+      return new mathMLTree.MathNode("mtd", [buildMathML_buildGroup(cell, options)]);
+    }));
+  })); // Set column alignment, row spacing, column spacing, and
+  // array lines by setting attributes on the table element.
+  // Set the row spacing. In MathML, we specify a gap distance.
+  // We do not use rowGap[] because MathML automatically increases
+  // cell height with the height/depth of the element content.
+  // LaTeX \arraystretch multiplies the row baseline-to-baseline distance.
+  // We simulate this by adding (arraystretch - 1)em to the gap. This
+  // does a reasonable job of adjusting arrays containing 1 em tall content.
+  // The 0.16 and 0.09 values are found emprically. They produce an array
+  // similar to LaTeX and in which content does not interfere with \hines.
+
+  var gap = group.arraystretch === 0.5 ? 0.1 // {smallmatrix}, {subarray}
+  : 0.16 + group.arraystretch - 1 + (group.addJot ? 0.09 : 0);
+  table.setAttribute("rowspacing", gap + "em"); // MathML table lines go only between cells.
+  // To place a line on an edge we'll use <menclose>, if necessary.
+
+  var menclose = "";
+  var align = "";
+
+  if (group.cols && group.cols.length > 0) {
+    // Find column alignment, column spacing, and  vertical lines.
+    var cols = group.cols;
+    var columnLines = "";
+    var prevTypeWasAlign = false;
+    var iStart = 0;
+    var iEnd = cols.length;
+
+    if (cols[0].type === "separator") {
+      menclose += "top ";
+      iStart = 1;
+    }
+
+    if (cols[cols.length - 1].type === "separator") {
+      menclose += "bottom ";
+      iEnd -= 1;
+    }
+
+    for (var i = iStart; i < iEnd; i++) {
+      if (cols[i].type === "align") {
+        align += alignMap[cols[i].align];
+
+        if (prevTypeWasAlign) {
+          columnLines += "none ";
+        }
+
+        prevTypeWasAlign = true;
+      } else if (cols[i].type === "separator") {
+        // MathML accepts only single lines between cells.
+        // So we read only the first of consecutive separators.
+        if (prevTypeWasAlign) {
+          columnLines += cols[i].separator === "|" ? "solid " : "dashed ";
+          prevTypeWasAlign = false;
+        }
+      }
+    }
+
+    table.setAttribute("columnalign", align.trim());
+
+    if (/[sd]/.test(columnLines)) {
+      table.setAttribute("columnlines", columnLines.trim());
+    }
+  } // Set column spacing.
+
+
+  if (group.colSeparationType === "align") {
+    var _cols = group.cols || [];
+
+    var spacing = "";
+
+    for (var _i = 1; _i < _cols.length; _i++) {
+      spacing += _i % 2 ? "0em " : "1em ";
+    }
+
+    table.setAttribute("columnspacing", spacing.trim());
+  } else if (group.colSeparationType === "alignat") {
+    table.setAttribute("columnspacing", "0em");
+  } else if (group.colSeparationType === "small") {
+    table.setAttribute("columnspacing", "0.2778em");
+  } else {
+    table.setAttribute("columnspacing", "1em");
+  } // Address \hline and \hdashline
+
+
+  var rowLines = "";
+  var hlines = group.hLinesBeforeRow;
+  menclose += hlines[0].length > 0 ? "left " : "";
+  menclose += hlines[hlines.length - 1].length > 0 ? "right " : "";
+
+  for (var _i2 = 1; _i2 < hlines.length - 1; _i2++) {
+    rowLines += hlines[_i2].length === 0 ? "none " // MathML accepts only a single line between rows. Read one element.
+    : hlines[_i2][0] ? "dashed " : "solid ";
+  }
+
+  if (/[sd]/.test(rowLines)) {
+    table.setAttribute("rowlines", rowLines.trim());
+  }
+
+  if (menclose !== "") {
+    table = new mathMLTree.MathNode("menclose", [table]);
+    table.setAttribute("notation", menclose.trim());
+  }
+
+  if (group.arraystretch && group.arraystretch < 1) {
+    // A small array. Wrap in scriptstyle so row gap is not too large.
+    table = new mathMLTree.MathNode("mstyle", [table]);
+    table.setAttribute("scriptlevel", "1");
+  }
+
+  return table;
+}; // Convenience function for aligned and alignedat environments.
+
+
+var array_alignedHandler = function alignedHandler(context, args) {
+  var cols = [];
+  var res = parseArray(context.parser, {
+    cols: cols,
+    addJot: true
+  }, "display"); // Determining number of columns.
+  // 1. If the first argument is given, we use it as a number of columns,
+  //    and makes sure that each row doesn't exceed that number.
+  // 2. Otherwise, just count number of columns = maximum number
+  //    of cells in each row ("aligned" mode -- isAligned will be true).
+  //
+  // At the same time, prepend empty group {} at beginning of every second
+  // cell in each row (starting with second cell) so that operators become
+  // binary.  This behavior is implemented in amsmath's \start@aligned.
+
+  var numMaths;
+  var numCols = 0;
+  var emptyGroup = {
+    type: "ordgroup",
+    mode: context.mode,
+    body: []
+  };
+
+  if (args[0] && args[0].type === "ordgroup") {
+    var arg0 = "";
+
+    for (var i = 0; i < args[0].body.length; i++) {
+      var textord = assertNodeType(args[0].body[i], "textord");
+      arg0 += textord.text;
+    }
+
+    numMaths = Number(arg0);
+    numCols = numMaths * 2;
+  }
+
+  var isAligned = !numCols;
+  res.body.forEach(function (row) {
+    for (var _i3 = 1; _i3 < row.length; _i3 += 2) {
+      // Modify ordgroup node within styling node
+      var styling = assertNodeType(row[_i3], "styling");
+      var ordgroup = assertNodeType(styling.body[0], "ordgroup");
+      ordgroup.body.unshift(emptyGroup);
+    }
+
+    if (!isAligned) {
+      // Case 1
+      var curMaths = row.length / 2;
+
+      if (numMaths < curMaths) {
+        throw new src_ParseError("Too many math in a row: " + ("expected " + numMaths + ", but got " + curMaths), row[0]);
+      }
+    } else if (numCols < row.length) {
+      // Case 2
+      numCols = row.length;
+    }
+  }); // Adjusting alignment.
+  // In aligned mode, we add one \qquad between columns;
+  // otherwise we add nothing.
+
+  for (var _i4 = 0; _i4 < numCols; ++_i4) {
+    var align = "r";
+    var pregap = 0;
+
+    if (_i4 % 2 === 1) {
+      align = "l";
+    } else if (_i4 > 0 && isAligned) {
+      // "aligned" mode.
+      pregap = 1; // add one \quad
+    }
+
+    cols[_i4] = {
+      type: "align",
+      align: align,
+      pregap: pregap,
+      postgap: 0
+    };
+  }
+
+  res.colSeparationType = isAligned ? "align" : "alignat";
+  return res;
+}; // Arrays are part of LaTeX, defined in lttab.dtx so its documentation
+// is part of the source2e.pdf file of LaTeX2e source documentation.
+// {darray} is an {array} environment where cells are set in \displaystyle,
+// as defined in nccmath.sty.
+
+
+defineEnvironment({
+  type: "array",
+  names: ["array", "darray"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    // Since no types are specified above, the two possibilities are
+    // - The argument is wrapped in {} or [], in which case Parser's
+    //   parseGroup() returns an "ordgroup" wrapping some symbol node.
+    // - The argument is a bare symbol node.
+    var symNode = checkSymbolNodeType(args[0]);
+    var colalign = symNode ? [args[0]] : assertNodeType(args[0], "ordgroup").body;
+    var cols = colalign.map(function (nde) {
+      var node = assertSymbolNodeType(nde);
+      var ca = node.text;
+
+      if ("lcr".indexOf(ca) !== -1) {
+        return {
+          type: "align",
+          align: ca
+        };
+      } else if (ca === "|") {
+        return {
+          type: "separator",
+          separator: "|"
+        };
+      } else if (ca === ":") {
+        return {
+          type: "separator",
+          separator: ":"
+        };
+      }
+
+      throw new src_ParseError("Unknown column alignment: " + ca, nde);
+    });
+    var res = {
+      cols: cols,
+      hskipBeforeAndAfter: true // \@preamble in lttab.dtx
+
+    };
+    return parseArray(context.parser, res, dCellStyle(context.envName));
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // The matrix environments of amsmath builds on the array environment
+// of LaTeX, which is discussed above.
+
+defineEnvironment({
+  type: "array",
+  names: ["matrix", "pmatrix", "bmatrix", "Bmatrix", "vmatrix", "Vmatrix"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(context) {
+    var delimiters = {
+      "matrix": null,
+      "pmatrix": ["(", ")"],
+      "bmatrix": ["[", "]"],
+      "Bmatrix": ["\\{", "\\}"],
+      "vmatrix": ["|", "|"],
+      "Vmatrix": ["\\Vert", "\\Vert"]
+    }[context.envName]; // \hskip -\arraycolsep in amsmath
+
+    var payload = {
+      hskipBeforeAndAfter: false
+    };
+    var res = parseArray(context.parser, payload, dCellStyle(context.envName));
+    return delimiters ? {
+      type: "leftright",
+      mode: context.mode,
+      body: [res],
+      left: delimiters[0],
+      right: delimiters[1],
+      rightColor: undefined // \right uninfluenced by \color in array
+
+    } : res;
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+});
+defineEnvironment({
+  type: "array",
+  names: ["smallmatrix"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(context) {
+    var payload = {
+      arraystretch: 0.5
+    };
+    var res = parseArray(context.parser, payload, "script");
+    res.colSeparationType = "small";
+    return res;
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+});
+defineEnvironment({
+  type: "array",
+  names: ["subarray"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(context, args) {
+    // Parsing of {subarray} is similar to {array}
+    var symNode = checkSymbolNodeType(args[0]);
+    var colalign = symNode ? [args[0]] : assertNodeType(args[0], "ordgroup").body;
+    var cols = colalign.map(function (nde) {
+      var node = assertSymbolNodeType(nde);
+      var ca = node.text; // {subarray} only recognizes "l" & "c"
+
+      if ("lc".indexOf(ca) !== -1) {
+        return {
+          type: "align",
+          align: ca
+        };
+      }
+
+      throw new src_ParseError("Unknown column alignment: " + ca, nde);
+    });
+
+    if (cols.length > 1) {
+      throw new src_ParseError("{subarray} can contain only one column");
+    }
+
+    var res = {
+      cols: cols,
+      hskipBeforeAndAfter: false,
+      arraystretch: 0.5
+    };
+    res = parseArray(context.parser, res, "script");
+
+    if (res.body.length > 0 && res.body[0].length > 1) {
+      throw new src_ParseError("{subarray} can contain only one column");
+    }
+
+    return res;
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // A cases environment (in amsmath.sty) is almost equivalent to
+// \def\arraystretch{1.2}%
+// \left\{\begin{array}{@{}l@{\quad}l@{}} … \end{array}\right.
+// {dcases} is a {cases} environment where cells are set in \displaystyle,
+// as defined in mathtools.sty.
+// {rcases} is another mathtools environment. It's brace is on the right side.
+
+defineEnvironment({
+  type: "array",
+  names: ["cases", "dcases", "rcases", "drcases"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(context) {
+    var payload = {
+      arraystretch: 1.2,
+      cols: [{
+        type: "align",
+        align: "l",
+        pregap: 0,
+        // TODO(kevinb) get the current style.
+        // For now we use the metrics for TEXT style which is what we were
+        // doing before.  Before attempting to get the current style we
+        // should look at TeX's behavior especially for \over and matrices.
+        postgap: 1.0
+        /* 1em quad */
+
+      }, {
+        type: "align",
+        align: "l",
+        pregap: 0,
+        postgap: 0
+      }]
+    };
+    var res = parseArray(context.parser, payload, dCellStyle(context.envName));
+    return {
+      type: "leftright",
+      mode: context.mode,
+      body: [res],
+      left: context.envName.indexOf("r") > -1 ? "." : "\\{",
+      right: context.envName.indexOf("r") > -1 ? "\\}" : ".",
+      rightColor: undefined
+    };
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // An aligned environment is like the align* environment
+// except it operates within math mode.
+// Note that we assume \nomallineskiplimit to be zero,
+// so that \strut@ is the same as \strut.
+
+defineEnvironment({
+  type: "array",
+  names: ["aligned"],
+  props: {
+    numArgs: 0
+  },
+  handler: array_alignedHandler,
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // A gathered environment is like an array environment with one centered
+// column, but where rows are considered lines so get \jot line spacing
+// and contents are set in \displaystyle.
+
+defineEnvironment({
+  type: "array",
+  names: ["gathered"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(context) {
+    var res = {
+      cols: [{
+        type: "align",
+        align: "c"
+      }],
+      addJot: true
+    };
+    return parseArray(context.parser, res, "display");
+  },
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // alignat environment is like an align environment, but one must explicitly
+// specify maximum number of columns in each row, and can adjust spacing between
+// each columns.
+
+defineEnvironment({
+  type: "array",
+  names: ["alignedat"],
+  // One for numbered and for unnumbered;
+  // but, KaTeX doesn't supports math numbering yet,
+  // they make no difference for now.
+  props: {
+    numArgs: 1
+  },
+  handler: array_alignedHandler,
+  htmlBuilder: array_htmlBuilder,
+  mathmlBuilder: array_mathmlBuilder
+}); // Catch \hline outside array environment
+
+defineFunction({
+  type: "text",
+  // Doesn't matter what this is.
+  names: ["\\hline", "\\hdashline"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: true
+  },
+  handler: function handler(context, args) {
+    throw new src_ParseError(context.funcName + " valid only within array environment");
+  }
+});
+// CONCATENATED MODULE: ./src/environments.js
+
+var environments = _environments;
+/* harmony default export */ var src_environments = (environments); // All environment definitions should be imported below
+
+
+// CONCATENATED MODULE: ./src/functions/environment.js
+
+
+
+ // Environment delimiters. HTML/MathML rendering is defined in the corresponding
+// defineEnvironment definitions.
+// $FlowFixMe, "environment" handler returns an environment ParseNode
+
+defineFunction({
+  type: "environment",
+  names: ["\\begin", "\\end"],
+  props: {
+    numArgs: 1,
+    argTypes: ["text"]
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var nameGroup = args[0];
+
+    if (nameGroup.type !== "ordgroup") {
+      throw new src_ParseError("Invalid environment name", nameGroup);
+    }
+
+    var envName = "";
+
+    for (var i = 0; i < nameGroup.body.length; ++i) {
+      envName += assertNodeType(nameGroup.body[i], "textord").text;
+    }
+
+    if (funcName === "\\begin") {
+      // begin...end is similar to left...right
+      if (!src_environments.hasOwnProperty(envName)) {
+        throw new src_ParseError("No such environment: " + envName, nameGroup);
+      } // Build the environment object. Arguments and other information will
+      // be made available to the begin and end methods using properties.
+
+
+      var env = src_environments[envName];
+
+      var _parser$parseArgument = parser.parseArguments("\\begin{" + envName + "}", env),
+          _args = _parser$parseArgument.args,
+          optArgs = _parser$parseArgument.optArgs;
+
+      var context = {
+        mode: parser.mode,
+        envName: envName,
+        parser: parser
+      };
+      var result = env.handler(context, _args, optArgs);
+      parser.expect("\\end", false);
+      var endNameToken = parser.nextToken;
+      var end = assertNodeType(parser.parseFunction(), "environment");
+
+      if (end.name !== envName) {
+        throw new src_ParseError("Mismatch: \\begin{" + envName + "} matched by \\end{" + end.name + "}", endNameToken);
+      }
+
+      return result;
+    }
+
+    return {
+      type: "environment",
+      mode: parser.mode,
+      name: envName,
+      nameGroup: nameGroup
+    };
+  }
+});
+// CONCATENATED MODULE: ./src/functions/mclass.js
+
+
+
+
+
+
+var mclass_makeSpan = buildCommon.makeSpan;
+
+function mclass_htmlBuilder(group, options) {
+  var elements = buildHTML_buildExpression(group.body, options, true);
+  return mclass_makeSpan([group.mclass], elements, options);
+}
+
+function mclass_mathmlBuilder(group, options) {
+  var node;
+  var inner = buildMathML_buildExpression(group.body, options);
+
+  if (group.mclass === "minner") {
+    return mathMLTree.newDocumentFragment(inner);
+  } else if (group.mclass === "mord") {
+    if (group.isCharacterBox) {
+      node = inner[0];
+      node.type = "mi";
+    } else {
+      node = new mathMLTree.MathNode("mi", inner);
+    }
+  } else {
+    if (group.isCharacterBox) {
+      node = inner[0];
+      node.type = "mo";
+    } else {
+      node = new mathMLTree.MathNode("mo", inner);
+    } // Set spacing based on what is the most likely adjacent atom type.
+    // See TeXbook p170.
+
+
+    if (group.mclass === "mbin") {
+      node.attributes.lspace = "0.22em"; // medium space
+
+      node.attributes.rspace = "0.22em";
+    } else if (group.mclass === "mpunct") {
+      node.attributes.lspace = "0em";
+      node.attributes.rspace = "0.17em"; // thinspace
+    } else if (group.mclass === "mopen" || group.mclass === "mclose") {
+      node.attributes.lspace = "0em";
+      node.attributes.rspace = "0em";
+    } // MathML <mo> default space is 5/18 em, so <mrel> needs no action.
+    // Ref: https://developer.mozilla.org/en-US/docs/Web/MathML/Element/mo
+
+  }
+
+  return node;
+} // Math class commands except \mathop
+
+
+defineFunction({
+  type: "mclass",
+  names: ["\\mathord", "\\mathbin", "\\mathrel", "\\mathopen", "\\mathclose", "\\mathpunct", "\\mathinner"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: "m" + funcName.substr(5),
+      // TODO(kevinb): don't prefix with 'm'
+      body: ordargument(body),
+      isCharacterBox: utils.isCharacterBox(body)
+    };
+  },
+  htmlBuilder: mclass_htmlBuilder,
+  mathmlBuilder: mclass_mathmlBuilder
+});
+var binrelClass = function binrelClass(arg) {
+  // \binrel@ spacing varies with (bin|rel|ord) of the atom in the argument.
+  // (by rendering separately and with {}s before and after, and measuring
+  // the change in spacing).  We'll do roughly the same by detecting the
+  // atom type directly.
+  var atom = arg.type === "ordgroup" && arg.body.length ? arg.body[0] : arg;
+
+  if (atom.type === "atom" && (atom.family === "bin" || atom.family === "rel")) {
+    return "m" + atom.family;
+  } else {
+    return "mord";
+  }
+}; // \@binrel{x}{y} renders like y but as mbin/mrel/mord if x is mbin/mrel/mord.
+// This is equivalent to \binrel@{x}\binrel@@{y} in AMSTeX.
+
+defineFunction({
+  type: "mclass",
+  names: ["\\@binrel"],
+  props: {
+    numArgs: 2
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: binrelClass(args[0]),
+      body: [args[1]],
+      isCharacterBox: utils.isCharacterBox(args[1])
+    };
+  }
+}); // Build a relation or stacked op by placing one symbol on top of another
+
+defineFunction({
+  type: "mclass",
+  names: ["\\stackrel", "\\overset", "\\underset"],
+  props: {
+    numArgs: 2
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    var baseArg = args[1];
+    var shiftedArg = args[0];
+    var mclass;
+
+    if (funcName !== "\\stackrel") {
+      // LaTeX applies \binrel spacing to \overset and \underset.
+      mclass = binrelClass(baseArg);
+    } else {
+      mclass = "mrel"; // for \stackrel
+    }
+
+    var baseOp = {
+      type: "op",
+      mode: baseArg.mode,
+      limits: true,
+      alwaysHandleSupSub: true,
+      parentIsSupSub: false,
+      symbol: false,
+      suppressBaseShift: funcName !== "\\stackrel",
+      body: ordargument(baseArg)
+    };
+    var supsub = {
+      type: "supsub",
+      mode: shiftedArg.mode,
+      base: baseOp,
+      sup: funcName === "\\underset" ? null : shiftedArg,
+      sub: funcName === "\\underset" ? shiftedArg : null
+    };
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: mclass,
+      body: [supsub],
+      isCharacterBox: utils.isCharacterBox(supsub)
+    };
+  },
+  htmlBuilder: mclass_htmlBuilder,
+  mathmlBuilder: mclass_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/font.js
+// TODO(kevinb): implement \\sl and \\sc
+
+
+
+
+
+
+var font_htmlBuilder = function htmlBuilder(group, options) {
+  var font = group.font;
+  var newOptions = options.withFont(font);
+  return buildHTML_buildGroup(group.body, newOptions);
+};
+
+var font_mathmlBuilder = function mathmlBuilder(group, options) {
+  var font = group.font;
+  var newOptions = options.withFont(font);
+  return buildMathML_buildGroup(group.body, newOptions);
+};
+
+var fontAliases = {
+  "\\Bbb": "\\mathbb",
+  "\\bold": "\\mathbf",
+  "\\frak": "\\mathfrak",
+  "\\bm": "\\boldsymbol"
+};
+defineFunction({
+  type: "font",
+  names: [// styles, except \boldsymbol defined below
+  "\\mathrm", "\\mathit", "\\mathbf", "\\mathnormal", // families
+  "\\mathbb", "\\mathcal", "\\mathfrak", "\\mathscr", "\\mathsf", "\\mathtt", // aliases, except \bm defined below
+  "\\Bbb", "\\bold", "\\frak"],
+  props: {
+    numArgs: 1,
+    greediness: 2
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    var func = funcName;
+
+    if (func in fontAliases) {
+      func = fontAliases[func];
+    }
+
+    return {
+      type: "font",
+      mode: parser.mode,
+      font: func.slice(1),
+      body: body
+    };
+  },
+  htmlBuilder: font_htmlBuilder,
+  mathmlBuilder: font_mathmlBuilder
+});
+defineFunction({
+  type: "mclass",
+  names: ["\\boldsymbol", "\\bm"],
+  props: {
+    numArgs: 1,
+    greediness: 2
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    var body = args[0];
+    var isCharacterBox = utils.isCharacterBox(body); // amsbsy.sty's \boldsymbol uses \binrel spacing to inherit the
+    // argument's bin|rel|ord status
+
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: binrelClass(body),
+      body: [{
+        type: "font",
+        mode: parser.mode,
+        font: "boldsymbol",
+        body: body
+      }],
+      isCharacterBox: isCharacterBox
+    };
+  }
+}); // Old font changing functions
+
+defineFunction({
+  type: "font",
+  names: ["\\rm", "\\sf", "\\tt", "\\bf", "\\it", "\\cal"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser,
+        funcName = _ref3.funcName,
+        breakOnTokenText = _ref3.breakOnTokenText;
+    var mode = parser.mode;
+    var body = parser.parseExpression(true, breakOnTokenText);
+    var style = "math" + funcName.slice(1);
+    return {
+      type: "font",
+      mode: mode,
+      font: style,
+      body: {
+        type: "ordgroup",
+        mode: parser.mode,
+        body: body
+      }
+    };
+  },
+  htmlBuilder: font_htmlBuilder,
+  mathmlBuilder: font_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/genfrac.js
+
+
+
+
+
+
+
+
+
+
+
+var genfrac_adjustStyle = function adjustStyle(size, originalStyle) {
+  // Figure out what style this fraction should be in based on the
+  // function used
+  var style = originalStyle;
+
+  if (size === "display") {
+    // Get display style as a default.
+    // If incoming style is sub/sup, use style.text() to get correct size.
+    style = style.id >= src_Style.SCRIPT.id ? style.text() : src_Style.DISPLAY;
+  } else if (size === "text" && style.size === src_Style.DISPLAY.size) {
+    // We're in a \tfrac but incoming style is displaystyle, so:
+    style = src_Style.TEXT;
+  } else if (size === "script") {
+    style = src_Style.SCRIPT;
+  } else if (size === "scriptscript") {
+    style = src_Style.SCRIPTSCRIPT;
+  }
+
+  return style;
+};
+
+var genfrac_htmlBuilder = function htmlBuilder(group, options) {
+  // Fractions are handled in the TeXbook on pages 444-445, rules 15(a-e).
+  var style = genfrac_adjustStyle(group.size, options.style);
+  var nstyle = style.fracNum();
+  var dstyle = style.fracDen();
+  var newOptions;
+  newOptions = options.havingStyle(nstyle);
+  var numerm = buildHTML_buildGroup(group.numer, newOptions, options);
+
+  if (group.continued) {
+    // \cfrac inserts a \strut into the numerator.
+    // Get \strut dimensions from TeXbook page 353.
+    var hStrut = 8.5 / options.fontMetrics().ptPerEm;
+    var dStrut = 3.5 / options.fontMetrics().ptPerEm;
+    numerm.height = numerm.height < hStrut ? hStrut : numerm.height;
+    numerm.depth = numerm.depth < dStrut ? dStrut : numerm.depth;
+  }
+
+  newOptions = options.havingStyle(dstyle);
+  var denomm = buildHTML_buildGroup(group.denom, newOptions, options);
+  var rule;
+  var ruleWidth;
+  var ruleSpacing;
+
+  if (group.hasBarLine) {
+    if (group.barSize) {
+      ruleWidth = units_calculateSize(group.barSize, options);
+      rule = buildCommon.makeLineSpan("frac-line", options, ruleWidth);
+    } else {
+      rule = buildCommon.makeLineSpan("frac-line", options);
+    }
+
+    ruleWidth = rule.height;
+    ruleSpacing = rule.height;
+  } else {
+    rule = null;
+    ruleWidth = 0;
+    ruleSpacing = options.fontMetrics().defaultRuleThickness;
+  } // Rule 15b
+
+
+  var numShift;
+  var clearance;
+  var denomShift;
+
+  if (style.size === src_Style.DISPLAY.size || group.size === "display") {
+    numShift = options.fontMetrics().num1;
+
+    if (ruleWidth > 0) {
+      clearance = 3 * ruleSpacing;
+    } else {
+      clearance = 7 * ruleSpacing;
+    }
+
+    denomShift = options.fontMetrics().denom1;
+  } else {
+    if (ruleWidth > 0) {
+      numShift = options.fontMetrics().num2;
+      clearance = ruleSpacing;
+    } else {
+      numShift = options.fontMetrics().num3;
+      clearance = 3 * ruleSpacing;
+    }
+
+    denomShift = options.fontMetrics().denom2;
+  }
+
+  var frac;
+
+  if (!rule) {
+    // Rule 15c
+    var candidateClearance = numShift - numerm.depth - (denomm.height - denomShift);
+
+    if (candidateClearance < clearance) {
+      numShift += 0.5 * (clearance - candidateClearance);
+      denomShift += 0.5 * (clearance - candidateClearance);
+    }
+
+    frac = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [{
+        type: "elem",
+        elem: denomm,
+        shift: denomShift
+      }, {
+        type: "elem",
+        elem: numerm,
+        shift: -numShift
+      }]
+    }, options);
+  } else {
+    // Rule 15d
+    var axisHeight = options.fontMetrics().axisHeight;
+
+    if (numShift - numerm.depth - (axisHeight + 0.5 * ruleWidth) < clearance) {
+      numShift += clearance - (numShift - numerm.depth - (axisHeight + 0.5 * ruleWidth));
+    }
+
+    if (axisHeight - 0.5 * ruleWidth - (denomm.height - denomShift) < clearance) {
+      denomShift += clearance - (axisHeight - 0.5 * ruleWidth - (denomm.height - denomShift));
+    }
+
+    var midShift = -(axisHeight - 0.5 * ruleWidth);
+    frac = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [{
+        type: "elem",
+        elem: denomm,
+        shift: denomShift
+      }, {
+        type: "elem",
+        elem: rule,
+        shift: midShift
+      }, {
+        type: "elem",
+        elem: numerm,
+        shift: -numShift
+      }]
+    }, options);
+  } // Since we manually change the style sometimes (with \dfrac or \tfrac),
+  // account for the possible size change here.
+
+
+  newOptions = options.havingStyle(style);
+  frac.height *= newOptions.sizeMultiplier / options.sizeMultiplier;
+  frac.depth *= newOptions.sizeMultiplier / options.sizeMultiplier; // Rule 15e
+
+  var delimSize;
+
+  if (style.size === src_Style.DISPLAY.size) {
+    delimSize = options.fontMetrics().delim1;
+  } else {
+    delimSize = options.fontMetrics().delim2;
+  }
+
+  var leftDelim;
+  var rightDelim;
+
+  if (group.leftDelim == null) {
+    leftDelim = makeNullDelimiter(options, ["mopen"]);
+  } else {
+    leftDelim = delimiter.customSizedDelim(group.leftDelim, delimSize, true, options.havingStyle(style), group.mode, ["mopen"]);
+  }
+
+  if (group.continued) {
+    rightDelim = buildCommon.makeSpan([]); // zero width for \cfrac
+  } else if (group.rightDelim == null) {
+    rightDelim = makeNullDelimiter(options, ["mclose"]);
+  } else {
+    rightDelim = delimiter.customSizedDelim(group.rightDelim, delimSize, true, options.havingStyle(style), group.mode, ["mclose"]);
+  }
+
+  return buildCommon.makeSpan(["mord"].concat(newOptions.sizingClasses(options)), [leftDelim, buildCommon.makeSpan(["mfrac"], [frac]), rightDelim], options);
+};
+
+var genfrac_mathmlBuilder = function mathmlBuilder(group, options) {
+  var node = new mathMLTree.MathNode("mfrac", [buildMathML_buildGroup(group.numer, options), buildMathML_buildGroup(group.denom, options)]);
+
+  if (!group.hasBarLine) {
+    node.setAttribute("linethickness", "0px");
+  } else if (group.barSize) {
+    var ruleWidth = units_calculateSize(group.barSize, options);
+    node.setAttribute("linethickness", ruleWidth + "em");
+  }
+
+  var style = genfrac_adjustStyle(group.size, options.style);
+
+  if (style.size !== options.style.size) {
+    node = new mathMLTree.MathNode("mstyle", [node]);
+    var isDisplay = style.size === src_Style.DISPLAY.size ? "true" : "false";
+    node.setAttribute("displaystyle", isDisplay);
+    node.setAttribute("scriptlevel", "0");
+  }
+
+  if (group.leftDelim != null || group.rightDelim != null) {
+    var withDelims = [];
+
+    if (group.leftDelim != null) {
+      var leftOp = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(group.leftDelim.replace("\\", ""))]);
+      leftOp.setAttribute("fence", "true");
+      withDelims.push(leftOp);
+    }
+
+    withDelims.push(node);
+
+    if (group.rightDelim != null) {
+      var rightOp = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(group.rightDelim.replace("\\", ""))]);
+      rightOp.setAttribute("fence", "true");
+      withDelims.push(rightOp);
+    }
+
+    return buildMathML_makeRow(withDelims);
+  }
+
+  return node;
+};
+
+defineFunction({
+  type: "genfrac",
+  names: ["\\cfrac", "\\dfrac", "\\frac", "\\tfrac", "\\dbinom", "\\binom", "\\tbinom", "\\\\atopfrac", // can’t be entered directly
+  "\\\\bracefrac", "\\\\brackfrac"],
+  props: {
+    numArgs: 2,
+    greediness: 2
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var numer = args[0];
+    var denom = args[1];
+    var hasBarLine;
+    var leftDelim = null;
+    var rightDelim = null;
+    var size = "auto";
+
+    switch (funcName) {
+      case "\\cfrac":
+      case "\\dfrac":
+      case "\\frac":
+      case "\\tfrac":
+        hasBarLine = true;
+        break;
+
+      case "\\\\atopfrac":
+        hasBarLine = false;
+        break;
+
+      case "\\dbinom":
+      case "\\binom":
+      case "\\tbinom":
+        hasBarLine = false;
+        leftDelim = "(";
+        rightDelim = ")";
+        break;
+
+      case "\\\\bracefrac":
+        hasBarLine = false;
+        leftDelim = "\\{";
+        rightDelim = "\\}";
+        break;
+
+      case "\\\\brackfrac":
+        hasBarLine = false;
+        leftDelim = "[";
+        rightDelim = "]";
+        break;
+
+      default:
+        throw new Error("Unrecognized genfrac command");
+    }
+
+    switch (funcName) {
+      case "\\cfrac":
+      case "\\dfrac":
+      case "\\dbinom":
+        size = "display";
+        break;
+
+      case "\\tfrac":
+      case "\\tbinom":
+        size = "text";
+        break;
+    }
+
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      continued: funcName === "\\cfrac",
+      numer: numer,
+      denom: denom,
+      hasBarLine: hasBarLine,
+      leftDelim: leftDelim,
+      rightDelim: rightDelim,
+      size: size,
+      barSize: null
+    };
+  },
+  htmlBuilder: genfrac_htmlBuilder,
+  mathmlBuilder: genfrac_mathmlBuilder
+}); // Infix generalized fractions -- these are not rendered directly, but replaced
+// immediately by one of the variants above.
+
+defineFunction({
+  type: "infix",
+  names: ["\\over", "\\choose", "\\atop", "\\brace", "\\brack"],
+  props: {
+    numArgs: 0,
+    infix: true
+  },
+  handler: function handler(_ref2) {
+    var parser = _ref2.parser,
+        funcName = _ref2.funcName,
+        token = _ref2.token;
+    var replaceWith;
+
+    switch (funcName) {
+      case "\\over":
+        replaceWith = "\\frac";
+        break;
+
+      case "\\choose":
+        replaceWith = "\\binom";
+        break;
+
+      case "\\atop":
+        replaceWith = "\\\\atopfrac";
+        break;
+
+      case "\\brace":
+        replaceWith = "\\\\bracefrac";
+        break;
+
+      case "\\brack":
+        replaceWith = "\\\\brackfrac";
+        break;
+
+      default:
+        throw new Error("Unrecognized infix genfrac command");
+    }
+
+    return {
+      type: "infix",
+      mode: parser.mode,
+      replaceWith: replaceWith,
+      token: token
+    };
+  }
+});
+var stylArray = ["display", "text", "script", "scriptscript"];
+
+var delimFromValue = function delimFromValue(delimString) {
+  var delim = null;
+
+  if (delimString.length > 0) {
+    delim = delimString;
+    delim = delim === "." ? null : delim;
+  }
+
+  return delim;
+};
+
+defineFunction({
+  type: "genfrac",
+  names: ["\\genfrac"],
+  props: {
+    numArgs: 6,
+    greediness: 6,
+    argTypes: ["math", "math", "size", "text", "math", "math"]
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser;
+    var numer = args[4];
+    var denom = args[5]; // Look into the parse nodes to get the desired delimiters.
+
+    var leftDelim = args[0].type === "atom" && args[0].family === "open" ? delimFromValue(args[0].text) : null;
+    var rightDelim = args[1].type === "atom" && args[1].family === "close" ? delimFromValue(args[1].text) : null;
+    var barNode = assertNodeType(args[2], "size");
+    var hasBarLine;
+    var barSize = null;
+
+    if (barNode.isBlank) {
+      // \genfrac acts differently than \above.
+      // \genfrac treats an empty size group as a signal to use a
+      // standard bar size. \above would see size = 0 and omit the bar.
+      hasBarLine = true;
+    } else {
+      barSize = barNode.value;
+      hasBarLine = barSize.number > 0;
+    } // Find out if we want displaystyle, textstyle, etc.
+
+
+    var size = "auto";
+    var styl = args[3];
+
+    if (styl.type === "ordgroup") {
+      if (styl.body.length > 0) {
+        var textOrd = assertNodeType(styl.body[0], "textord");
+        size = stylArray[Number(textOrd.text)];
+      }
+    } else {
+      styl = assertNodeType(styl, "textord");
+      size = stylArray[Number(styl.text)];
+    }
+
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      numer: numer,
+      denom: denom,
+      continued: false,
+      hasBarLine: hasBarLine,
+      barSize: barSize,
+      leftDelim: leftDelim,
+      rightDelim: rightDelim,
+      size: size
+    };
+  },
+  htmlBuilder: genfrac_htmlBuilder,
+  mathmlBuilder: genfrac_mathmlBuilder
+}); // \above is an infix fraction that also defines a fraction bar size.
+
+defineFunction({
+  type: "infix",
+  names: ["\\above"],
+  props: {
+    numArgs: 1,
+    argTypes: ["size"],
+    infix: true
+  },
+  handler: function handler(_ref4, args) {
+    var parser = _ref4.parser,
+        funcName = _ref4.funcName,
+        token = _ref4.token;
+    return {
+      type: "infix",
+      mode: parser.mode,
+      replaceWith: "\\\\abovefrac",
+      size: assertNodeType(args[0], "size").value,
+      token: token
+    };
+  }
+});
+defineFunction({
+  type: "genfrac",
+  names: ["\\\\abovefrac"],
+  props: {
+    numArgs: 3,
+    argTypes: ["math", "size", "math"]
+  },
+  handler: function handler(_ref5, args) {
+    var parser = _ref5.parser,
+        funcName = _ref5.funcName;
+    var numer = args[0];
+    var barSize = assert(assertNodeType(args[1], "infix").size);
+    var denom = args[2];
+    var hasBarLine = barSize.number > 0;
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      numer: numer,
+      denom: denom,
+      continued: false,
+      hasBarLine: hasBarLine,
+      barSize: barSize,
+      leftDelim: null,
+      rightDelim: null,
+      size: "auto"
+    };
+  },
+  htmlBuilder: genfrac_htmlBuilder,
+  mathmlBuilder: genfrac_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/horizBrace.js
+
+
+
+
+
+
+
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only "horizBrace", but
+var horizBrace_htmlBuilder = function htmlBuilder(grp, options) {
+  var style = options.style; // Pull out the `ParseNode<"horizBrace">` if `grp` is a "supsub" node.
+
+  var supSubGroup;
+  var group;
+
+  if (grp.type === "supsub") {
+    // Ref: LaTeX source2e: }}}}\limits}
+    // i.e. LaTeX treats the brace similar to an op and passes it
+    // with \limits, so we need to assign supsub style.
+    supSubGroup = grp.sup ? buildHTML_buildGroup(grp.sup, options.havingStyle(style.sup()), options) : buildHTML_buildGroup(grp.sub, options.havingStyle(style.sub()), options);
+    group = assertNodeType(grp.base, "horizBrace");
+  } else {
+    group = assertNodeType(grp, "horizBrace");
+  } // Build the base group
+
+
+  var body = buildHTML_buildGroup(group.base, options.havingBaseStyle(src_Style.DISPLAY)); // Create the stretchy element
+
+  var braceBody = stretchy.svgSpan(group, options); // Generate the vlist, with the appropriate kerns        ┏━━━━━━━━┓
+  // This first vlist contains the content and the brace:   equation
+
+  var vlist;
+
+  if (group.isOver) {
+    vlist = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "kern",
+        size: 0.1
+      }, {
+        type: "elem",
+        elem: braceBody
+      }]
+    }, options); // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+    vlist.children[0].children[0].children[1].classes.push("svg-align");
+  } else {
+    vlist = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: body.depth + 0.1 + braceBody.height,
+      children: [{
+        type: "elem",
+        elem: braceBody
+      }, {
+        type: "kern",
+        size: 0.1
+      }, {
+        type: "elem",
+        elem: body
+      }]
+    }, options); // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+    vlist.children[0].children[0].children[0].classes.push("svg-align");
+  }
+
+  if (supSubGroup) {
+    // To write the supsub, wrap the first vlist in another vlist:
+    // They can't all go in the same vlist, because the note might be
+    // wider than the equation. We want the equation to control the
+    // brace width.
+    //      note          long note           long note
+    //   ┏━━━━━━━━┓   or    ┏━━━┓     not    ┏━━━━━━━━━┓
+    //    equation           eqn                 eqn
+    var vSpan = buildCommon.makeSpan(["mord", group.isOver ? "mover" : "munder"], [vlist], options);
+
+    if (group.isOver) {
+      vlist = buildCommon.makeVList({
+        positionType: "firstBaseline",
+        children: [{
+          type: "elem",
+          elem: vSpan
+        }, {
+          type: "kern",
+          size: 0.2
+        }, {
+          type: "elem",
+          elem: supSubGroup
+        }]
+      }, options);
+    } else {
+      vlist = buildCommon.makeVList({
+        positionType: "bottom",
+        positionData: vSpan.depth + 0.2 + supSubGroup.height + supSubGroup.depth,
+        children: [{
+          type: "elem",
+          elem: supSubGroup
+        }, {
+          type: "kern",
+          size: 0.2
+        }, {
+          type: "elem",
+          elem: vSpan
+        }]
+      }, options);
+    }
+  }
+
+  return buildCommon.makeSpan(["mord", group.isOver ? "mover" : "munder"], [vlist], options);
+};
+
+var horizBrace_mathmlBuilder = function mathmlBuilder(group, options) {
+  var accentNode = stretchy.mathMLnode(group.label);
+  return new mathMLTree.MathNode(group.isOver ? "mover" : "munder", [buildMathML_buildGroup(group.base, options), accentNode]);
+}; // Horizontal stretchy braces
+
+
+defineFunction({
+  type: "horizBrace",
+  names: ["\\overbrace", "\\underbrace"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    return {
+      type: "horizBrace",
+      mode: parser.mode,
+      label: funcName,
+      isOver: /^\\over/.test(funcName),
+      base: args[0]
+    };
+  },
+  htmlBuilder: horizBrace_htmlBuilder,
+  mathmlBuilder: horizBrace_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/href.js
+
+
+
+
+
+
+defineFunction({
+  type: "href",
+  names: ["\\href"],
+  props: {
+    numArgs: 2,
+    argTypes: ["url", "original"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var body = args[1];
+    var href = assertNodeType(args[0], "url").url;
+
+    if (!parser.settings.isTrusted({
+      command: "\\href",
+      url: href
+    })) {
+      return parser.formatUnsupportedCmd("\\href");
+    }
+
+    return {
+      type: "href",
+      mode: parser.mode,
+      href: href,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var elements = buildHTML_buildExpression(group.body, options, false);
+    return buildCommon.makeAnchor(group.href, [], elements, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var math = buildExpressionRow(group.body, options);
+
+    if (!(math instanceof mathMLTree_MathNode)) {
+      math = new mathMLTree_MathNode("mrow", [math]);
+    }
+
+    math.setAttribute("href", group.href);
+    return math;
+  }
+});
+defineFunction({
+  type: "href",
+  names: ["\\url"],
+  props: {
+    numArgs: 1,
+    argTypes: ["url"],
+    allowedInText: true
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    var href = assertNodeType(args[0], "url").url;
+
+    if (!parser.settings.isTrusted({
+      command: "\\url",
+      url: href
+    })) {
+      return parser.formatUnsupportedCmd("\\url");
+    }
+
+    var chars = [];
+
+    for (var i = 0; i < href.length; i++) {
+      var c = href[i];
+
+      if (c === "~") {
+        c = "\\textasciitilde";
+      }
+
+      chars.push({
+        type: "textord",
+        mode: "text",
+        text: c
+      });
+    }
+
+    var body = {
+      type: "text",
+      mode: parser.mode,
+      font: "\\texttt",
+      body: chars
+    };
+    return {
+      type: "href",
+      mode: parser.mode,
+      href: href,
+      body: ordargument(body)
+    };
+  }
+});
+// CONCATENATED MODULE: ./src/functions/html.js
+
+
+
+
+
+
+defineFunction({
+  type: "html",
+  names: ["\\htmlClass", "\\htmlId", "\\htmlStyle", "\\htmlData"],
+  props: {
+    numArgs: 2,
+    argTypes: ["raw", "original"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName,
+        token = _ref.token;
+    var value = assertNodeType(args[0], "raw").string;
+    var body = args[1];
+
+    if (parser.settings.strict) {
+      parser.settings.reportNonstrict("htmlExtension", "HTML extension is disabled on strict mode");
+    }
+
+    var trustContext;
+    var attributes = {};
+
+    switch (funcName) {
+      case "\\htmlClass":
+        attributes.class = value;
+        trustContext = {
+          command: "\\htmlClass",
+          class: value
+        };
+        break;
+
+      case "\\htmlId":
+        attributes.id = value;
+        trustContext = {
+          command: "\\htmlId",
+          id: value
+        };
+        break;
+
+      case "\\htmlStyle":
+        attributes.style = value;
+        trustContext = {
+          command: "\\htmlStyle",
+          style: value
+        };
+        break;
+
+      case "\\htmlData":
+        {
+          var data = value.split(",");
+
+          for (var i = 0; i < data.length; i++) {
+            var keyVal = data[i].split("=");
+
+            if (keyVal.length !== 2) {
+              throw new src_ParseError("Error parsing key-value for \\htmlData");
+            }
+
+            attributes["data-" + keyVal[0].trim()] = keyVal[1].trim();
+          }
+
+          trustContext = {
+            command: "\\htmlData",
+            attributes: attributes
+          };
+          break;
+        }
+
+      default:
+        throw new Error("Unrecognized html command");
+    }
+
+    if (!parser.settings.isTrusted(trustContext)) {
+      return parser.formatUnsupportedCmd(funcName);
+    }
+
+    return {
+      type: "html",
+      mode: parser.mode,
+      attributes: attributes,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var elements = buildHTML_buildExpression(group.body, options, false);
+    var classes = ["enclosing"];
+
+    if (group.attributes.class) {
+      classes.push.apply(classes, group.attributes.class.trim().split(/\s+/));
+    }
+
+    var span = buildCommon.makeSpan(classes, elements, options);
+
+    for (var attr in group.attributes) {
+      if (attr !== "class" && group.attributes.hasOwnProperty(attr)) {
+        span.setAttribute(attr, group.attributes[attr]);
+      }
+    }
+
+    return span;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    return buildExpressionRow(group.body, options);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/htmlmathml.js
+
+
+
+
+defineFunction({
+  type: "htmlmathml",
+  names: ["\\html@mathml"],
+  props: {
+    numArgs: 2,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    return {
+      type: "htmlmathml",
+      mode: parser.mode,
+      html: ordargument(args[0]),
+      mathml: ordargument(args[1])
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var elements = buildHTML_buildExpression(group.html, options, false);
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    return buildExpressionRow(group.mathml, options);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/includegraphics.js
+
+
+
+
+
+
+
+var includegraphics_sizeData = function sizeData(str) {
+  if (/^[-+]? *(\d+(\.\d*)?|\.\d+)$/.test(str)) {
+    // str is a number with no unit specified.
+    // default unit is bp, per graphix package.
+    return {
+      number: +str,
+      unit: "bp"
+    };
+  } else {
+    var match = /([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(str);
+
+    if (!match) {
+      throw new src_ParseError("Invalid size: '" + str + "' in \\includegraphics");
+    }
+
+    var data = {
+      number: +(match[1] + match[2]),
+      // sign + magnitude, cast to number
+      unit: match[3]
+    };
+
+    if (!validUnit(data)) {
+      throw new src_ParseError("Invalid unit: '" + data.unit + "' in \\includegraphics.");
+    }
+
+    return data;
+  }
+};
+
+defineFunction({
+  type: "includegraphics",
+  names: ["\\includegraphics"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1,
+    argTypes: ["raw", "url"],
+    allowedInText: false
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser;
+    var width = {
+      number: 0,
+      unit: "em"
+    };
+    var height = {
+      number: 0.9,
+      unit: "em"
+    }; // sorta character sized.
+
+    var totalheight = {
+      number: 0,
+      unit: "em"
+    };
+    var alt = "";
+
+    if (optArgs[0]) {
+      var attributeStr = assertNodeType(optArgs[0], "raw").string; // Parser.js does not parse key/value pairs. We get a string.
+
+      var attributes = attributeStr.split(",");
+
+      for (var i = 0; i < attributes.length; i++) {
+        var keyVal = attributes[i].split("=");
+
+        if (keyVal.length === 2) {
+          var str = keyVal[1].trim();
+
+          switch (keyVal[0].trim()) {
+            case "alt":
+              alt = str;
+              break;
+
+            case "width":
+              width = includegraphics_sizeData(str);
+              break;
+
+            case "height":
+              height = includegraphics_sizeData(str);
+              break;
+
+            case "totalheight":
+              totalheight = includegraphics_sizeData(str);
+              break;
+
+            default:
+              throw new src_ParseError("Invalid key: '" + keyVal[0] + "' in \\includegraphics.");
+          }
+        }
+      }
+    }
+
+    var src = assertNodeType(args[0], "url").url;
+
+    if (alt === "") {
+      // No alt given. Use the file name. Strip away the path.
+      alt = src;
+      alt = alt.replace(/^.*[\\/]/, '');
+      alt = alt.substring(0, alt.lastIndexOf('.'));
+    }
+
+    if (!parser.settings.isTrusted({
+      command: "\\includegraphics",
+      url: src
+    })) {
+      return parser.formatUnsupportedCmd("\\includegraphics");
+    }
+
+    return {
+      type: "includegraphics",
+      mode: parser.mode,
+      alt: alt,
+      width: width,
+      height: height,
+      totalheight: totalheight,
+      src: src
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var height = units_calculateSize(group.height, options);
+    var depth = 0;
+
+    if (group.totalheight.number > 0) {
+      depth = units_calculateSize(group.totalheight, options) - height;
+      depth = Number(depth.toFixed(2));
+    }
+
+    var width = 0;
+
+    if (group.width.number > 0) {
+      width = units_calculateSize(group.width, options);
+    }
+
+    var style = {
+      height: height + depth + "em"
+    };
+
+    if (width > 0) {
+      style.width = width + "em";
+    }
+
+    if (depth > 0) {
+      style.verticalAlign = -depth + "em";
+    }
+
+    var node = new domTree_Img(group.src, group.alt, style);
+    node.height = height;
+    node.depth = depth;
+    return node;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mglyph", []);
+    node.setAttribute("alt", group.alt);
+    var height = units_calculateSize(group.height, options);
+    var depth = 0;
+
+    if (group.totalheight.number > 0) {
+      depth = units_calculateSize(group.totalheight, options) - height;
+      depth = depth.toFixed(2);
+      node.setAttribute("valign", "-" + depth + "em");
+    }
+
+    node.setAttribute("height", height + depth + "em");
+
+    if (group.width.number > 0) {
+      var width = units_calculateSize(group.width, options);
+      node.setAttribute("width", width + "em");
+    }
+
+    node.setAttribute("src", group.src);
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/kern.js
+// Horizontal spacing commands
+
+
+
+
+ // TODO: \hskip and \mskip should support plus and minus in lengths
+
+defineFunction({
+  type: "kern",
+  names: ["\\kern", "\\mkern", "\\hskip", "\\mskip"],
+  props: {
+    numArgs: 1,
+    argTypes: ["size"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var size = assertNodeType(args[0], "size");
+
+    if (parser.settings.strict) {
+      var mathFunction = funcName[1] === 'm'; // \mkern, \mskip
+
+      var muUnit = size.value.unit === 'mu';
+
+      if (mathFunction) {
+        if (!muUnit) {
+          parser.settings.reportNonstrict("mathVsTextUnits", "LaTeX's " + funcName + " supports only mu units, " + ("not " + size.value.unit + " units"));
+        }
+
+        if (parser.mode !== "math") {
+          parser.settings.reportNonstrict("mathVsTextUnits", "LaTeX's " + funcName + " works only in math mode");
+        }
+      } else {
+        // !mathFunction
+        if (muUnit) {
+          parser.settings.reportNonstrict("mathVsTextUnits", "LaTeX's " + funcName + " doesn't support mu units");
+        }
+      }
+    }
+
+    return {
+      type: "kern",
+      mode: parser.mode,
+      dimension: size.value
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    return buildCommon.makeGlue(group.dimension, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var dimension = units_calculateSize(group.dimension, options);
+    return new mathMLTree.SpaceNode(dimension);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/lap.js
+// Horizontal overlap functions
+
+
+
+
+
+defineFunction({
+  type: "lap",
+  names: ["\\mathllap", "\\mathrlap", "\\mathclap"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    return {
+      type: "lap",
+      mode: parser.mode,
+      alignment: funcName.slice(5),
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // mathllap, mathrlap, mathclap
+    var inner;
+
+    if (group.alignment === "clap") {
+      // ref: https://www.math.lsu.edu/~aperlis/publications/mathclap/
+      inner = buildCommon.makeSpan([], [buildHTML_buildGroup(group.body, options)]); // wrap, since CSS will center a .clap > .inner > span
+
+      inner = buildCommon.makeSpan(["inner"], [inner], options);
+    } else {
+      inner = buildCommon.makeSpan(["inner"], [buildHTML_buildGroup(group.body, options)]);
+    }
+
+    var fix = buildCommon.makeSpan(["fix"], []);
+    var node = buildCommon.makeSpan([group.alignment], [inner, fix], options); // At this point, we have correctly set horizontal alignment of the
+    // two items involved in the lap.
+    // Next, use a strut to set the height of the HTML bounding box.
+    // Otherwise, a tall argument may be misplaced.
+    // This code resolved issue #1153
+
+    var strut = buildCommon.makeSpan(["strut"]);
+    strut.style.height = node.height + node.depth + "em";
+    strut.style.verticalAlign = -node.depth + "em";
+    node.children.unshift(strut); // Next, prevent vertical misplacement when next to something tall.
+    // This code resolves issue #1234
+
+    node = buildCommon.makeSpan(["thinbox"], [node], options);
+    return buildCommon.makeSpan(["mord", "vbox"], [node], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    // mathllap, mathrlap, mathclap
+    var node = new mathMLTree.MathNode("mpadded", [buildMathML_buildGroup(group.body, options)]);
+
+    if (group.alignment !== "rlap") {
+      var offset = group.alignment === "llap" ? "-1" : "-0.5";
+      node.setAttribute("lspace", offset + "width");
+    }
+
+    node.setAttribute("width", "0px");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/math.js
+
+ // Switching from text mode back to math mode
+
+defineFunction({
+  type: "styling",
+  names: ["\\(", "$"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: false
+  },
+  handler: function handler(_ref, args) {
+    var funcName = _ref.funcName,
+        parser = _ref.parser;
+    var outerMode = parser.mode;
+    parser.switchMode("math");
+    var close = funcName === "\\(" ? "\\)" : "$";
+    var body = parser.parseExpression(false, close);
+    parser.expect(close);
+    parser.switchMode(outerMode);
+    return {
+      type: "styling",
+      mode: parser.mode,
+      style: "text",
+      body: body
+    };
+  }
+}); // Check for extra closing math delimiters
+
+defineFunction({
+  type: "text",
+  // Doesn't matter what this is.
+  names: ["\\)", "\\]"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: false
+  },
+  handler: function handler(context, args) {
+    throw new src_ParseError("Mismatched " + context.funcName);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/mathchoice.js
+
+
+
+
+
+
+var mathchoice_chooseMathStyle = function chooseMathStyle(group, options) {
+  switch (options.style.size) {
+    case src_Style.DISPLAY.size:
+      return group.display;
+
+    case src_Style.TEXT.size:
+      return group.text;
+
+    case src_Style.SCRIPT.size:
+      return group.script;
+
+    case src_Style.SCRIPTSCRIPT.size:
+      return group.scriptscript;
+
+    default:
+      return group.text;
+  }
+};
+
+defineFunction({
+  type: "mathchoice",
+  names: ["\\mathchoice"],
+  props: {
+    numArgs: 4
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    return {
+      type: "mathchoice",
+      mode: parser.mode,
+      display: ordargument(args[0]),
+      text: ordargument(args[1]),
+      script: ordargument(args[2]),
+      scriptscript: ordargument(args[3])
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var body = mathchoice_chooseMathStyle(group, options);
+    var elements = buildHTML_buildExpression(body, options, false);
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var body = mathchoice_chooseMathStyle(group, options);
+    return buildExpressionRow(body, options);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/utils/assembleSupSub.js
+
+
+// For an operator with limits, assemble the base, sup, and sub into a span.
+var assembleSupSub_assembleSupSub = function assembleSupSub(base, supGroup, subGroup, options, style, slant, baseShift) {
+  base = buildCommon.makeSpan([], [base]);
+  var sub;
+  var sup; // We manually have to handle the superscripts and subscripts. This,
+  // aside from the kern calculations, is copied from supsub.
+
+  if (supGroup) {
+    var elem = buildHTML_buildGroup(supGroup, options.havingStyle(style.sup()), options);
+    sup = {
+      elem: elem,
+      kern: Math.max(options.fontMetrics().bigOpSpacing1, options.fontMetrics().bigOpSpacing3 - elem.depth)
+    };
+  }
+
+  if (subGroup) {
+    var _elem = buildHTML_buildGroup(subGroup, options.havingStyle(style.sub()), options);
+
+    sub = {
+      elem: _elem,
+      kern: Math.max(options.fontMetrics().bigOpSpacing2, options.fontMetrics().bigOpSpacing4 - _elem.height)
+    };
+  } // Build the final group as a vlist of the possible subscript, base,
+  // and possible superscript.
+
+
+  var finalGroup;
+
+  if (sup && sub) {
+    var bottom = options.fontMetrics().bigOpSpacing5 + sub.elem.height + sub.elem.depth + sub.kern + base.depth + baseShift;
+    finalGroup = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: bottom,
+      children: [{
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }, {
+        type: "elem",
+        elem: sub.elem,
+        marginLeft: -slant + "em"
+      }, {
+        type: "kern",
+        size: sub.kern
+      }, {
+        type: "elem",
+        elem: base
+      }, {
+        type: "kern",
+        size: sup.kern
+      }, {
+        type: "elem",
+        elem: sup.elem,
+        marginLeft: slant + "em"
+      }, {
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }]
+    }, options);
+  } else if (sub) {
+    var top = base.height - baseShift; // Shift the limits by the slant of the symbol. Note
+    // that we are supposed to shift the limits by 1/2 of the slant,
+    // but since we are centering the limits adding a full slant of
+    // margin will shift by 1/2 that.
+
+    finalGroup = buildCommon.makeVList({
+      positionType: "top",
+      positionData: top,
+      children: [{
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }, {
+        type: "elem",
+        elem: sub.elem,
+        marginLeft: -slant + "em"
+      }, {
+        type: "kern",
+        size: sub.kern
+      }, {
+        type: "elem",
+        elem: base
+      }]
+    }, options);
+  } else if (sup) {
+    var _bottom = base.depth + baseShift;
+
+    finalGroup = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: _bottom,
+      children: [{
+        type: "elem",
+        elem: base
+      }, {
+        type: "kern",
+        size: sup.kern
+      }, {
+        type: "elem",
+        elem: sup.elem,
+        marginLeft: slant + "em"
+      }, {
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }]
+    }, options);
+  } else {
+    // This case probably shouldn't occur (this would mean the
+    // supsub was sending us a group with no superscript or
+    // subscript) but be safe.
+    return base;
+  }
+
+  return buildCommon.makeSpan(["mop", "op-limits"], [finalGroup], options);
+};
+// CONCATENATED MODULE: ./src/functions/op.js
+// Limits, symbols
+
+
+
+
+
+
+
+
+
+
+// Most operators have a large successor symbol, but these don't.
+var noSuccessor = ["\\smallint"]; // NOTE: Unlike most `htmlBuilder`s, this one handles not only "op", but also
+// "supsub" since some of them (like \int) can affect super/subscripting.
+
+var op_htmlBuilder = function htmlBuilder(grp, options) {
+  // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+  var supGroup;
+  var subGroup;
+  var hasLimits = false;
+  var group;
+
+  if (grp.type === "supsub") {
+    // If we have limits, supsub will pass us its group to handle. Pull
+    // out the superscript and subscript and set the group to the op in
+    // its base.
+    supGroup = grp.sup;
+    subGroup = grp.sub;
+    group = assertNodeType(grp.base, "op");
+    hasLimits = true;
+  } else {
+    group = assertNodeType(grp, "op");
+  }
+
+  var style = options.style;
+  var large = false;
+
+  if (style.size === src_Style.DISPLAY.size && group.symbol && !utils.contains(noSuccessor, group.name)) {
+    // Most symbol operators get larger in displaystyle (rule 13)
+    large = true;
+  }
+
+  var base;
+
+  if (group.symbol) {
+    // If this is a symbol, create the symbol.
+    var fontName = large ? "Size2-Regular" : "Size1-Regular";
+    var stash = "";
+
+    if (group.name === "\\oiint" || group.name === "\\oiiint") {
+      // No font glyphs yet, so use a glyph w/o the oval.
+      // TODO: When font glyphs are available, delete this code.
+      stash = group.name.substr(1); // $FlowFixMe
+
+      group.name = stash === "oiint" ? "\\iint" : "\\iiint";
+    }
+
+    base = buildCommon.makeSymbol(group.name, fontName, "math", options, ["mop", "op-symbol", large ? "large-op" : "small-op"]);
+
+    if (stash.length > 0) {
+      // We're in \oiint or \oiiint. Overlay the oval.
+      // TODO: When font glyphs are available, delete this code.
+      var italic = base.italic;
+      var oval = buildCommon.staticSvg(stash + "Size" + (large ? "2" : "1"), options);
+      base = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: base,
+          shift: 0
+        }, {
+          type: "elem",
+          elem: oval,
+          shift: large ? 0.08 : 0
+        }]
+      }, options); // $FlowFixMe
+
+      group.name = "\\" + stash;
+      base.classes.unshift("mop"); // $FlowFixMe
+
+      base.italic = italic;
+    }
+  } else if (group.body) {
+    // If this is a list, compose that list.
+    var inner = buildHTML_buildExpression(group.body, options, true);
+
+    if (inner.length === 1 && inner[0] instanceof domTree_SymbolNode) {
+      base = inner[0];
+      base.classes[0] = "mop"; // replace old mclass
+    } else {
+      base = buildCommon.makeSpan(["mop"], buildCommon.tryCombineChars(inner), options);
+    }
+  } else {
+    // Otherwise, this is a text operator. Build the text from the
+    // operator's name.
+    // TODO(emily): Add a space in the middle of some of these
+    // operators, like \limsup
+    var output = [];
+
+    for (var i = 1; i < group.name.length; i++) {
+      output.push(buildCommon.mathsym(group.name[i], group.mode, options));
+    }
+
+    base = buildCommon.makeSpan(["mop"], output, options);
+  } // If content of op is a single symbol, shift it vertically.
+
+
+  var baseShift = 0;
+  var slant = 0;
+
+  if ((base instanceof domTree_SymbolNode || group.name === "\\oiint" || group.name === "\\oiiint") && !group.suppressBaseShift) {
+    // We suppress the shift of the base of \overset and \underset. Otherwise,
+    // shift the symbol so its center lies on the axis (rule 13). It
+    // appears that our fonts have the centers of the symbols already
+    // almost on the axis, so these numbers are very small. Note we
+    // don't actually apply this here, but instead it is used either in
+    // the vlist creation or separately when there are no limits.
+    baseShift = (base.height - base.depth) / 2 - options.fontMetrics().axisHeight; // The slant of the symbol is just its italic correction.
+    // $FlowFixMe
+
+    slant = base.italic;
+  }
+
+  if (hasLimits) {
+    return assembleSupSub_assembleSupSub(base, supGroup, subGroup, options, style, slant, baseShift);
+  } else {
+    if (baseShift) {
+      base.style.position = "relative";
+      base.style.top = baseShift + "em";
+    }
+
+    return base;
+  }
+};
+
+var op_mathmlBuilder = function mathmlBuilder(group, options) {
+  var node;
+
+  if (group.symbol) {
+    // This is a symbol. Just add the symbol.
+    node = new mathMLTree_MathNode("mo", [buildMathML_makeText(group.name, group.mode)]);
+
+    if (utils.contains(noSuccessor, group.name)) {
+      node.setAttribute("largeop", "false");
+    }
+  } else if (group.body) {
+    // This is an operator with children. Add them.
+    node = new mathMLTree_MathNode("mo", buildMathML_buildExpression(group.body, options));
+  } else {
+    // This is a text operator. Add all of the characters from the
+    // operator's name.
+    node = new mathMLTree_MathNode("mi", [new mathMLTree_TextNode(group.name.slice(1))]); // Append an <mo>&ApplyFunction;</mo>.
+    // ref: https://www.w3.org/TR/REC-MathML/chap3_2.html#sec3.2.4
+
+    var operator = new mathMLTree_MathNode("mo", [buildMathML_makeText("\u2061", "text")]);
+
+    if (group.parentIsSupSub) {
+      node = new mathMLTree_MathNode("mo", [node, operator]);
+    } else {
+      node = newDocumentFragment([node, operator]);
+    }
+  }
+
+  return node;
+};
+
+var singleCharBigOps = {
+  "\u220F": "\\prod",
+  "\u2210": "\\coprod",
+  "\u2211": "\\sum",
+  "\u22C0": "\\bigwedge",
+  "\u22C1": "\\bigvee",
+  "\u22C2": "\\bigcap",
+  "\u22C3": "\\bigcup",
+  "\u2A00": "\\bigodot",
+  "\u2A01": "\\bigoplus",
+  "\u2A02": "\\bigotimes",
+  "\u2A04": "\\biguplus",
+  "\u2A06": "\\bigsqcup"
+};
+defineFunction({
+  type: "op",
+  names: ["\\coprod", "\\bigvee", "\\bigwedge", "\\biguplus", "\\bigcap", "\\bigcup", "\\intop", "\\prod", "\\sum", "\\bigotimes", "\\bigoplus", "\\bigodot", "\\bigsqcup", "\\smallint", "\u220F", "\u2210", "\u2211", "\u22C0", "\u22C1", "\u22C2", "\u22C3", "\u2A00", "\u2A01", "\u2A02", "\u2A04", "\u2A06"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var fName = funcName;
+
+    if (fName.length === 1) {
+      fName = singleCharBigOps[fName];
+    }
+
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: true,
+      parentIsSupSub: false,
+      symbol: true,
+      name: fName
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+}); // Note: calling defineFunction with a type that's already been defined only
+// works because the same htmlBuilder and mathmlBuilder are being used.
+
+defineFunction({
+  type: "op",
+  names: ["\\mathop"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    var body = args[0];
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: false,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+}); // There are 2 flags for operators; whether they produce limits in
+// displaystyle, and whether they are symbols and should grow in
+// displaystyle. These four groups cover the four possible choices.
+
+var singleCharIntegrals = {
+  "\u222B": "\\int",
+  "\u222C": "\\iint",
+  "\u222D": "\\iiint",
+  "\u222E": "\\oint",
+  "\u222F": "\\oiint",
+  "\u2230": "\\oiiint"
+}; // No limits, not symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\arcsin", "\\arccos", "\\arctan", "\\arctg", "\\arcctg", "\\arg", "\\ch", "\\cos", "\\cosec", "\\cosh", "\\cot", "\\cotg", "\\coth", "\\csc", "\\ctg", "\\cth", "\\deg", "\\dim", "\\exp", "\\hom", "\\ker", "\\lg", "\\ln", "\\log", "\\sec", "\\sin", "\\sinh", "\\sh", "\\tan", "\\tanh", "\\tg", "\\th"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(_ref3) {
+    var parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: false,
+      name: funcName
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+}); // Limits, not symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\det", "\\gcd", "\\inf", "\\lim", "\\max", "\\min", "\\Pr", "\\sup"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(_ref4) {
+    var parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: true,
+      parentIsSupSub: false,
+      symbol: false,
+      name: funcName
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+}); // No limits, symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\int", "\\iint", "\\iiint", "\\oint", "\\oiint", "\\oiiint", "\u222B", "\u222C", "\u222D", "\u222E", "\u222F", "\u2230"],
+  props: {
+    numArgs: 0
+  },
+  handler: function handler(_ref5) {
+    var parser = _ref5.parser,
+        funcName = _ref5.funcName;
+    var fName = funcName;
+
+    if (fName.length === 1) {
+      fName = singleCharIntegrals[fName];
+    }
+
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: true,
+      name: fName
+    };
+  },
+  htmlBuilder: op_htmlBuilder,
+  mathmlBuilder: op_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/operatorname.js
+
+
+
+
+
+
+
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only
+// "operatorname", but also  "supsub" since \operatorname* can
+var operatorname_htmlBuilder = function htmlBuilder(grp, options) {
+  // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+  var supGroup;
+  var subGroup;
+  var hasLimits = false;
+  var group;
+
+  if (grp.type === "supsub") {
+    // If we have limits, supsub will pass us its group to handle. Pull
+    // out the superscript and subscript and set the group to the op in
+    // its base.
+    supGroup = grp.sup;
+    subGroup = grp.sub;
+    group = assertNodeType(grp.base, "operatorname");
+    hasLimits = true;
+  } else {
+    group = assertNodeType(grp, "operatorname");
+  }
+
+  var base;
+
+  if (group.body.length > 0) {
+    var body = group.body.map(function (child) {
+      // $FlowFixMe: Check if the node has a string `text` property.
+      var childText = child.text;
+
+      if (typeof childText === "string") {
+        return {
+          type: "textord",
+          mode: child.mode,
+          text: childText
+        };
+      } else {
+        return child;
+      }
+    }); // Consolidate function names into symbol characters.
+
+    var expression = buildHTML_buildExpression(body, options.withFont("mathrm"), true);
+
+    for (var i = 0; i < expression.length; i++) {
+      var child = expression[i];
+
+      if (child instanceof domTree_SymbolNode) {
+        // Per amsopn package,
+        // change minus to hyphen and \ast to asterisk
+        child.text = child.text.replace(/\u2212/, "-").replace(/\u2217/, "*");
+      }
+    }
+
+    base = buildCommon.makeSpan(["mop"], expression, options);
+  } else {
+    base = buildCommon.makeSpan(["mop"], [], options);
+  }
+
+  if (hasLimits) {
+    return assembleSupSub_assembleSupSub(base, supGroup, subGroup, options, options.style, 0, 0);
+  } else {
+    return base;
+  }
+};
+
+var operatorname_mathmlBuilder = function mathmlBuilder(group, options) {
+  // The steps taken here are similar to the html version.
+  var expression = buildMathML_buildExpression(group.body, options.withFont("mathrm")); // Is expression a string or has it something like a fraction?
+
+  var isAllString = true; // default
+
+  for (var i = 0; i < expression.length; i++) {
+    var node = expression[i];
+
+    if (node instanceof mathMLTree.SpaceNode) {// Do nothing
+    } else if (node instanceof mathMLTree.MathNode) {
+      switch (node.type) {
+        case "mi":
+        case "mn":
+        case "ms":
+        case "mspace":
+        case "mtext":
+          break;
+        // Do nothing yet.
+
+        case "mo":
+          {
+            var child = node.children[0];
+
+            if (node.children.length === 1 && child instanceof mathMLTree.TextNode) {
+              child.text = child.text.replace(/\u2212/, "-").replace(/\u2217/, "*");
+            } else {
+              isAllString = false;
+            }
+
+            break;
+          }
+
+        default:
+          isAllString = false;
+      }
+    } else {
+      isAllString = false;
+    }
+  }
+
+  if (isAllString) {
+    // Write a single TextNode instead of multiple nested tags.
+    var word = expression.map(function (node) {
+      return node.toText();
+    }).join("");
+    expression = [new mathMLTree.TextNode(word)];
+  }
+
+  var identifier = new mathMLTree.MathNode("mi", expression);
+  identifier.setAttribute("mathvariant", "normal"); // \u2061 is the same as &ApplyFunction;
+  // ref: https://www.w3schools.com/charsets/ref_html_entities_a.asp
+
+  var operator = new mathMLTree.MathNode("mo", [buildMathML_makeText("\u2061", "text")]);
+
+  if (group.parentIsSupSub) {
+    return new mathMLTree.MathNode("mo", [identifier, operator]);
+  } else {
+    return mathMLTree.newDocumentFragment([identifier, operator]);
+  }
+}; // \operatorname
+// amsopn.dtx: \mathop{#1\kern\z@\operator@font#3}\newmcodes@
+
+
+defineFunction({
+  type: "operatorname",
+  names: ["\\operatorname", "\\operatorname*"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    return {
+      type: "operatorname",
+      mode: parser.mode,
+      body: ordargument(body),
+      alwaysHandleSupSub: funcName === "\\operatorname*",
+      limits: false,
+      parentIsSupSub: false
+    };
+  },
+  htmlBuilder: operatorname_htmlBuilder,
+  mathmlBuilder: operatorname_mathmlBuilder
+});
+// CONCATENATED MODULE: ./src/functions/ordgroup.js
+
+
+
+
+defineFunctionBuilders({
+  type: "ordgroup",
+  htmlBuilder: function htmlBuilder(group, options) {
+    if (group.semisimple) {
+      return buildCommon.makeFragment(buildHTML_buildExpression(group.body, options, false));
+    }
+
+    return buildCommon.makeSpan(["mord"], buildHTML_buildExpression(group.body, options, true), options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    return buildExpressionRow(group.body, options, true);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/overline.js
+
+
+
+
+
+defineFunction({
+  type: "overline",
+  names: ["\\overline"],
+  props: {
+    numArgs: 1
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var body = args[0];
+    return {
+      type: "overline",
+      mode: parser.mode,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Overlines are handled in the TeXbook pg 443, Rule 9.
+    // Build the inner group in the cramped style.
+    var innerGroup = buildHTML_buildGroup(group.body, options.havingCrampedStyle()); // Create the line above the body
+
+    var line = buildCommon.makeLineSpan("overline-line", options); // Generate the vlist, with the appropriate kerns
+
+    var defaultRuleThickness = options.fontMetrics().defaultRuleThickness;
+    var vlist = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: innerGroup
+      }, {
+        type: "kern",
+        size: 3 * defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: line
+      }, {
+        type: "kern",
+        size: defaultRuleThickness
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "overline"], [vlist], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var operator = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode("\u203E")]);
+    operator.setAttribute("stretchy", "true");
+    var node = new mathMLTree.MathNode("mover", [buildMathML_buildGroup(group.body, options), operator]);
+    node.setAttribute("accent", "true");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/phantom.js
+
+
+
+
+
+defineFunction({
+  type: "phantom",
+  names: ["\\phantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var body = args[0];
+    return {
+      type: "phantom",
+      mode: parser.mode,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var elements = buildHTML_buildExpression(group.body, options.withPhantom(), false); // \phantom isn't supposed to affect the elements it contains.
+    // See "color" for more details.
+
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var inner = buildMathML_buildExpression(group.body, options);
+    return new mathMLTree.MathNode("mphantom", inner);
+  }
+});
+defineFunction({
+  type: "hphantom",
+  names: ["\\hphantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref2, args) {
+    var parser = _ref2.parser;
+    var body = args[0];
+    return {
+      type: "hphantom",
+      mode: parser.mode,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var node = buildCommon.makeSpan([], [buildHTML_buildGroup(group.body, options.withPhantom())]);
+    node.height = 0;
+    node.depth = 0;
+
+    if (node.children) {
+      for (var i = 0; i < node.children.length; i++) {
+        node.children[i].height = 0;
+        node.children[i].depth = 0;
+      }
+    } // See smash for comment re: use of makeVList
+
+
+    node = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: node
+      }]
+    }, options); // For spacing, TeX treats \smash as a math group (same spacing as ord).
+
+    return buildCommon.makeSpan(["mord"], [node], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var inner = buildMathML_buildExpression(ordargument(group.body), options);
+    var phantom = new mathMLTree.MathNode("mphantom", inner);
+    var node = new mathMLTree.MathNode("mpadded", [phantom]);
+    node.setAttribute("height", "0px");
+    node.setAttribute("depth", "0px");
+    return node;
+  }
+});
+defineFunction({
+  type: "vphantom",
+  names: ["\\vphantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref3, args) {
+    var parser = _ref3.parser;
+    var body = args[0];
+    return {
+      type: "vphantom",
+      mode: parser.mode,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var inner = buildCommon.makeSpan(["inner"], [buildHTML_buildGroup(group.body, options.withPhantom())]);
+    var fix = buildCommon.makeSpan(["fix"], []);
+    return buildCommon.makeSpan(["mord", "rlap"], [inner, fix], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var inner = buildMathML_buildExpression(ordargument(group.body), options);
+    var phantom = new mathMLTree.MathNode("mphantom", inner);
+    var node = new mathMLTree.MathNode("mpadded", [phantom]);
+    node.setAttribute("width", "0px");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/raisebox.js
+
+
+
+
+
+
+ // Box manipulation
+
+defineFunction({
+  type: "raisebox",
+  names: ["\\raisebox"],
+  props: {
+    numArgs: 2,
+    argTypes: ["size", "hbox"],
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    var amount = assertNodeType(args[0], "size").value;
+    var body = args[1];
+    return {
+      type: "raisebox",
+      mode: parser.mode,
+      dy: amount,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var body = buildHTML_buildGroup(group.body, options);
+    var dy = units_calculateSize(group.dy, options);
+    return buildCommon.makeVList({
+      positionType: "shift",
+      positionData: -dy,
+      children: [{
+        type: "elem",
+        elem: body
+      }]
+    }, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mpadded", [buildMathML_buildGroup(group.body, options)]);
+    var dy = group.dy.number + group.dy.unit;
+    node.setAttribute("voffset", dy);
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/rule.js
+
+
+
+
+
+defineFunction({
+  type: "rule",
+  names: ["\\rule"],
+  props: {
+    numArgs: 2,
+    numOptionalArgs: 1,
+    argTypes: ["size", "size", "size"]
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser;
+    var shift = optArgs[0];
+    var width = assertNodeType(args[0], "size");
+    var height = assertNodeType(args[1], "size");
+    return {
+      type: "rule",
+      mode: parser.mode,
+      shift: shift && assertNodeType(shift, "size").value,
+      width: width.value,
+      height: height.value
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Make an empty span for the rule
+    var rule = buildCommon.makeSpan(["mord", "rule"], [], options); // Calculate the shift, width, and height of the rule, and account for units
+
+    var width = units_calculateSize(group.width, options);
+    var height = units_calculateSize(group.height, options);
+    var shift = group.shift ? units_calculateSize(group.shift, options) : 0; // Style the rule to the right size
+
+    rule.style.borderRightWidth = width + "em";
+    rule.style.borderTopWidth = height + "em";
+    rule.style.bottom = shift + "em"; // Record the height and width
+
+    rule.width = width;
+    rule.height = height + shift;
+    rule.depth = -shift; // Font size is the number large enough that the browser will
+    // reserve at least `absHeight` space above the baseline.
+    // The 1.125 factor was empirically determined
+
+    rule.maxFontSize = height * 1.125 * options.sizeMultiplier;
+    return rule;
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var width = units_calculateSize(group.width, options);
+    var height = units_calculateSize(group.height, options);
+    var shift = group.shift ? units_calculateSize(group.shift, options) : 0;
+    var color = options.color && options.getColor() || "black";
+    var rule = new mathMLTree.MathNode("mspace");
+    rule.setAttribute("mathbackground", color);
+    rule.setAttribute("width", width + "em");
+    rule.setAttribute("height", height + "em");
+    var wrapper = new mathMLTree.MathNode("mpadded", [rule]);
+
+    if (shift >= 0) {
+      wrapper.setAttribute("height", "+" + shift + "em");
+    } else {
+      wrapper.setAttribute("height", shift + "em");
+      wrapper.setAttribute("depth", "+" + -shift + "em");
+    }
+
+    wrapper.setAttribute("voffset", shift + "em");
+    return wrapper;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/sizing.js
+
+
+
+
+
+function sizingGroup(value, options, baseOptions) {
+  var inner = buildHTML_buildExpression(value, options, false);
+  var multiplier = options.sizeMultiplier / baseOptions.sizeMultiplier; // Add size-resetting classes to the inner list and set maxFontSize
+  // manually. Handle nested size changes.
+
+  for (var i = 0; i < inner.length; i++) {
+    var pos = inner[i].classes.indexOf("sizing");
+
+    if (pos < 0) {
+      Array.prototype.push.apply(inner[i].classes, options.sizingClasses(baseOptions));
+    } else if (inner[i].classes[pos + 1] === "reset-size" + options.size) {
+      // This is a nested size change: e.g., inner[i] is the "b" in
+      // `\Huge a \small b`. Override the old size (the `reset-` class)
+      // but not the new size.
+      inner[i].classes[pos + 1] = "reset-size" + baseOptions.size;
+    }
+
+    inner[i].height *= multiplier;
+    inner[i].depth *= multiplier;
+  }
+
+  return buildCommon.makeFragment(inner);
+}
+var sizeFuncs = ["\\tiny", "\\sixptsize", "\\scriptsize", "\\footnotesize", "\\small", "\\normalsize", "\\large", "\\Large", "\\LARGE", "\\huge", "\\Huge"];
+var sizing_htmlBuilder = function htmlBuilder(group, options) {
+  // Handle sizing operators like \Huge. Real TeX doesn't actually allow
+  // these functions inside of math expressions, so we do some special
+  // handling.
+  var newOptions = options.havingSize(group.size);
+  return sizingGroup(group.body, newOptions, options);
+};
+defineFunction({
+  type: "sizing",
+  names: sizeFuncs,
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var breakOnTokenText = _ref.breakOnTokenText,
+        funcName = _ref.funcName,
+        parser = _ref.parser;
+    var body = parser.parseExpression(false, breakOnTokenText);
+    return {
+      type: "sizing",
+      mode: parser.mode,
+      // Figure out what size to use based on the list of functions above
+      size: sizeFuncs.indexOf(funcName) + 1,
+      body: body
+    };
+  },
+  htmlBuilder: sizing_htmlBuilder,
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var newOptions = options.havingSize(group.size);
+    var inner = buildMathML_buildExpression(group.body, newOptions);
+    var node = new mathMLTree.MathNode("mstyle", inner); // TODO(emily): This doesn't produce the correct size for nested size
+    // changes, because we don't keep state of what style we're currently
+    // in, so we can't reset the size to normal before changing it.  Now
+    // that we're passing an options parameter we should be able to fix
+    // this.
+
+    node.setAttribute("mathsize", newOptions.sizeMultiplier + "em");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/smash.js
+// smash, with optional [tb], as in AMS
+
+
+
+
+
+
+defineFunction({
+  type: "smash",
+  names: ["\\smash"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser;
+    var smashHeight = false;
+    var smashDepth = false;
+    var tbArg = optArgs[0] && assertNodeType(optArgs[0], "ordgroup");
+
+    if (tbArg) {
+      // Optional [tb] argument is engaged.
+      // ref: amsmath: \renewcommand{\smash}[1][tb]{%
+      //               def\mb@t{\ht}\def\mb@b{\dp}\def\mb@tb{\ht\z@\z@\dp}%
+      var letter = "";
+
+      for (var i = 0; i < tbArg.body.length; ++i) {
+        var node = tbArg.body[i]; // $FlowFixMe: Not every node type has a `text` property.
+
+        letter = node.text;
+
+        if (letter === "t") {
+          smashHeight = true;
+        } else if (letter === "b") {
+          smashDepth = true;
+        } else {
+          smashHeight = false;
+          smashDepth = false;
+          break;
+        }
+      }
+    } else {
+      smashHeight = true;
+      smashDepth = true;
+    }
+
+    var body = args[0];
+    return {
+      type: "smash",
+      mode: parser.mode,
+      body: body,
+      smashHeight: smashHeight,
+      smashDepth: smashDepth
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var node = buildCommon.makeSpan([], [buildHTML_buildGroup(group.body, options)]);
+
+    if (!group.smashHeight && !group.smashDepth) {
+      return node;
+    }
+
+    if (group.smashHeight) {
+      node.height = 0; // In order to influence makeVList, we have to reset the children.
+
+      if (node.children) {
+        for (var i = 0; i < node.children.length; i++) {
+          node.children[i].height = 0;
+        }
+      }
+    }
+
+    if (group.smashDepth) {
+      node.depth = 0;
+
+      if (node.children) {
+        for (var _i = 0; _i < node.children.length; _i++) {
+          node.children[_i].depth = 0;
+        }
+      }
+    } // At this point, we've reset the TeX-like height and depth values.
+    // But the span still has an HTML line height.
+    // makeVList applies "display: table-cell", which prevents the browser
+    // from acting on that line height. So we'll call makeVList now.
+
+
+    var smashedNode = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: node
+      }]
+    }, options); // For spacing, TeX treats \hphantom as a math group (same spacing as ord).
+
+    return buildCommon.makeSpan(["mord"], [smashedNode], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mpadded", [buildMathML_buildGroup(group.body, options)]);
+
+    if (group.smashHeight) {
+      node.setAttribute("height", "0px");
+    }
+
+    if (group.smashDepth) {
+      node.setAttribute("depth", "0px");
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/sqrt.js
+
+
+
+
+
+
+
+defineFunction({
+  type: "sqrt",
+  names: ["\\sqrt"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1
+  },
+  handler: function handler(_ref, args, optArgs) {
+    var parser = _ref.parser;
+    var index = optArgs[0];
+    var body = args[0];
+    return {
+      type: "sqrt",
+      mode: parser.mode,
+      body: body,
+      index: index
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Square roots are handled in the TeXbook pg. 443, Rule 11.
+    // First, we do the same steps as in overline to build the inner group
+    // and line
+    var inner = buildHTML_buildGroup(group.body, options.havingCrampedStyle());
+
+    if (inner.height === 0) {
+      // Render a small surd.
+      inner.height = options.fontMetrics().xHeight;
+    } // Some groups can return document fragments.  Handle those by wrapping
+    // them in a span.
+
+
+    inner = buildCommon.wrapFragment(inner, options); // Calculate the minimum size for the \surd delimiter
+
+    var metrics = options.fontMetrics();
+    var theta = metrics.defaultRuleThickness;
+    var phi = theta;
+
+    if (options.style.id < src_Style.TEXT.id) {
+      phi = options.fontMetrics().xHeight;
+    } // Calculate the clearance between the body and line
+
+
+    var lineClearance = theta + phi / 4;
+    var minDelimiterHeight = inner.height + inner.depth + lineClearance + theta; // Create a sqrt SVG of the required minimum size
+
+    var _delimiter$sqrtImage = delimiter.sqrtImage(minDelimiterHeight, options),
+        img = _delimiter$sqrtImage.span,
+        ruleWidth = _delimiter$sqrtImage.ruleWidth,
+        advanceWidth = _delimiter$sqrtImage.advanceWidth;
+
+    var delimDepth = img.height - ruleWidth; // Adjust the clearance based on the delimiter size
+
+    if (delimDepth > inner.height + inner.depth + lineClearance) {
+      lineClearance = (lineClearance + delimDepth - inner.height - inner.depth) / 2;
+    } // Shift the sqrt image
+
+
+    var imgShift = img.height - inner.height - lineClearance - ruleWidth;
+    inner.style.paddingLeft = advanceWidth + "em"; // Overlay the image and the argument.
+
+    var body = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: inner,
+        wrapperClasses: ["svg-align"]
+      }, {
+        type: "kern",
+        size: -(inner.height + imgShift)
+      }, {
+        type: "elem",
+        elem: img
+      }, {
+        type: "kern",
+        size: ruleWidth
+      }]
+    }, options);
+
+    if (!group.index) {
+      return buildCommon.makeSpan(["mord", "sqrt"], [body], options);
+    } else {
+      // Handle the optional root index
+      // The index is always in scriptscript style
+      var newOptions = options.havingStyle(src_Style.SCRIPTSCRIPT);
+      var rootm = buildHTML_buildGroup(group.index, newOptions, options); // The amount the index is shifted by. This is taken from the TeX
+      // source, in the definition of `\r@@t`.
+
+      var toShift = 0.6 * (body.height - body.depth); // Build a VList with the superscript shifted up correctly
+
+      var rootVList = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: -toShift,
+        children: [{
+          type: "elem",
+          elem: rootm
+        }]
+      }, options); // Add a class surrounding it so we can add on the appropriate
+      // kerning
+
+      var rootVListWrap = buildCommon.makeSpan(["root"], [rootVList]);
+      return buildCommon.makeSpan(["mord", "sqrt"], [rootVListWrap, body], options);
+    }
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var body = group.body,
+        index = group.index;
+    return index ? new mathMLTree.MathNode("mroot", [buildMathML_buildGroup(body, options), buildMathML_buildGroup(index, options)]) : new mathMLTree.MathNode("msqrt", [buildMathML_buildGroup(body, options)]);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/styling.js
+
+
+
+
+
+var styling_styleMap = {
+  "display": src_Style.DISPLAY,
+  "text": src_Style.TEXT,
+  "script": src_Style.SCRIPT,
+  "scriptscript": src_Style.SCRIPTSCRIPT
+};
+defineFunction({
+  type: "styling",
+  names: ["\\displaystyle", "\\textstyle", "\\scriptstyle", "\\scriptscriptstyle"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var breakOnTokenText = _ref.breakOnTokenText,
+        funcName = _ref.funcName,
+        parser = _ref.parser;
+    // parse out the implicit body
+    var body = parser.parseExpression(true, breakOnTokenText); // TODO: Refactor to avoid duplicating styleMap in multiple places (e.g.
+    // here and in buildHTML and de-dupe the enumeration of all the styles).
+    // $FlowFixMe: The names above exactly match the styles.
+
+    var style = funcName.slice(1, funcName.length - 5);
+    return {
+      type: "styling",
+      mode: parser.mode,
+      // Figure out what style to use by pulling out the style from
+      // the function name
+      style: style,
+      body: body
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Style changes are handled in the TeXbook on pg. 442, Rule 3.
+    var newStyle = styling_styleMap[group.style];
+    var newOptions = options.havingStyle(newStyle).withFont('');
+    return sizingGroup(group.body, newOptions, options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    // Figure out what style we're changing to.
+    var newStyle = styling_styleMap[group.style];
+    var newOptions = options.havingStyle(newStyle);
+    var inner = buildMathML_buildExpression(group.body, newOptions);
+    var node = new mathMLTree.MathNode("mstyle", inner);
+    var styleAttributes = {
+      "display": ["0", "true"],
+      "text": ["0", "false"],
+      "script": ["1", "false"],
+      "scriptscript": ["2", "false"]
+    };
+    var attr = styleAttributes[group.style];
+    node.setAttribute("scriptlevel", attr[0]);
+    node.setAttribute("displaystyle", attr[1]);
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/supsub.js
+
+
+
+
+
+
+
+
+
+
+
+
+
+/**
+ * Sometimes, groups perform special rules when they have superscripts or
+ * subscripts attached to them. This function lets the `supsub` group know that
+ * Sometimes, groups perform special rules when they have superscripts or
+ * its inner element should handle the superscripts and subscripts instead of
+ * handling them itself.
+ */
+var supsub_htmlBuilderDelegate = function htmlBuilderDelegate(group, options) {
+  var base = group.base;
+
+  if (!base) {
+    return null;
+  } else if (base.type === "op") {
+    // Operators handle supsubs differently when they have limits
+    // (e.g. `\displaystyle\sum_2^3`)
+    var delegate = base.limits && (options.style.size === src_Style.DISPLAY.size || base.alwaysHandleSupSub);
+    return delegate ? op_htmlBuilder : null;
+  } else if (base.type === "operatorname") {
+    var _delegate = base.alwaysHandleSupSub && (options.style.size === src_Style.DISPLAY.size || base.limits);
+
+    return _delegate ? operatorname_htmlBuilder : null;
+  } else if (base.type === "accent") {
+    return utils.isCharacterBox(base.base) ? accent_htmlBuilder : null;
+  } else if (base.type === "horizBrace") {
+    var isSup = !group.sub;
+    return isSup === base.isOver ? horizBrace_htmlBuilder : null;
+  } else {
+    return null;
+  }
+}; // Super scripts and subscripts, whose precise placement can depend on other
+// functions that precede them.
+
+
+defineFunctionBuilders({
+  type: "supsub",
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Superscript and subscripts are handled in the TeXbook on page
+    // 445-446, rules 18(a-f).
+    // Here is where we defer to the inner group if it should handle
+    // superscripts and subscripts itself.
+    var builderDelegate = supsub_htmlBuilderDelegate(group, options);
+
+    if (builderDelegate) {
+      return builderDelegate(group, options);
+    }
+
+    var valueBase = group.base,
+        valueSup = group.sup,
+        valueSub = group.sub;
+    var base = buildHTML_buildGroup(valueBase, options);
+    var supm;
+    var subm;
+    var metrics = options.fontMetrics(); // Rule 18a
+
+    var supShift = 0;
+    var subShift = 0;
+    var isCharacterBox = valueBase && utils.isCharacterBox(valueBase);
+
+    if (valueSup) {
+      var newOptions = options.havingStyle(options.style.sup());
+      supm = buildHTML_buildGroup(valueSup, newOptions, options);
+
+      if (!isCharacterBox) {
+        supShift = base.height - newOptions.fontMetrics().supDrop * newOptions.sizeMultiplier / options.sizeMultiplier;
+      }
+    }
+
+    if (valueSub) {
+      var _newOptions = options.havingStyle(options.style.sub());
+
+      subm = buildHTML_buildGroup(valueSub, _newOptions, options);
+
+      if (!isCharacterBox) {
+        subShift = base.depth + _newOptions.fontMetrics().subDrop * _newOptions.sizeMultiplier / options.sizeMultiplier;
+      }
+    } // Rule 18c
+
+
+    var minSupShift;
+
+    if (options.style === src_Style.DISPLAY) {
+      minSupShift = metrics.sup1;
+    } else if (options.style.cramped) {
+      minSupShift = metrics.sup3;
+    } else {
+      minSupShift = metrics.sup2;
+    } // scriptspace is a font-size-independent size, so scale it
+    // appropriately for use as the marginRight.
+
+
+    var multiplier = options.sizeMultiplier;
+    var marginRight = 0.5 / metrics.ptPerEm / multiplier + "em";
+    var marginLeft = null;
+
+    if (subm) {
+      // Subscripts shouldn't be shifted by the base's italic correction.
+      // Account for that by shifting the subscript back the appropriate
+      // amount. Note we only do this when the base is a single symbol.
+      var isOiint = group.base && group.base.type === "op" && group.base.name && (group.base.name === "\\oiint" || group.base.name === "\\oiiint");
+
+      if (base instanceof domTree_SymbolNode || isOiint) {
+        // $FlowFixMe
+        marginLeft = -base.italic + "em";
+      }
+    }
+
+    var supsub;
+
+    if (supm && subm) {
+      supShift = Math.max(supShift, minSupShift, supm.depth + 0.25 * metrics.xHeight);
+      subShift = Math.max(subShift, metrics.sub2);
+      var ruleWidth = metrics.defaultRuleThickness; // Rule 18e
+
+      var maxWidth = 4 * ruleWidth;
+
+      if (supShift - supm.depth - (subm.height - subShift) < maxWidth) {
+        subShift = maxWidth - (supShift - supm.depth) + subm.height;
+        var psi = 0.8 * metrics.xHeight - (supShift - supm.depth);
+
+        if (psi > 0) {
+          supShift += psi;
+          subShift -= psi;
+        }
+      }
+
+      var vlistElem = [{
+        type: "elem",
+        elem: subm,
+        shift: subShift,
+        marginRight: marginRight,
+        marginLeft: marginLeft
+      }, {
+        type: "elem",
+        elem: supm,
+        shift: -supShift,
+        marginRight: marginRight
+      }];
+      supsub = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: vlistElem
+      }, options);
+    } else if (subm) {
+      // Rule 18b
+      subShift = Math.max(subShift, metrics.sub1, subm.height - 0.8 * metrics.xHeight);
+      var _vlistElem = [{
+        type: "elem",
+        elem: subm,
+        marginLeft: marginLeft,
+        marginRight: marginRight
+      }];
+      supsub = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: subShift,
+        children: _vlistElem
+      }, options);
+    } else if (supm) {
+      // Rule 18c, d
+      supShift = Math.max(supShift, minSupShift, supm.depth + 0.25 * metrics.xHeight);
+      supsub = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: -supShift,
+        children: [{
+          type: "elem",
+          elem: supm,
+          marginRight: marginRight
+        }]
+      }, options);
+    } else {
+      throw new Error("supsub must have either sup or sub.");
+    } // Wrap the supsub vlist in a span.msupsub to reset text-align.
+
+
+    var mclass = getTypeOfDomTree(base, "right") || "mord";
+    return buildCommon.makeSpan([mclass], [base, buildCommon.makeSpan(["msupsub"], [supsub])], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    // Is the inner group a relevant horizonal brace?
+    var isBrace = false;
+    var isOver;
+    var isSup;
+
+    if (group.base && group.base.type === "horizBrace") {
+      isSup = !!group.sup;
+
+      if (isSup === group.base.isOver) {
+        isBrace = true;
+        isOver = group.base.isOver;
+      }
+    }
+
+    if (group.base && (group.base.type === "op" || group.base.type === "operatorname")) {
+      group.base.parentIsSupSub = true;
+    }
+
+    var children = [buildMathML_buildGroup(group.base, options)];
+
+    if (group.sub) {
+      children.push(buildMathML_buildGroup(group.sub, options));
+    }
+
+    if (group.sup) {
+      children.push(buildMathML_buildGroup(group.sup, options));
+    }
+
+    var nodeType;
+
+    if (isBrace) {
+      nodeType = isOver ? "mover" : "munder";
+    } else if (!group.sub) {
+      var base = group.base;
+
+      if (base && base.type === "op" && base.limits && (options.style === src_Style.DISPLAY || base.alwaysHandleSupSub)) {
+        nodeType = "mover";
+      } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub && (base.limits || options.style === src_Style.DISPLAY)) {
+        nodeType = "mover";
+      } else {
+        nodeType = "msup";
+      }
+    } else if (!group.sup) {
+      var _base = group.base;
+
+      if (_base && _base.type === "op" && _base.limits && (options.style === src_Style.DISPLAY || _base.alwaysHandleSupSub)) {
+        nodeType = "munder";
+      } else if (_base && _base.type === "operatorname" && _base.alwaysHandleSupSub && (_base.limits || options.style === src_Style.DISPLAY)) {
+        nodeType = "munder";
+      } else {
+        nodeType = "msub";
+      }
+    } else {
+      var _base2 = group.base;
+
+      if (_base2 && _base2.type === "op" && _base2.limits && options.style === src_Style.DISPLAY) {
+        nodeType = "munderover";
+      } else if (_base2 && _base2.type === "operatorname" && _base2.alwaysHandleSupSub && (options.style === src_Style.DISPLAY || _base2.limits)) {
+        nodeType = "munderover";
+      } else {
+        nodeType = "msubsup";
+      }
+    }
+
+    var node = new mathMLTree.MathNode(nodeType, children);
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/symbolsOp.js
+
+
+
+ // Operator ParseNodes created in Parser.js from symbol Groups in src/symbols.js.
+
+defineFunctionBuilders({
+  type: "atom",
+  htmlBuilder: function htmlBuilder(group, options) {
+    return buildCommon.mathsym(group.text, group.mode, options, ["m" + group.family]);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mo", [buildMathML_makeText(group.text, group.mode)]);
+
+    if (group.family === "bin") {
+      var variant = buildMathML_getVariant(group, options);
+
+      if (variant === "bold-italic") {
+        node.setAttribute("mathvariant", variant);
+      }
+    } else if (group.family === "punct") {
+      node.setAttribute("separator", "true");
+    } else if (group.family === "open" || group.family === "close") {
+      // Delims built here should not stretch vertically.
+      // See delimsizing.js for stretchy delims.
+      node.setAttribute("stretchy", "false");
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/symbolsOrd.js
+
+
+
+
+// "mathord" and "textord" ParseNodes created in Parser.js from symbol Groups in
+var defaultVariant = {
+  "mi": "italic",
+  "mn": "normal",
+  "mtext": "normal"
+};
+defineFunctionBuilders({
+  type: "mathord",
+  htmlBuilder: function htmlBuilder(group, options) {
+    return buildCommon.makeOrd(group, options, "mathord");
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node = new mathMLTree.MathNode("mi", [buildMathML_makeText(group.text, group.mode, options)]);
+    var variant = buildMathML_getVariant(group, options) || "italic";
+
+    if (variant !== defaultVariant[node.type]) {
+      node.setAttribute("mathvariant", variant);
+    }
+
+    return node;
+  }
+});
+defineFunctionBuilders({
+  type: "textord",
+  htmlBuilder: function htmlBuilder(group, options) {
+    return buildCommon.makeOrd(group, options, "textord");
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var text = buildMathML_makeText(group.text, group.mode, options);
+    var variant = buildMathML_getVariant(group, options) || "normal";
+    var node;
+
+    if (group.mode === 'text') {
+      node = new mathMLTree.MathNode("mtext", [text]);
+    } else if (/[0-9]/.test(group.text)) {
+      // TODO(kevinb) merge adjacent <mn> nodes
+      // do it as a post processing step
+      node = new mathMLTree.MathNode("mn", [text]);
+    } else if (group.text === "\\prime") {
+      node = new mathMLTree.MathNode("mo", [text]);
+    } else {
+      node = new mathMLTree.MathNode("mi", [text]);
+    }
+
+    if (variant !== defaultVariant[node.type]) {
+      node.setAttribute("mathvariant", variant);
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/symbolsSpacing.js
+
+
+
+ // A map of CSS-based spacing functions to their CSS class.
+
+var cssSpace = {
+  "\\nobreak": "nobreak",
+  "\\allowbreak": "allowbreak"
+}; // A lookup table to determine whether a spacing function/symbol should be
+// treated like a regular space character.  If a symbol or command is a key
+// in this table, then it should be a regular space character.  Furthermore,
+// the associated value may have a `className` specifying an extra CSS class
+// to add to the created `span`.
+
+var regularSpace = {
+  " ": {},
+  "\\ ": {},
+  "~": {
+    className: "nobreak"
+  },
+  "\\space": {},
+  "\\nobreakspace": {
+    className: "nobreak"
+  }
+}; // ParseNode<"spacing"> created in Parser.js from the "spacing" symbol Groups in
+// src/symbols.js.
+
+defineFunctionBuilders({
+  type: "spacing",
+  htmlBuilder: function htmlBuilder(group, options) {
+    if (regularSpace.hasOwnProperty(group.text)) {
+      var className = regularSpace[group.text].className || ""; // Spaces are generated by adding an actual space. Each of these
+      // things has an entry in the symbols table, so these will be turned
+      // into appropriate outputs.
+
+      if (group.mode === "text") {
+        var ord = buildCommon.makeOrd(group, options, "textord");
+        ord.classes.push(className);
+        return ord;
+      } else {
+        return buildCommon.makeSpan(["mspace", className], [buildCommon.mathsym(group.text, group.mode, options)], options);
+      }
+    } else if (cssSpace.hasOwnProperty(group.text)) {
+      // Spaces based on just a CSS class.
+      return buildCommon.makeSpan(["mspace", cssSpace[group.text]], [], options);
+    } else {
+      throw new src_ParseError("Unknown type of space \"" + group.text + "\"");
+    }
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var node;
+
+    if (regularSpace.hasOwnProperty(group.text)) {
+      node = new mathMLTree.MathNode("mtext", [new mathMLTree.TextNode("\xA0")]);
+    } else if (cssSpace.hasOwnProperty(group.text)) {
+      // CSS-based MathML spaces (\nobreak, \allowbreak) are ignored
+      return new mathMLTree.MathNode("mspace");
+    } else {
+      throw new src_ParseError("Unknown type of space \"" + group.text + "\"");
+    }
+
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/tag.js
+
+
+
+
+var tag_pad = function pad() {
+  var padNode = new mathMLTree.MathNode("mtd", []);
+  padNode.setAttribute("width", "50%");
+  return padNode;
+};
+
+defineFunctionBuilders({
+  type: "tag",
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var table = new mathMLTree.MathNode("mtable", [new mathMLTree.MathNode("mtr", [tag_pad(), new mathMLTree.MathNode("mtd", [buildExpressionRow(group.body, options)]), tag_pad(), new mathMLTree.MathNode("mtd", [buildExpressionRow(group.tag, options)])])]);
+    table.setAttribute("width", "100%");
+    return table; // TODO: Left-aligned tags.
+    // Currently, the group and options passed here do not contain
+    // enough info to set tag alignment. `leqno` is in Settings but it is
+    // not passed to Options. On the HTML side, leqno is
+    // set by a CSS class applied in buildTree.js. That would have worked
+    // in MathML if browsers supported <mlabeledtr>. Since they don't, we
+    // need to rewrite the way this function is called.
+  }
+});
+// CONCATENATED MODULE: ./src/functions/text.js
+
+
+
+ // Non-mathy text, possibly in a font
+
+var textFontFamilies = {
+  "\\text": undefined,
+  "\\textrm": "textrm",
+  "\\textsf": "textsf",
+  "\\texttt": "texttt",
+  "\\textnormal": "textrm"
+};
+var textFontWeights = {
+  "\\textbf": "textbf",
+  "\\textmd": "textmd"
+};
+var textFontShapes = {
+  "\\textit": "textit",
+  "\\textup": "textup"
+};
+
+var optionsWithFont = function optionsWithFont(group, options) {
+  var font = group.font; // Checks if the argument is a font family or a font style.
+
+  if (!font) {
+    return options;
+  } else if (textFontFamilies[font]) {
+    return options.withTextFontFamily(textFontFamilies[font]);
+  } else if (textFontWeights[font]) {
+    return options.withTextFontWeight(textFontWeights[font]);
+  } else {
+    return options.withTextFontShape(textFontShapes[font]);
+  }
+};
+
+defineFunction({
+  type: "text",
+  names: [// Font families
+  "\\text", "\\textrm", "\\textsf", "\\texttt", "\\textnormal", // Font weights
+  "\\textbf", "\\textmd", // Font Shapes
+  "\\textit", "\\textup"],
+  props: {
+    numArgs: 1,
+    argTypes: ["text"],
+    greediness: 2,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser,
+        funcName = _ref.funcName;
+    var body = args[0];
+    return {
+      type: "text",
+      mode: parser.mode,
+      body: ordargument(body),
+      font: funcName
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var newOptions = optionsWithFont(group, options);
+    var inner = buildHTML_buildExpression(group.body, newOptions, true);
+    return buildCommon.makeSpan(["mord", "text"], buildCommon.tryCombineChars(inner), newOptions);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var newOptions = optionsWithFont(group, options);
+    return buildExpressionRow(group.body, newOptions);
+  }
+});
+// CONCATENATED MODULE: ./src/functions/underline.js
+
+
+
+
+
+defineFunction({
+  type: "underline",
+  names: ["\\underline"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: function handler(_ref, args) {
+    var parser = _ref.parser;
+    return {
+      type: "underline",
+      mode: parser.mode,
+      body: args[0]
+    };
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    // Underlines are handled in the TeXbook pg 443, Rule 10.
+    // Build the inner group.
+    var innerGroup = buildHTML_buildGroup(group.body, options); // Create the line to go below the body
+
+    var line = buildCommon.makeLineSpan("underline-line", options); // Generate the vlist, with the appropriate kerns
+
+    var defaultRuleThickness = options.fontMetrics().defaultRuleThickness;
+    var vlist = buildCommon.makeVList({
+      positionType: "top",
+      positionData: innerGroup.height,
+      children: [{
+        type: "kern",
+        size: defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: line
+      }, {
+        type: "kern",
+        size: 3 * defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: innerGroup
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "underline"], [vlist], options);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var operator = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode("\u203E")]);
+    operator.setAttribute("stretchy", "true");
+    var node = new mathMLTree.MathNode("munder", [buildMathML_buildGroup(group.body, options), operator]);
+    node.setAttribute("accentunder", "true");
+    return node;
+  }
+});
+// CONCATENATED MODULE: ./src/functions/verb.js
+
+
+
+
+defineFunction({
+  type: "verb",
+  names: ["\\verb"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: function handler(context, args, optArgs) {
+    // \verb and \verb* are dealt with directly in Parser.js.
+    // If we end up here, it's because of a failure to match the two delimiters
+    // in the regex in Lexer.js.  LaTeX raises the following error when \verb is
+    // terminated by end of line (or file).
+    throw new src_ParseError("\\verb ended by end of line instead of matching delimiter");
+  },
+  htmlBuilder: function htmlBuilder(group, options) {
+    var text = makeVerb(group);
+    var body = []; // \verb enters text mode and therefore is sized like \textstyle
+
+    var newOptions = options.havingStyle(options.style.text());
+
+    for (var i = 0; i < text.length; i++) {
+      var c = text[i];
+
+      if (c === '~') {
+        c = '\\textasciitilde';
+      }
+
+      body.push(buildCommon.makeSymbol(c, "Typewriter-Regular", group.mode, newOptions, ["mord", "texttt"]));
+    }
+
+    return buildCommon.makeSpan(["mord", "text"].concat(newOptions.sizingClasses(options)), buildCommon.tryCombineChars(body), newOptions);
+  },
+  mathmlBuilder: function mathmlBuilder(group, options) {
+    var text = new mathMLTree.TextNode(makeVerb(group));
+    var node = new mathMLTree.MathNode("mtext", [text]);
+    node.setAttribute("mathvariant", "monospace");
+    return node;
+  }
+});
+/**
+ * Converts verb group into body string.
+ *
+ * \verb* replaces each space with an open box \u2423
+ * \verb replaces each space with a no-break space \xA0
+ */
+
+var makeVerb = function makeVerb(group) {
+  return group.body.replace(/ /g, group.star ? "\u2423" : '\xA0');
+};
+// CONCATENATED MODULE: ./src/functions.js
+/** Include this to ensure that all functions are defined. */
+
+var functions = _functions;
+/* harmony default export */ var src_functions = (functions); // TODO(kevinb): have functions return an object and call defineFunction with
+// that object in this file instead of relying on side-effects.
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+// CONCATENATED MODULE: ./src/Lexer.js
+/**
+ * The Lexer class handles tokenizing the input in various ways. Since our
+ * parser expects us to be able to backtrack, the lexer allows lexing from any
+ * given starting point.
+ *
+ * Its main exposed function is the `lex` function, which takes a position to
+ * lex from and a type of token to lex. It defers to the appropriate `_innerLex`
+ * function.
+ *
+ * The various `_innerLex` functions perform the actual lexing of different
+ * kinds.
+ */
+
+
+
+
+/* The following tokenRegex
+ * - matches typical whitespace (but not NBSP etc.) using its first group
+ * - does not match any control character \x00-\x1f except whitespace
+ * - does not match a bare backslash
+ * - matches any ASCII character except those just mentioned
+ * - does not match the BMP private use area \uE000-\uF8FF
+ * - does not match bare surrogate code units
+ * - matches any BMP character except for those just described
+ * - matches any valid Unicode surrogate pair
+ * - matches a backslash followed by one or more letters
+ * - matches a backslash followed by any BMP character, including newline
+ * Just because the Lexer matches something doesn't mean it's valid input:
+ * If there is no matching function or symbol definition, the Parser will
+ * still reject the input.
+ */
+var spaceRegexString = "[ \r\n\t]";
+var controlWordRegexString = "\\\\[a-zA-Z@]+";
+var controlSymbolRegexString = "\\\\[^\uD800-\uDFFF]";
+var controlWordWhitespaceRegexString = "" + controlWordRegexString + spaceRegexString + "*";
+var controlWordWhitespaceRegex = new RegExp("^(" + controlWordRegexString + ")" + spaceRegexString + "*$");
+var combiningDiacriticalMarkString = "[\u0300-\u036F]";
+var combiningDiacriticalMarksEndRegex = new RegExp(combiningDiacriticalMarkString + "+$");
+var tokenRegexString = "(" + spaceRegexString + "+)|" + // whitespace
+"([!-\\[\\]-\u2027\u202A-\uD7FF\uF900-\uFFFF]" + ( // single codepoint
+combiningDiacriticalMarkString + "*") + // ...plus accents
+"|[\uD800-\uDBFF][\uDC00-\uDFFF]" + ( // surrogate pair
+combiningDiacriticalMarkString + "*") + // ...plus accents
+"|\\\\verb\\*([^]).*?\\3" + // \verb*
+"|\\\\verb([^*a-zA-Z]).*?\\4" + // \verb unstarred
+"|\\\\operatorname\\*" + ( // \operatorname*
+"|" + controlWordWhitespaceRegexString) + ( // \macroName + spaces
+"|" + controlSymbolRegexString + ")"); // \\, \', etc.
+
+/** Main Lexer class */
+
+var Lexer_Lexer =
+/*#__PURE__*/
+function () {
+  // category codes, only supports comment characters (14) for now
+  function Lexer(input, settings) {
+    this.input = void 0;
+    this.settings = void 0;
+    this.tokenRegex = void 0;
+    this.catcodes = void 0;
+    // Separate accents from characters
+    this.input = input;
+    this.settings = settings;
+    this.tokenRegex = new RegExp(tokenRegexString, 'g');
+    this.catcodes = {
+      "%": 14 // comment character
+
+    };
+  }
+
+  var _proto = Lexer.prototype;
+
+  _proto.setCatcode = function setCatcode(char, code) {
+    this.catcodes[char] = code;
+  }
+  /**
+   * This function lexes a single token.
+   */
+  ;
+
+  _proto.lex = function lex() {
+    var input = this.input;
+    var pos = this.tokenRegex.lastIndex;
+
+    if (pos === input.length) {
+      return new Token_Token("EOF", new SourceLocation(this, pos, pos));
+    }
+
+    var match = this.tokenRegex.exec(input);
+
+    if (match === null || match.index !== pos) {
+      throw new src_ParseError("Unexpected character: '" + input[pos] + "'", new Token_Token(input[pos], new SourceLocation(this, pos, pos + 1)));
+    }
+
+    var text = match[2] || " ";
+
+    if (this.catcodes[text] === 14) {
+      // comment character
+      var nlIndex = input.indexOf('\n', this.tokenRegex.lastIndex);
+
+      if (nlIndex === -1) {
+        this.tokenRegex.lastIndex = input.length; // EOF
+
+        this.settings.reportNonstrict("commentAtEnd", "% comment has no terminating newline; LaTeX would " + "fail because of commenting the end of math mode (e.g. $)");
+      } else {
+        this.tokenRegex.lastIndex = nlIndex + 1;
+      }
+
+      return this.lex();
+    } // Trim any trailing whitespace from control word match
+
+
+    var controlMatch = text.match(controlWordWhitespaceRegex);
+
+    if (controlMatch) {
+      text = controlMatch[1];
+    }
+
+    return new Token_Token(text, new SourceLocation(this, pos, this.tokenRegex.lastIndex));
+  };
+
+  return Lexer;
+}();
+
+
+// CONCATENATED MODULE: ./src/Namespace.js
+/**
+ * A `Namespace` refers to a space of nameable things like macros or lengths,
+ * which can be `set` either globally or local to a nested group, using an
+ * undo stack similar to how TeX implements this functionality.
+ * Performance-wise, `get` and local `set` take constant time, while global
+ * `set` takes time proportional to the depth of group nesting.
+ */
+
+
+var Namespace_Namespace =
+/*#__PURE__*/
+function () {
+  /**
+   * Both arguments are optional.  The first argument is an object of
+   * built-in mappings which never change.  The second argument is an object
+   * of initial (global-level) mappings, which will constantly change
+   * according to any global/top-level `set`s done.
+   */
+  function Namespace(builtins, globalMacros) {
+    if (builtins === void 0) {
+      builtins = {};
+    }
+
+    if (globalMacros === void 0) {
+      globalMacros = {};
+    }
+
+    this.current = void 0;
+    this.builtins = void 0;
+    this.undefStack = void 0;
+    this.current = globalMacros;
+    this.builtins = builtins;
+    this.undefStack = [];
+  }
+  /**
+   * Start a new nested group, affecting future local `set`s.
+   */
+
+
+  var _proto = Namespace.prototype;
+
+  _proto.beginGroup = function beginGroup() {
+    this.undefStack.push({});
+  }
+  /**
+   * End current nested group, restoring values before the group began.
+   */
+  ;
+
+  _proto.endGroup = function endGroup() {
+    if (this.undefStack.length === 0) {
+      throw new src_ParseError("Unbalanced namespace destruction: attempt " + "to pop global namespace; please report this as a bug");
+    }
+
+    var undefs = this.undefStack.pop();
+
+    for (var undef in undefs) {
+      if (undefs.hasOwnProperty(undef)) {
+        if (undefs[undef] === undefined) {
+          delete this.current[undef];
+        } else {
+          this.current[undef] = undefs[undef];
+        }
+      }
+    }
+  }
+  /**
+   * Detect whether `name` has a definition.  Equivalent to
+   * `get(name) != null`.
+   */
+  ;
+
+  _proto.has = function has(name) {
+    return this.current.hasOwnProperty(name) || this.builtins.hasOwnProperty(name);
+  }
+  /**
+   * Get the current value of a name, or `undefined` if there is no value.
+   *
+   * Note: Do not use `if (namespace.get(...))` to detect whether a macro
+   * is defined, as the definition may be the empty string which evaluates
+   * to `false` in JavaScript.  Use `if (namespace.get(...) != null)` or
+   * `if (namespace.has(...))`.
+   */
+  ;
+
+  _proto.get = function get(name) {
+    if (this.current.hasOwnProperty(name)) {
+      return this.current[name];
+    } else {
+      return this.builtins[name];
+    }
+  }
+  /**
+   * Set the current value of a name, and optionally set it globally too.
+   * Local set() sets the current value and (when appropriate) adds an undo
+   * operation to the undo stack.  Global set() may change the undo
+   * operation at every level, so takes time linear in their number.
+   */
+  ;
+
+  _proto.set = function set(name, value, global) {
+    if (global === void 0) {
+      global = false;
+    }
+
+    if (global) {
+      // Global set is equivalent to setting in all groups.  Simulate this
+      // by destroying any undos currently scheduled for this name,
+      // and adding an undo with the *new* value (in case it later gets
+      // locally reset within this environment).
+      for (var i = 0; i < this.undefStack.length; i++) {
+        delete this.undefStack[i][name];
+      }
+
+      if (this.undefStack.length > 0) {
+        this.undefStack[this.undefStack.length - 1][name] = value;
+      }
+    } else {
+      // Undo this set at end of this group (possibly to `undefined`),
+      // unless an undo is already in place, in which case that older
+      // value is the correct one.
+      var top = this.undefStack[this.undefStack.length - 1];
+
+      if (top && !top.hasOwnProperty(name)) {
+        top[name] = this.current[name];
+      }
+    }
+
+    this.current[name] = value;
+  };
+
+  return Namespace;
+}();
+
+
+// CONCATENATED MODULE: ./src/macros.js
+/**
+ * Predefined macros for KaTeX.
+ * This can be used to define some commands in terms of others.
+ */
+
+
+
+
+
+
+var builtinMacros = {};
+/* harmony default export */ var macros = (builtinMacros); // This function might one day accept an additional argument and do more things.
+
+function defineMacro(name, body) {
+  builtinMacros[name] = body;
+} //////////////////////////////////////////////////////////////////////
+// macro tools
+
+defineMacro("\\noexpand", function (context) {
+  // The expansion is the token itself; but that token is interpreted
+  // as if its meaning were ‘\relax’ if it is a control sequence that
+  // would ordinarily be expanded by TeX’s expansion rules.
+  var t = context.popToken();
+
+  if (context.isExpandable(t.text)) {
+    t.noexpand = true;
+    t.treatAsRelax = true;
+  }
+
+  return {
+    tokens: [t],
+    numArgs: 0
+  };
+});
+defineMacro("\\expandafter", function (context) {
+  // TeX first reads the token that comes immediately after \expandafter,
+  // without expanding it; let’s call this token t. Then TeX reads the
+  // token that comes after t (and possibly more tokens, if that token
+  // has an argument), replacing it by its expansion. Finally TeX puts
+  // t back in front of that expansion.
+  var t = context.popToken();
+  context.expandOnce(true); // expand only an expandable token
+
+  return {
+    tokens: [t],
+    numArgs: 0
+  };
+}); // LaTeX's \@firstoftwo{#1}{#2} expands to #1, skipping #2
+// TeX source: \long\def\@firstoftwo#1#2{#1}
+
+defineMacro("\\@firstoftwo", function (context) {
+  var args = context.consumeArgs(2);
+  return {
+    tokens: args[0],
+    numArgs: 0
+  };
+}); // LaTeX's \@secondoftwo{#1}{#2} expands to #2, skipping #1
+// TeX source: \long\def\@secondoftwo#1#2{#2}
+
+defineMacro("\\@secondoftwo", function (context) {
+  var args = context.consumeArgs(2);
+  return {
+    tokens: args[1],
+    numArgs: 0
+  };
+}); // LaTeX's \@ifnextchar{#1}{#2}{#3} looks ahead to the next (unexpanded)
+// symbol that isn't a space, consuming any spaces but not consuming the
+// first nonspace character.  If that nonspace character matches #1, then
+// the macro expands to #2; otherwise, it expands to #3.
+
+defineMacro("\\@ifnextchar", function (context) {
+  var args = context.consumeArgs(3); // symbol, if, else
+
+  context.consumeSpaces();
+  var nextToken = context.future();
+
+  if (args[0].length === 1 && args[0][0].text === nextToken.text) {
+    return {
+      tokens: args[1],
+      numArgs: 0
+    };
+  } else {
+    return {
+      tokens: args[2],
+      numArgs: 0
+    };
+  }
+}); // LaTeX's \@ifstar{#1}{#2} looks ahead to the next (unexpanded) symbol.
+// If it is `*`, then it consumes the symbol, and the macro expands to #1;
+// otherwise, the macro expands to #2 (without consuming the symbol).
+// TeX source: \def\@ifstar#1{\@ifnextchar *{\@firstoftwo{#1}}}
+
+defineMacro("\\@ifstar", "\\@ifnextchar *{\\@firstoftwo{#1}}"); // LaTeX's \TextOrMath{#1}{#2} expands to #1 in text mode, #2 in math mode
+
+defineMacro("\\TextOrMath", function (context) {
+  var args = context.consumeArgs(2);
+
+  if (context.mode === 'text') {
+    return {
+      tokens: args[0],
+      numArgs: 0
+    };
+  } else {
+    return {
+      tokens: args[1],
+      numArgs: 0
+    };
+  }
+}); // Lookup table for parsing numbers in base 8 through 16
+
+var digitToNumber = {
+  "0": 0,
+  "1": 1,
+  "2": 2,
+  "3": 3,
+  "4": 4,
+  "5": 5,
+  "6": 6,
+  "7": 7,
+  "8": 8,
+  "9": 9,
+  "a": 10,
+  "A": 10,
+  "b": 11,
+  "B": 11,
+  "c": 12,
+  "C": 12,
+  "d": 13,
+  "D": 13,
+  "e": 14,
+  "E": 14,
+  "f": 15,
+  "F": 15
+}; // TeX \char makes a literal character (catcode 12) using the following forms:
+// (see The TeXBook, p. 43)
+//   \char123  -- decimal
+//   \char'123 -- octal
+//   \char"123 -- hex
+//   \char`x   -- character that can be written (i.e. isn't active)
+//   \char`\x  -- character that cannot be written (e.g. %)
+// These all refer to characters from the font, so we turn them into special
+// calls to a function \@char dealt with in the Parser.
+
+defineMacro("\\char", function (context) {
+  var token = context.popToken();
+  var base;
+  var number = '';
+
+  if (token.text === "'") {
+    base = 8;
+    token = context.popToken();
+  } else if (token.text === '"') {
+    base = 16;
+    token = context.popToken();
+  } else if (token.text === "`") {
+    token = context.popToken();
+
+    if (token.text[0] === "\\") {
+      number = token.text.charCodeAt(1);
+    } else if (token.text === "EOF") {
+      throw new src_ParseError("\\char` missing argument");
+    } else {
+      number = token.text.charCodeAt(0);
+    }
+  } else {
+    base = 10;
+  }
+
+  if (base) {
+    // Parse a number in the given base, starting with first `token`.
+    number = digitToNumber[token.text];
+
+    if (number == null || number >= base) {
+      throw new src_ParseError("Invalid base-" + base + " digit " + token.text);
+    }
+
+    var digit;
+
+    while ((digit = digitToNumber[context.future().text]) != null && digit < base) {
+      number *= base;
+      number += digit;
+      context.popToken();
+    }
+  }
+
+  return "\\@char{" + number + "}";
+}); // \newcommand{\macro}[args]{definition}
+// \renewcommand{\macro}[args]{definition}
+// TODO: Optional arguments: \newcommand{\macro}[args][default]{definition}
+
+var macros_newcommand = function newcommand(context, existsOK, nonexistsOK) {
+  var arg = context.consumeArgs(1)[0];
+
+  if (arg.length !== 1) {
+    throw new src_ParseError("\\newcommand's first argument must be a macro name");
+  }
+
+  var name = arg[0].text;
+  var exists = context.isDefined(name);
+
+  if (exists && !existsOK) {
+    throw new src_ParseError("\\newcommand{" + name + "} attempting to redefine " + (name + "; use \\renewcommand"));
+  }
+
+  if (!exists && !nonexistsOK) {
+    throw new src_ParseError("\\renewcommand{" + name + "} when command " + name + " " + "does not yet exist; use \\newcommand");
+  }
+
+  var numArgs = 0;
+  arg = context.consumeArgs(1)[0];
+
+  if (arg.length === 1 && arg[0].text === "[") {
+    var argText = '';
+    var token = context.expandNextToken();
+
+    while (token.text !== "]" && token.text !== "EOF") {
+      // TODO: Should properly expand arg, e.g., ignore {}s
+      argText += token.text;
+      token = context.expandNextToken();
+    }
+
+    if (!argText.match(/^\s*[0-9]+\s*$/)) {
+      throw new src_ParseError("Invalid number of arguments: " + argText);
+    }
+
+    numArgs = parseInt(argText);
+    arg = context.consumeArgs(1)[0];
+  } // Final arg is the expansion of the macro
+
+
+  context.macros.set(name, {
+    tokens: arg,
+    numArgs: numArgs
+  });
+  return '';
+};
+
+defineMacro("\\newcommand", function (context) {
+  return macros_newcommand(context, false, true);
+});
+defineMacro("\\renewcommand", function (context) {
+  return macros_newcommand(context, true, false);
+});
+defineMacro("\\providecommand", function (context) {
+  return macros_newcommand(context, true, true);
+}); // terminal (console) tools
+
+defineMacro("\\message", function (context) {
+  var arg = context.consumeArgs(1)[0]; // eslint-disable-next-line no-console
+
+  console.log(arg.reverse().map(function (token) {
+    return token.text;
+  }).join(""));
+  return '';
+});
+defineMacro("\\errmessage", function (context) {
+  var arg = context.consumeArgs(1)[0]; // eslint-disable-next-line no-console
+
+  console.error(arg.reverse().map(function (token) {
+    return token.text;
+  }).join(""));
+  return '';
+});
+defineMacro("\\show", function (context) {
+  var tok = context.popToken();
+  var name = tok.text; // eslint-disable-next-line no-console
+
+  console.log(tok, context.macros.get(name), src_functions[name], src_symbols.math[name], src_symbols.text[name]);
+  return '';
+}); //////////////////////////////////////////////////////////////////////
+// Grouping
+// \let\bgroup={ \let\egroup=}
+
+defineMacro("\\bgroup", "{");
+defineMacro("\\egroup", "}"); // Symbols from latex.ltx:
+// \def\lq{`}
+// \def\rq{'}
+// \def \aa {\r a}
+// \def \AA {\r A}
+
+defineMacro("\\lq", "`");
+defineMacro("\\rq", "'");
+defineMacro("\\aa", "\\r a");
+defineMacro("\\AA", "\\r A"); // Copyright (C) and registered (R) symbols. Use raw symbol in MathML.
+// \DeclareTextCommandDefault{\textcopyright}{\textcircled{c}}
+// \DeclareTextCommandDefault{\textregistered}{\textcircled{%
+//      \check@mathfonts\fontsize\sf@size\z@\math@fontsfalse\selectfont R}}
+// \DeclareRobustCommand{\copyright}{%
+//    \ifmmode{\nfss@text{\textcopyright}}\else\textcopyright\fi}
+
+defineMacro("\\textcopyright", "\\html@mathml{\\textcircled{c}}{\\char`©}");
+defineMacro("\\copyright", "\\TextOrMath{\\textcopyright}{\\text{\\textcopyright}}");
+defineMacro("\\textregistered", "\\html@mathml{\\textcircled{\\scriptsize R}}{\\char`®}"); // Characters omitted from Unicode range 1D400–1D7FF
+
+defineMacro("\u212C", "\\mathscr{B}"); // script
+
+defineMacro("\u2130", "\\mathscr{E}");
+defineMacro("\u2131", "\\mathscr{F}");
+defineMacro("\u210B", "\\mathscr{H}");
+defineMacro("\u2110", "\\mathscr{I}");
+defineMacro("\u2112", "\\mathscr{L}");
+defineMacro("\u2133", "\\mathscr{M}");
+defineMacro("\u211B", "\\mathscr{R}");
+defineMacro("\u212D", "\\mathfrak{C}"); // Fraktur
+
+defineMacro("\u210C", "\\mathfrak{H}");
+defineMacro("\u2128", "\\mathfrak{Z}"); // Define \Bbbk with a macro that works in both HTML and MathML.
+
+defineMacro("\\Bbbk", "\\Bbb{k}"); // Unicode middle dot
+// The KaTeX fonts do not contain U+00B7. Instead, \cdotp displays
+// the dot at U+22C5 and gives it punct spacing.
+
+defineMacro("\xB7", "\\cdotp"); // \llap and \rlap render their contents in text mode
+
+defineMacro("\\llap", "\\mathllap{\\textrm{#1}}");
+defineMacro("\\rlap", "\\mathrlap{\\textrm{#1}}");
+defineMacro("\\clap", "\\mathclap{\\textrm{#1}}"); // \not is defined by base/fontmath.ltx via
+// \DeclareMathSymbol{\not}{\mathrel}{symbols}{"36}
+// It's thus treated like a \mathrel, but defined by a symbol that has zero
+// width but extends to the right.  We use \rlap to get that spacing.
+// For MathML we write U+0338 here. buildMathML.js will then do the overlay.
+
+defineMacro("\\not", '\\html@mathml{\\mathrel{\\mathrlap\\@not}}{\\char"338}'); // Negated symbols from base/fontmath.ltx:
+// \def\neq{\not=} \let\ne=\neq
+// \DeclareRobustCommand
+//   \notin{\mathrel{\m@th\mathpalette\c@ncel\in}}
+// \def\c@ncel#1#2{\m@th\ooalign{$\hfil#1\mkern1mu/\hfil$\crcr$#1#2$}}
+
+defineMacro("\\neq", "\\html@mathml{\\mathrel{\\not=}}{\\mathrel{\\char`≠}}");
+defineMacro("\\ne", "\\neq");
+defineMacro("\u2260", "\\neq");
+defineMacro("\\notin", "\\html@mathml{\\mathrel{{\\in}\\mathllap{/\\mskip1mu}}}" + "{\\mathrel{\\char`∉}}");
+defineMacro("\u2209", "\\notin"); // Unicode stacked relations
+
+defineMacro("\u2258", "\\html@mathml{" + "\\mathrel{=\\kern{-1em}\\raisebox{0.4em}{$\\scriptsize\\frown$}}" + "}{\\mathrel{\\char`\u2258}}");
+defineMacro("\u2259", "\\html@mathml{\\stackrel{\\tiny\\wedge}{=}}{\\mathrel{\\char`\u2258}}");
+defineMacro("\u225A", "\\html@mathml{\\stackrel{\\tiny\\vee}{=}}{\\mathrel{\\char`\u225A}}");
+defineMacro("\u225B", "\\html@mathml{\\stackrel{\\scriptsize\\star}{=}}" + "{\\mathrel{\\char`\u225B}}");
+defineMacro("\u225D", "\\html@mathml{\\stackrel{\\tiny\\mathrm{def}}{=}}" + "{\\mathrel{\\char`\u225D}}");
+defineMacro("\u225E", "\\html@mathml{\\stackrel{\\tiny\\mathrm{m}}{=}}" + "{\\mathrel{\\char`\u225E}}");
+defineMacro("\u225F", "\\html@mathml{\\stackrel{\\tiny?}{=}}{\\mathrel{\\char`\u225F}}"); // Misc Unicode
+
+defineMacro("\u27C2", "\\perp");
+defineMacro("\u203C", "\\mathclose{!\\mkern-0.8mu!}");
+defineMacro("\u220C", "\\notni");
+defineMacro("\u231C", "\\ulcorner");
+defineMacro("\u231D", "\\urcorner");
+defineMacro("\u231E", "\\llcorner");
+defineMacro("\u231F", "\\lrcorner");
+defineMacro("\xA9", "\\copyright");
+defineMacro("\xAE", "\\textregistered");
+defineMacro("\uFE0F", "\\textregistered"); // The KaTeX fonts have corners at codepoints that don't match Unicode.
+// For MathML purposes, use the Unicode code point.
+
+defineMacro("\\ulcorner", "\\html@mathml{\\@ulcorner}{\\mathop{\\char\"231c}}");
+defineMacro("\\urcorner", "\\html@mathml{\\@urcorner}{\\mathop{\\char\"231d}}");
+defineMacro("\\llcorner", "\\html@mathml{\\@llcorner}{\\mathop{\\char\"231e}}");
+defineMacro("\\lrcorner", "\\html@mathml{\\@lrcorner}{\\mathop{\\char\"231f}}"); //////////////////////////////////////////////////////////////////////
+// LaTeX_2ε
+// \vdots{\vbox{\baselineskip4\p@  \lineskiplimit\z@
+// \kern6\p@\hbox{.}\hbox{.}\hbox{.}}}
+// We'll call \varvdots, which gets a glyph from symbols.js.
+// The zero-width rule gets us an equivalent to the vertical 6pt kern.
+
+defineMacro("\\vdots", "\\mathord{\\varvdots\\rule{0pt}{15pt}}");
+defineMacro("\u22EE", "\\vdots"); //////////////////////////////////////////////////////////////////////
+// amsmath.sty
+// http://mirrors.concertpass.com/tex-archive/macros/latex/required/amsmath/amsmath.pdf
+// Italic Greek capital letters.  AMS defines these with \DeclareMathSymbol,
+// but they are equivalent to \mathit{\Letter}.
+
+defineMacro("\\varGamma", "\\mathit{\\Gamma}");
+defineMacro("\\varDelta", "\\mathit{\\Delta}");
+defineMacro("\\varTheta", "\\mathit{\\Theta}");
+defineMacro("\\varLambda", "\\mathit{\\Lambda}");
+defineMacro("\\varXi", "\\mathit{\\Xi}");
+defineMacro("\\varPi", "\\mathit{\\Pi}");
+defineMacro("\\varSigma", "\\mathit{\\Sigma}");
+defineMacro("\\varUpsilon", "\\mathit{\\Upsilon}");
+defineMacro("\\varPhi", "\\mathit{\\Phi}");
+defineMacro("\\varPsi", "\\mathit{\\Psi}");
+defineMacro("\\varOmega", "\\mathit{\\Omega}"); //\newcommand{\substack}[1]{\subarray{c}#1\endsubarray}
+
+defineMacro("\\substack", "\\begin{subarray}{c}#1\\end{subarray}"); // \renewcommand{\colon}{\nobreak\mskip2mu\mathpunct{}\nonscript
+// \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
+
+defineMacro("\\colon", "\\nobreak\\mskip2mu\\mathpunct{}" + "\\mathchoice{\\mkern-3mu}{\\mkern-3mu}{}{}{:}\\mskip6mu"); // \newcommand{\boxed}[1]{\fbox{\m@th$\displaystyle#1$}}
+
+defineMacro("\\boxed", "\\fbox{$\\displaystyle{#1}$}"); // \def\iff{\DOTSB\;\Longleftrightarrow\;}
+// \def\implies{\DOTSB\;\Longrightarrow\;}
+// \def\impliedby{\DOTSB\;\Longleftarrow\;}
+
+defineMacro("\\iff", "\\DOTSB\\;\\Longleftrightarrow\\;");
+defineMacro("\\implies", "\\DOTSB\\;\\Longrightarrow\\;");
+defineMacro("\\impliedby", "\\DOTSB\\;\\Longleftarrow\\;"); // AMSMath's automatic \dots, based on \mdots@@ macro.
+
+var dotsByToken = {
+  ',': '\\dotsc',
+  '\\not': '\\dotsb',
+  // \keybin@ checks for the following:
+  '+': '\\dotsb',
+  '=': '\\dotsb',
+  '<': '\\dotsb',
+  '>': '\\dotsb',
+  '-': '\\dotsb',
+  '*': '\\dotsb',
+  ':': '\\dotsb',
+  // Symbols whose definition starts with \DOTSB:
+  '\\DOTSB': '\\dotsb',
+  '\\coprod': '\\dotsb',
+  '\\bigvee': '\\dotsb',
+  '\\bigwedge': '\\dotsb',
+  '\\biguplus': '\\dotsb',
+  '\\bigcap': '\\dotsb',
+  '\\bigcup': '\\dotsb',
+  '\\prod': '\\dotsb',
+  '\\sum': '\\dotsb',
+  '\\bigotimes': '\\dotsb',
+  '\\bigoplus': '\\dotsb',
+  '\\bigodot': '\\dotsb',
+  '\\bigsqcup': '\\dotsb',
+  '\\And': '\\dotsb',
+  '\\longrightarrow': '\\dotsb',
+  '\\Longrightarrow': '\\dotsb',
+  '\\longleftarrow': '\\dotsb',
+  '\\Longleftarrow': '\\dotsb',
+  '\\longleftrightarrow': '\\dotsb',
+  '\\Longleftrightarrow': '\\dotsb',
+  '\\mapsto': '\\dotsb',
+  '\\longmapsto': '\\dotsb',
+  '\\hookrightarrow': '\\dotsb',
+  '\\doteq': '\\dotsb',
+  // Symbols whose definition starts with \mathbin:
+  '\\mathbin': '\\dotsb',
+  // Symbols whose definition starts with \mathrel:
+  '\\mathrel': '\\dotsb',
+  '\\relbar': '\\dotsb',
+  '\\Relbar': '\\dotsb',
+  '\\xrightarrow': '\\dotsb',
+  '\\xleftarrow': '\\dotsb',
+  // Symbols whose definition starts with \DOTSI:
+  '\\DOTSI': '\\dotsi',
+  '\\int': '\\dotsi',
+  '\\oint': '\\dotsi',
+  '\\iint': '\\dotsi',
+  '\\iiint': '\\dotsi',
+  '\\iiiint': '\\dotsi',
+  '\\idotsint': '\\dotsi',
+  // Symbols whose definition starts with \DOTSX:
+  '\\DOTSX': '\\dotsx'
+};
+defineMacro("\\dots", function (context) {
+  // TODO: If used in text mode, should expand to \textellipsis.
+  // However, in KaTeX, \textellipsis and \ldots behave the same
+  // (in text mode), and it's unlikely we'd see any of the math commands
+  // that affect the behavior of \dots when in text mode.  So fine for now
+  // (until we support \ifmmode ... \else ... \fi).
+  var thedots = '\\dotso';
+  var next = context.expandAfterFuture().text;
+
+  if (next in dotsByToken) {
+    thedots = dotsByToken[next];
+  } else if (next.substr(0, 4) === '\\not') {
+    thedots = '\\dotsb';
+  } else if (next in src_symbols.math) {
+    if (utils.contains(['bin', 'rel'], src_symbols.math[next].group)) {
+      thedots = '\\dotsb';
+    }
+  }
+
+  return thedots;
+});
+var spaceAfterDots = {
+  // \rightdelim@ checks for the following:
+  ')': true,
+  ']': true,
+  '\\rbrack': true,
+  '\\}': true,
+  '\\rbrace': true,
+  '\\rangle': true,
+  '\\rceil': true,
+  '\\rfloor': true,
+  '\\rgroup': true,
+  '\\rmoustache': true,
+  '\\right': true,
+  '\\bigr': true,
+  '\\biggr': true,
+  '\\Bigr': true,
+  '\\Biggr': true,
+  // \extra@ also tests for the following:
+  '$': true,
+  // \extrap@ checks for the following:
+  ';': true,
+  '.': true,
+  ',': true
+};
+defineMacro("\\dotso", function (context) {
+  var next = context.future().text;
+
+  if (next in spaceAfterDots) {
+    return "\\ldots\\,";
+  } else {
+    return "\\ldots";
+  }
+});
+defineMacro("\\dotsc", function (context) {
+  var next = context.future().text; // \dotsc uses \extra@ but not \extrap@, instead specially checking for
+  // ';' and '.', but doesn't check for ','.
+
+  if (next in spaceAfterDots && next !== ',') {
+    return "\\ldots\\,";
+  } else {
+    return "\\ldots";
+  }
+});
+defineMacro("\\cdots", function (context) {
+  var next = context.future().text;
+
+  if (next in spaceAfterDots) {
+    return "\\@cdots\\,";
+  } else {
+    return "\\@cdots";
+  }
+});
+defineMacro("\\dotsb", "\\cdots");
+defineMacro("\\dotsm", "\\cdots");
+defineMacro("\\dotsi", "\\!\\cdots"); // amsmath doesn't actually define \dotsx, but \dots followed by a macro
+// starting with \DOTSX implies \dotso, and then \extra@ detects this case
+// and forces the added `\,`.
+
+defineMacro("\\dotsx", "\\ldots\\,"); // \let\DOTSI\relax
+// \let\DOTSB\relax
+// \let\DOTSX\relax
+
+defineMacro("\\DOTSI", "\\relax");
+defineMacro("\\DOTSB", "\\relax");
+defineMacro("\\DOTSX", "\\relax"); // Spacing, based on amsmath.sty's override of LaTeX defaults
+// \DeclareRobustCommand{\tmspace}[3]{%
+//   \ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
+
+defineMacro("\\tmspace", "\\TextOrMath{\\kern#1#3}{\\mskip#1#2}\\relax"); // \renewcommand{\,}{\tmspace+\thinmuskip{.1667em}}
+// TODO: math mode should use \thinmuskip
+
+defineMacro("\\,", "\\tmspace+{3mu}{.1667em}"); // \let\thinspace\,
+
+defineMacro("\\thinspace", "\\,"); // \def\>{\mskip\medmuskip}
+// \renewcommand{\:}{\tmspace+\medmuskip{.2222em}}
+// TODO: \> and math mode of \: should use \medmuskip = 4mu plus 2mu minus 4mu
+
+defineMacro("\\>", "\\mskip{4mu}");
+defineMacro("\\:", "\\tmspace+{4mu}{.2222em}"); // \let\medspace\:
+
+defineMacro("\\medspace", "\\:"); // \renewcommand{\;}{\tmspace+\thickmuskip{.2777em}}
+// TODO: math mode should use \thickmuskip = 5mu plus 5mu
+
+defineMacro("\\;", "\\tmspace+{5mu}{.2777em}"); // \let\thickspace\;
+
+defineMacro("\\thickspace", "\\;"); // \renewcommand{\!}{\tmspace-\thinmuskip{.1667em}}
+// TODO: math mode should use \thinmuskip
+
+defineMacro("\\!", "\\tmspace-{3mu}{.1667em}"); // \let\negthinspace\!
+
+defineMacro("\\negthinspace", "\\!"); // \newcommand{\negmedspace}{\tmspace-\medmuskip{.2222em}}
+// TODO: math mode should use \medmuskip
+
+defineMacro("\\negmedspace", "\\tmspace-{4mu}{.2222em}"); // \newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}}
+// TODO: math mode should use \thickmuskip
+
+defineMacro("\\negthickspace", "\\tmspace-{5mu}{.277em}"); // \def\enspace{\kern.5em }
+
+defineMacro("\\enspace", "\\kern.5em "); // \def\enskip{\hskip.5em\relax}
+
+defineMacro("\\enskip", "\\hskip.5em\\relax"); // \def\quad{\hskip1em\relax}
+
+defineMacro("\\quad", "\\hskip1em\\relax"); // \def\qquad{\hskip2em\relax}
+
+defineMacro("\\qquad", "\\hskip2em\\relax"); // \tag@in@display form of \tag
+
+defineMacro("\\tag", "\\@ifstar\\tag@literal\\tag@paren");
+defineMacro("\\tag@paren", "\\tag@literal{({#1})}");
+defineMacro("\\tag@literal", function (context) {
+  if (context.macros.get("\\df@tag")) {
+    throw new src_ParseError("Multiple \\tag");
+  }
+
+  return "\\gdef\\df@tag{\\text{#1}}";
+}); // \renewcommand{\bmod}{\nonscript\mskip-\medmuskip\mkern5mu\mathbin
+//   {\operator@font mod}\penalty900
+//   \mkern5mu\nonscript\mskip-\medmuskip}
+// \newcommand{\pod}[1]{\allowbreak
+//   \if@display\mkern18mu\else\mkern8mu\fi(#1)}
+// \renewcommand{\pmod}[1]{\pod{{\operator@font mod}\mkern6mu#1}}
+// \newcommand{\mod}[1]{\allowbreak\if@display\mkern18mu
+//   \else\mkern12mu\fi{\operator@font mod}\,\,#1}
+// TODO: math mode should use \medmuskip = 4mu plus 2mu minus 4mu
+
+defineMacro("\\bmod", "\\mathchoice{\\mskip1mu}{\\mskip1mu}{\\mskip5mu}{\\mskip5mu}" + "\\mathbin{\\rm mod}" + "\\mathchoice{\\mskip1mu}{\\mskip1mu}{\\mskip5mu}{\\mskip5mu}");
+defineMacro("\\pod", "\\allowbreak" + "\\mathchoice{\\mkern18mu}{\\mkern8mu}{\\mkern8mu}{\\mkern8mu}(#1)");
+defineMacro("\\pmod", "\\pod{{\\rm mod}\\mkern6mu#1}");
+defineMacro("\\mod", "\\allowbreak" + "\\mathchoice{\\mkern18mu}{\\mkern12mu}{\\mkern12mu}{\\mkern12mu}" + "{\\rm mod}\\,\\,#1"); // \pmb    --   A simulation of bold.
+// The version in ambsy.sty works by typesetting three copies of the argument
+// with small offsets. We use two copies. We omit the vertical offset because
+// of rendering problems that makeVList encounters in Safari.
+
+defineMacro("\\pmb", "\\html@mathml{" + "\\@binrel{#1}{\\mathrlap{#1}\\kern0.5px#1}}" + "{\\mathbf{#1}}"); //////////////////////////////////////////////////////////////////////
+// LaTeX source2e
+// \\ defaults to \newline, but changes to \cr within array environment
+
+defineMacro("\\\\", "\\newline"); // \def\TeX{T\kern-.1667em\lower.5ex\hbox{E}\kern-.125emX\@}
+// TODO: Doesn't normally work in math mode because \@ fails.  KaTeX doesn't
+// support \@ yet, so that's omitted, and we add \text so that the result
+// doesn't look funny in math mode.
+
+defineMacro("\\TeX", "\\textrm{\\html@mathml{" + "T\\kern-.1667em\\raisebox{-.5ex}{E}\\kern-.125emX" + "}{TeX}}"); // \DeclareRobustCommand{\LaTeX}{L\kern-.36em%
+//         {\sbox\z@ T%
+//          \vbox to\ht\z@{\hbox{\check@mathfonts
+//                               \fontsize\sf@size\z@
+//                               \math@fontsfalse\selectfont
+//                               A}%
+//                         \vss}%
+//         }%
+//         \kern-.15em%
+//         \TeX}
+// This code aligns the top of the A with the T (from the perspective of TeX's
+// boxes, though visually the A appears to extend above slightly).
+// We compute the corresponding \raisebox when A is rendered in \normalsize
+// \scriptstyle, which has a scale factor of 0.7 (see Options.js).
+
+var latexRaiseA = fontMetricsData['Main-Regular']["T".charCodeAt(0)][1] - 0.7 * fontMetricsData['Main-Regular']["A".charCodeAt(0)][1] + "em";
+defineMacro("\\LaTeX", "\\textrm{\\html@mathml{" + ("L\\kern-.36em\\raisebox{" + latexRaiseA + "}{\\scriptstyle A}") + "\\kern-.15em\\TeX}{LaTeX}}"); // New KaTeX logo based on tweaking LaTeX logo
+
+defineMacro("\\KaTeX", "\\textrm{\\html@mathml{" + ("K\\kern-.17em\\raisebox{" + latexRaiseA + "}{\\scriptstyle A}") + "\\kern-.15em\\TeX}{KaTeX}}"); // \DeclareRobustCommand\hspace{\@ifstar\@hspacer\@hspace}
+// \def\@hspace#1{\hskip  #1\relax}
+// \def\@hspacer#1{\vrule \@width\z@\nobreak
+//                 \hskip #1\hskip \z@skip}
+
+defineMacro("\\hspace", "\\@ifstar\\@hspacer\\@hspace");
+defineMacro("\\@hspace", "\\hskip #1\\relax");
+defineMacro("\\@hspacer", "\\rule{0pt}{0pt}\\hskip #1\\relax"); //////////////////////////////////////////////////////////////////////
+// mathtools.sty
+//\providecommand\ordinarycolon{:}
+
+defineMacro("\\ordinarycolon", ":"); //\def\vcentcolon{\mathrel{\mathop\ordinarycolon}}
+//TODO(edemaine): Not yet centered. Fix via \raisebox or #726
+
+defineMacro("\\vcentcolon", "\\mathrel{\\mathop\\ordinarycolon}"); // \providecommand*\dblcolon{\vcentcolon\mathrel{\mkern-.9mu}\vcentcolon}
+
+defineMacro("\\dblcolon", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-.9mu}\\vcentcolon}}" + "{\\mathop{\\char\"2237}}"); // \providecommand*\coloneqq{\vcentcolon\mathrel{\mkern-1.2mu}=}
+
+defineMacro("\\coloneqq", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}=}}" + "{\\mathop{\\char\"2254}}"); // ≔
+// \providecommand*\Coloneqq{\dblcolon\mathrel{\mkern-1.2mu}=}
+
+defineMacro("\\Coloneqq", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}=}}" + "{\\mathop{\\char\"2237\\char\"3d}}"); // \providecommand*\coloneq{\vcentcolon\mathrel{\mkern-1.2mu}\mathrel{-}}
+
+defineMacro("\\coloneq", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\mathrel{-}}}" + "{\\mathop{\\char\"3a\\char\"2212}}"); // \providecommand*\Coloneq{\dblcolon\mathrel{\mkern-1.2mu}\mathrel{-}}
+
+defineMacro("\\Coloneq", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\mathrel{-}}}" + "{\\mathop{\\char\"2237\\char\"2212}}"); // \providecommand*\eqqcolon{=\mathrel{\mkern-1.2mu}\vcentcolon}
+
+defineMacro("\\eqqcolon", "\\html@mathml{" + "\\mathrel{=\\mathrel{\\mkern-1.2mu}\\vcentcolon}}" + "{\\mathop{\\char\"2255}}"); // ≕
+// \providecommand*\Eqqcolon{=\mathrel{\mkern-1.2mu}\dblcolon}
+
+defineMacro("\\Eqqcolon", "\\html@mathml{" + "\\mathrel{=\\mathrel{\\mkern-1.2mu}\\dblcolon}}" + "{\\mathop{\\char\"3d\\char\"2237}}"); // \providecommand*\eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\vcentcolon}
+
+defineMacro("\\eqcolon", "\\html@mathml{" + "\\mathrel{\\mathrel{-}\\mathrel{\\mkern-1.2mu}\\vcentcolon}}" + "{\\mathop{\\char\"2239}}"); // \providecommand*\Eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\dblcolon}
+
+defineMacro("\\Eqcolon", "\\html@mathml{" + "\\mathrel{\\mathrel{-}\\mathrel{\\mkern-1.2mu}\\dblcolon}}" + "{\\mathop{\\char\"2212\\char\"2237}}"); // \providecommand*\colonapprox{\vcentcolon\mathrel{\mkern-1.2mu}\approx}
+
+defineMacro("\\colonapprox", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\approx}}" + "{\\mathop{\\char\"3a\\char\"2248}}"); // \providecommand*\Colonapprox{\dblcolon\mathrel{\mkern-1.2mu}\approx}
+
+defineMacro("\\Colonapprox", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\approx}}" + "{\\mathop{\\char\"2237\\char\"2248}}"); // \providecommand*\colonsim{\vcentcolon\mathrel{\mkern-1.2mu}\sim}
+
+defineMacro("\\colonsim", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\sim}}" + "{\\mathop{\\char\"3a\\char\"223c}}"); // \providecommand*\Colonsim{\dblcolon\mathrel{\mkern-1.2mu}\sim}
+
+defineMacro("\\Colonsim", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\sim}}" + "{\\mathop{\\char\"2237\\char\"223c}}"); // Some Unicode characters are implemented with macros to mathtools functions.
+
+defineMacro("\u2237", "\\dblcolon"); // ::
+
+defineMacro("\u2239", "\\eqcolon"); // -:
+
+defineMacro("\u2254", "\\coloneqq"); // :=
+
+defineMacro("\u2255", "\\eqqcolon"); // =:
+
+defineMacro("\u2A74", "\\Coloneqq"); // ::=
+//////////////////////////////////////////////////////////////////////
+// colonequals.sty
+// Alternate names for mathtools's macros:
+
+defineMacro("\\ratio", "\\vcentcolon");
+defineMacro("\\coloncolon", "\\dblcolon");
+defineMacro("\\colonequals", "\\coloneqq");
+defineMacro("\\coloncolonequals", "\\Coloneqq");
+defineMacro("\\equalscolon", "\\eqqcolon");
+defineMacro("\\equalscoloncolon", "\\Eqqcolon");
+defineMacro("\\colonminus", "\\coloneq");
+defineMacro("\\coloncolonminus", "\\Coloneq");
+defineMacro("\\minuscolon", "\\eqcolon");
+defineMacro("\\minuscoloncolon", "\\Eqcolon"); // \colonapprox name is same in mathtools and colonequals.
+
+defineMacro("\\coloncolonapprox", "\\Colonapprox"); // \colonsim name is same in mathtools and colonequals.
+
+defineMacro("\\coloncolonsim", "\\Colonsim"); // Additional macros, implemented by analogy with mathtools definitions:
+
+defineMacro("\\simcolon", "\\mathrel{\\sim\\mathrel{\\mkern-1.2mu}\\vcentcolon}");
+defineMacro("\\simcoloncolon", "\\mathrel{\\sim\\mathrel{\\mkern-1.2mu}\\dblcolon}");
+defineMacro("\\approxcolon", "\\mathrel{\\approx\\mathrel{\\mkern-1.2mu}\\vcentcolon}");
+defineMacro("\\approxcoloncolon", "\\mathrel{\\approx\\mathrel{\\mkern-1.2mu}\\dblcolon}"); // Present in newtxmath, pxfonts and txfonts
+
+defineMacro("\\notni", "\\html@mathml{\\not\\ni}{\\mathrel{\\char`\u220C}}");
+defineMacro("\\limsup", "\\DOTSB\\operatorname*{lim\\,sup}");
+defineMacro("\\liminf", "\\DOTSB\\operatorname*{lim\\,inf}"); //////////////////////////////////////////////////////////////////////
+// MathML alternates for KaTeX glyphs in the Unicode private area
+
+defineMacro("\\gvertneqq", "\\html@mathml{\\@gvertneqq}{\u2269}");
+defineMacro("\\lvertneqq", "\\html@mathml{\\@lvertneqq}{\u2268}");
+defineMacro("\\ngeqq", "\\html@mathml{\\@ngeqq}{\u2271}");
+defineMacro("\\ngeqslant", "\\html@mathml{\\@ngeqslant}{\u2271}");
+defineMacro("\\nleqq", "\\html@mathml{\\@nleqq}{\u2270}");
+defineMacro("\\nleqslant", "\\html@mathml{\\@nleqslant}{\u2270}");
+defineMacro("\\nshortmid", "\\html@mathml{\\@nshortmid}{∤}");
+defineMacro("\\nshortparallel", "\\html@mathml{\\@nshortparallel}{∦}");
+defineMacro("\\nsubseteqq", "\\html@mathml{\\@nsubseteqq}{\u2288}");
+defineMacro("\\nsupseteqq", "\\html@mathml{\\@nsupseteqq}{\u2289}");
+defineMacro("\\varsubsetneq", "\\html@mathml{\\@varsubsetneq}{⊊}");
+defineMacro("\\varsubsetneqq", "\\html@mathml{\\@varsubsetneqq}{⫋}");
+defineMacro("\\varsupsetneq", "\\html@mathml{\\@varsupsetneq}{⊋}");
+defineMacro("\\varsupsetneqq", "\\html@mathml{\\@varsupsetneqq}{⫌}");
+defineMacro("\\imath", "\\html@mathml{\\@imath}{\u0131}");
+defineMacro("\\jmath", "\\html@mathml{\\@jmath}{\u0237}"); //////////////////////////////////////////////////////////////////////
+// stmaryrd and semantic
+// The stmaryrd and semantic packages render the next four items by calling a
+// glyph. Those glyphs do not exist in the KaTeX fonts. Hence the macros.
+
+defineMacro("\\llbracket", "\\html@mathml{" + "\\mathopen{[\\mkern-3.2mu[}}" + "{\\mathopen{\\char`\u27E6}}");
+defineMacro("\\rrbracket", "\\html@mathml{" + "\\mathclose{]\\mkern-3.2mu]}}" + "{\\mathclose{\\char`\u27E7}}");
+defineMacro("\u27E6", "\\llbracket"); // blackboard bold [
+
+defineMacro("\u27E7", "\\rrbracket"); // blackboard bold ]
+
+defineMacro("\\lBrace", "\\html@mathml{" + "\\mathopen{\\{\\mkern-3.2mu[}}" + "{\\mathopen{\\char`\u2983}}");
+defineMacro("\\rBrace", "\\html@mathml{" + "\\mathclose{]\\mkern-3.2mu\\}}}" + "{\\mathclose{\\char`\u2984}}");
+defineMacro("\u2983", "\\lBrace"); // blackboard bold {
+
+defineMacro("\u2984", "\\rBrace"); // blackboard bold }
+// TODO: Create variable sized versions of the last two items. I believe that
+// will require new font glyphs.
+// The stmaryrd function `\minuso` provides a "Plimsoll" symbol that
+// superimposes the characters \circ and \mathminus. Used in chemistry.
+
+defineMacro("\\minuso", "\\mathbin{\\html@mathml{" + "{\\mathrlap{\\mathchoice{\\kern{0.145em}}{\\kern{0.145em}}" + "{\\kern{0.1015em}}{\\kern{0.0725em}}\\circ}{-}}}" + "{\\char`⦵}}");
+defineMacro("⦵", "\\minuso"); //////////////////////////////////////////////////////////////////////
+// texvc.sty
+// The texvc package contains macros available in mediawiki pages.
+// We omit the functions deprecated at
+// https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax
+// We also omit texvc's \O, which conflicts with \text{\O}
+
+defineMacro("\\darr", "\\downarrow");
+defineMacro("\\dArr", "\\Downarrow");
+defineMacro("\\Darr", "\\Downarrow");
+defineMacro("\\lang", "\\langle");
+defineMacro("\\rang", "\\rangle");
+defineMacro("\\uarr", "\\uparrow");
+defineMacro("\\uArr", "\\Uparrow");
+defineMacro("\\Uarr", "\\Uparrow");
+defineMacro("\\N", "\\mathbb{N}");
+defineMacro("\\R", "\\mathbb{R}");
+defineMacro("\\Z", "\\mathbb{Z}");
+defineMacro("\\alef", "\\aleph");
+defineMacro("\\alefsym", "\\aleph");
+defineMacro("\\Alpha", "\\mathrm{A}");
+defineMacro("\\Beta", "\\mathrm{B}");
+defineMacro("\\bull", "\\bullet");
+defineMacro("\\Chi", "\\mathrm{X}");
+defineMacro("\\clubs", "\\clubsuit");
+defineMacro("\\cnums", "\\mathbb{C}");
+defineMacro("\\Complex", "\\mathbb{C}");
+defineMacro("\\Dagger", "\\ddagger");
+defineMacro("\\diamonds", "\\diamondsuit");
+defineMacro("\\empty", "\\emptyset");
+defineMacro("\\Epsilon", "\\mathrm{E}");
+defineMacro("\\Eta", "\\mathrm{H}");
+defineMacro("\\exist", "\\exists");
+defineMacro("\\harr", "\\leftrightarrow");
+defineMacro("\\hArr", "\\Leftrightarrow");
+defineMacro("\\Harr", "\\Leftrightarrow");
+defineMacro("\\hearts", "\\heartsuit");
+defineMacro("\\image", "\\Im");
+defineMacro("\\infin", "\\infty");
+defineMacro("\\Iota", "\\mathrm{I}");
+defineMacro("\\isin", "\\in");
+defineMacro("\\Kappa", "\\mathrm{K}");
+defineMacro("\\larr", "\\leftarrow");
+defineMacro("\\lArr", "\\Leftarrow");
+defineMacro("\\Larr", "\\Leftarrow");
+defineMacro("\\lrarr", "\\leftrightarrow");
+defineMacro("\\lrArr", "\\Leftrightarrow");
+defineMacro("\\Lrarr", "\\Leftrightarrow");
+defineMacro("\\Mu", "\\mathrm{M}");
+defineMacro("\\natnums", "\\mathbb{N}");
+defineMacro("\\Nu", "\\mathrm{N}");
+defineMacro("\\Omicron", "\\mathrm{O}");
+defineMacro("\\plusmn", "\\pm");
+defineMacro("\\rarr", "\\rightarrow");
+defineMacro("\\rArr", "\\Rightarrow");
+defineMacro("\\Rarr", "\\Rightarrow");
+defineMacro("\\real", "\\Re");
+defineMacro("\\reals", "\\mathbb{R}");
+defineMacro("\\Reals", "\\mathbb{R}");
+defineMacro("\\Rho", "\\mathrm{P}");
+defineMacro("\\sdot", "\\cdot");
+defineMacro("\\sect", "\\S");
+defineMacro("\\spades", "\\spadesuit");
+defineMacro("\\sub", "\\subset");
+defineMacro("\\sube", "\\subseteq");
+defineMacro("\\supe", "\\supseteq");
+defineMacro("\\Tau", "\\mathrm{T}");
+defineMacro("\\thetasym", "\\vartheta"); // TODO: defineMacro("\\varcoppa", "\\\mbox{\\coppa}");
+
+defineMacro("\\weierp", "\\wp");
+defineMacro("\\Zeta", "\\mathrm{Z}"); //////////////////////////////////////////////////////////////////////
+// statmath.sty
+// https://ctan.math.illinois.edu/macros/latex/contrib/statmath/statmath.pdf
+
+defineMacro("\\argmin", "\\DOTSB\\operatorname*{arg\\,min}");
+defineMacro("\\argmax", "\\DOTSB\\operatorname*{arg\\,max}");
+defineMacro("\\plim", "\\DOTSB\\mathop{\\operatorname{plim}}\\limits"); //////////////////////////////////////////////////////////////////////
+// braket.sty
+// http://ctan.math.washington.edu/tex-archive/macros/latex/contrib/braket/braket.pdf
+
+defineMacro("\\bra", "\\mathinner{\\langle{#1}|}");
+defineMacro("\\ket", "\\mathinner{|{#1}\\rangle}");
+defineMacro("\\braket", "\\mathinner{\\langle{#1}\\rangle}");
+defineMacro("\\Bra", "\\left\\langle#1\\right|");
+defineMacro("\\Ket", "\\left|#1\\right\\rangle"); // Custom Khan Academy colors, should be moved to an optional package
+
+defineMacro("\\blue", "\\textcolor{##6495ed}{#1}");
+defineMacro("\\orange", "\\textcolor{##ffa500}{#1}");
+defineMacro("\\pink", "\\textcolor{##ff00af}{#1}");
+defineMacro("\\red", "\\textcolor{##df0030}{#1}");
+defineMacro("\\green", "\\textcolor{##28ae7b}{#1}");
+defineMacro("\\gray", "\\textcolor{gray}{#1}");
+defineMacro("\\purple", "\\textcolor{##9d38bd}{#1}");
+defineMacro("\\blueA", "\\textcolor{##ccfaff}{#1}");
+defineMacro("\\blueB", "\\textcolor{##80f6ff}{#1}");
+defineMacro("\\blueC", "\\textcolor{##63d9ea}{#1}");
+defineMacro("\\blueD", "\\textcolor{##11accd}{#1}");
+defineMacro("\\blueE", "\\textcolor{##0c7f99}{#1}");
+defineMacro("\\tealA", "\\textcolor{##94fff5}{#1}");
+defineMacro("\\tealB", "\\textcolor{##26edd5}{#1}");
+defineMacro("\\tealC", "\\textcolor{##01d1c1}{#1}");
+defineMacro("\\tealD", "\\textcolor{##01a995}{#1}");
+defineMacro("\\tealE", "\\textcolor{##208170}{#1}");
+defineMacro("\\greenA", "\\textcolor{##b6ffb0}{#1}");
+defineMacro("\\greenB", "\\textcolor{##8af281}{#1}");
+defineMacro("\\greenC", "\\textcolor{##74cf70}{#1}");
+defineMacro("\\greenD", "\\textcolor{##1fab54}{#1}");
+defineMacro("\\greenE", "\\textcolor{##0d923f}{#1}");
+defineMacro("\\goldA", "\\textcolor{##ffd0a9}{#1}");
+defineMacro("\\goldB", "\\textcolor{##ffbb71}{#1}");
+defineMacro("\\goldC", "\\textcolor{##ff9c39}{#1}");
+defineMacro("\\goldD", "\\textcolor{##e07d10}{#1}");
+defineMacro("\\goldE", "\\textcolor{##a75a05}{#1}");
+defineMacro("\\redA", "\\textcolor{##fca9a9}{#1}");
+defineMacro("\\redB", "\\textcolor{##ff8482}{#1}");
+defineMacro("\\redC", "\\textcolor{##f9685d}{#1}");
+defineMacro("\\redD", "\\textcolor{##e84d39}{#1}");
+defineMacro("\\redE", "\\textcolor{##bc2612}{#1}");
+defineMacro("\\maroonA", "\\textcolor{##ffbde0}{#1}");
+defineMacro("\\maroonB", "\\textcolor{##ff92c6}{#1}");
+defineMacro("\\maroonC", "\\textcolor{##ed5fa6}{#1}");
+defineMacro("\\maroonD", "\\textcolor{##ca337c}{#1}");
+defineMacro("\\maroonE", "\\textcolor{##9e034e}{#1}");
+defineMacro("\\purpleA", "\\textcolor{##ddd7ff}{#1}");
+defineMacro("\\purpleB", "\\textcolor{##c6b9fc}{#1}");
+defineMacro("\\purpleC", "\\textcolor{##aa87ff}{#1}");
+defineMacro("\\purpleD", "\\textcolor{##7854ab}{#1}");
+defineMacro("\\purpleE", "\\textcolor{##543b78}{#1}");
+defineMacro("\\mintA", "\\textcolor{##f5f9e8}{#1}");
+defineMacro("\\mintB", "\\textcolor{##edf2df}{#1}");
+defineMacro("\\mintC", "\\textcolor{##e0e5cc}{#1}");
+defineMacro("\\grayA", "\\textcolor{##f6f7f7}{#1}");
+defineMacro("\\grayB", "\\textcolor{##f0f1f2}{#1}");
+defineMacro("\\grayC", "\\textcolor{##e3e5e6}{#1}");
+defineMacro("\\grayD", "\\textcolor{##d6d8da}{#1}");
+defineMacro("\\grayE", "\\textcolor{##babec2}{#1}");
+defineMacro("\\grayF", "\\textcolor{##888d93}{#1}");
+defineMacro("\\grayG", "\\textcolor{##626569}{#1}");
+defineMacro("\\grayH", "\\textcolor{##3b3e40}{#1}");
+defineMacro("\\grayI", "\\textcolor{##21242c}{#1}");
+defineMacro("\\kaBlue", "\\textcolor{##314453}{#1}");
+defineMacro("\\kaGreen", "\\textcolor{##71B307}{#1}");
+// CONCATENATED MODULE: ./src/MacroExpander.js
+/**
+ * This file contains the “gullet” where macros are expanded
+ * until only non-macro tokens remain.
+ */
+
+
+
+
+
+
+
+// List of commands that act like macros but aren't defined as a macro,
+// function, or symbol.  Used in `isDefined`.
+var implicitCommands = {
+  "\\relax": true,
+  // MacroExpander.js
+  "^": true,
+  // Parser.js
+  "_": true,
+  // Parser.js
+  "\\limits": true,
+  // Parser.js
+  "\\nolimits": true // Parser.js
+
+};
+
+var MacroExpander_MacroExpander =
+/*#__PURE__*/
+function () {
+  function MacroExpander(input, settings, mode) {
+    this.settings = void 0;
+    this.expansionCount = void 0;
+    this.lexer = void 0;
+    this.macros = void 0;
+    this.stack = void 0;
+    this.mode = void 0;
+    this.settings = settings;
+    this.expansionCount = 0;
+    this.feed(input); // Make new global namespace
+
+    this.macros = new Namespace_Namespace(macros, settings.macros);
+    this.mode = mode;
+    this.stack = []; // contains tokens in REVERSE order
+  }
+  /**
+   * Feed a new input string to the same MacroExpander
+   * (with existing macros etc.).
+   */
+
+
+  var _proto = MacroExpander.prototype;
+
+  _proto.feed = function feed(input) {
+    this.lexer = new Lexer_Lexer(input, this.settings);
+  }
+  /**
+   * Switches between "text" and "math" modes.
+   */
+  ;
+
+  _proto.switchMode = function switchMode(newMode) {
+    this.mode = newMode;
+  }
+  /**
+   * Start a new group nesting within all namespaces.
+   */
+  ;
+
+  _proto.beginGroup = function beginGroup() {
+    this.macros.beginGroup();
+  }
+  /**
+   * End current group nesting within all namespaces.
+   */
+  ;
+
+  _proto.endGroup = function endGroup() {
+    this.macros.endGroup();
+  }
+  /**
+   * Returns the topmost token on the stack, without expanding it.
+   * Similar in behavior to TeX's `\futurelet`.
+   */
+  ;
+
+  _proto.future = function future() {
+    if (this.stack.length === 0) {
+      this.pushToken(this.lexer.lex());
+    }
+
+    return this.stack[this.stack.length - 1];
+  }
+  /**
+   * Remove and return the next unexpanded token.
+   */
+  ;
+
+  _proto.popToken = function popToken() {
+    this.future(); // ensure non-empty stack
+
+    return this.stack.pop();
+  }
+  /**
+   * Add a given token to the token stack.  In particular, this get be used
+   * to put back a token returned from one of the other methods.
+   */
+  ;
+
+  _proto.pushToken = function pushToken(token) {
+    this.stack.push(token);
+  }
+  /**
+   * Append an array of tokens to the token stack.
+   */
+  ;
+
+  _proto.pushTokens = function pushTokens(tokens) {
+    var _this$stack;
+
+    (_this$stack = this.stack).push.apply(_this$stack, tokens);
+  }
+  /**
+   * Consume all following space tokens, without expansion.
+   */
+  ;
+
+  _proto.consumeSpaces = function consumeSpaces() {
+    for (;;) {
+      var token = this.future();
+
+      if (token.text === " ") {
+        this.stack.pop();
+      } else {
+        break;
+      }
+    }
+  }
+  /**
+   * Consume the specified number of arguments from the token stream,
+   * and return the resulting array of arguments.
+   */
+  ;
+
+  _proto.consumeArgs = function consumeArgs(numArgs) {
+    var args = []; // obtain arguments, either single token or balanced {…} group
+
+    for (var i = 0; i < numArgs; ++i) {
+      this.consumeSpaces(); // ignore spaces before each argument
+
+      var startOfArg = this.popToken();
+
+      if (startOfArg.text === "{") {
+        var arg = [];
+        var depth = 1;
+
+        while (depth !== 0) {
+          var tok = this.popToken();
+          arg.push(tok);
+
+          if (tok.text === "{") {
+            ++depth;
+          } else if (tok.text === "}") {
+            --depth;
+          } else if (tok.text === "EOF") {
+            throw new src_ParseError("End of input in macro argument", startOfArg);
+          }
+        }
+
+        arg.pop(); // remove last }
+
+        arg.reverse(); // like above, to fit in with stack order
+
+        args[i] = arg;
+      } else if (startOfArg.text === "EOF") {
+        throw new src_ParseError("End of input expecting macro argument");
+      } else {
+        args[i] = [startOfArg];
+      }
+    }
+
+    return args;
+  }
+  /**
+   * Expand the next token only once if possible.
+   *
+   * If the token is expanded, the resulting tokens will be pushed onto
+   * the stack in reverse order and will be returned as an array,
+   * also in reverse order.
+   *
+   * If not, the next token will be returned without removing it
+   * from the stack.  This case can be detected by a `Token` return value
+   * instead of an `Array` return value.
+   *
+   * In either case, the next token will be on the top of the stack,
+   * or the stack will be empty.
+   *
+   * Used to implement `expandAfterFuture` and `expandNextToken`.
+   *
+   * At the moment, macro expansion doesn't handle delimited macros,
+   * i.e. things like those defined by \def\foo#1\end{…}.
+   * See the TeX book page 202ff. for details on how those should behave.
+   *
+   * If expandableOnly, only expandable tokens are expanded and
+   * an undefined control sequence results in an error.
+   */
+  ;
+
+  _proto.expandOnce = function expandOnce(expandableOnly) {
+    var topToken = this.popToken();
+    var name = topToken.text;
+    var expansion = !topToken.noexpand ? this._getExpansion(name) : null;
+
+    if (expansion == null || expandableOnly && expansion.unexpandable) {
+      if (expandableOnly && expansion == null && name[0] === "\\" && !this.isDefined(name)) {
+        throw new src_ParseError("Undefined control sequence: " + name);
+      }
+
+      this.pushToken(topToken);
+      return topToken;
+    }
+
+    this.expansionCount++;
+
+    if (this.expansionCount > this.settings.maxExpand) {
+      throw new src_ParseError("Too many expansions: infinite loop or " + "need to increase maxExpand setting");
+    }
+
+    var tokens = expansion.tokens;
+
+    if (expansion.numArgs) {
+      var args = this.consumeArgs(expansion.numArgs); // paste arguments in place of the placeholders
+
+      tokens = tokens.slice(); // make a shallow copy
+
+      for (var i = tokens.length - 1; i >= 0; --i) {
+        var tok = tokens[i];
+
+        if (tok.text === "#") {
+          if (i === 0) {
+            throw new src_ParseError("Incomplete placeholder at end of macro body", tok);
+          }
+
+          tok = tokens[--i]; // next token on stack
+
+          if (tok.text === "#") {
+            // ## → #
+            tokens.splice(i + 1, 1); // drop first #
+          } else if (/^[1-9]$/.test(tok.text)) {
+            var _tokens;
+
+            // replace the placeholder with the indicated argument
+            (_tokens = tokens).splice.apply(_tokens, [i, 2].concat(args[+tok.text - 1]));
+          } else {
+            throw new src_ParseError("Not a valid argument number", tok);
+          }
+        }
+      }
+    } // Concatenate expansion onto top of stack.
+
+
+    this.pushTokens(tokens);
+    return tokens;
+  }
+  /**
+   * Expand the next token only once (if possible), and return the resulting
+   * top token on the stack (without removing anything from the stack).
+   * Similar in behavior to TeX's `\expandafter\futurelet`.
+   * Equivalent to expandOnce() followed by future().
+   */
+  ;
+
+  _proto.expandAfterFuture = function expandAfterFuture() {
+    this.expandOnce();
+    return this.future();
+  }
+  /**
+   * Recursively expand first token, then return first non-expandable token.
+   */
+  ;
+
+  _proto.expandNextToken = function expandNextToken() {
+    for (;;) {
+      var expanded = this.expandOnce(); // expandOnce returns Token if and only if it's fully expanded.
+
+      if (expanded instanceof Token_Token) {
+        // \relax stops the expansion, but shouldn't get returned (a
+        // null return value couldn't get implemented as a function).
+        // the token after \noexpand is interpreted as if its meaning
+        // were ‘\relax’
+        if (expanded.text === "\\relax" || expanded.treatAsRelax) {
+          this.stack.pop();
+        } else {
+          return this.stack.pop(); // === expanded
+        }
+      }
+    } // Flow unable to figure out that this pathway is impossible.
+    // https://github.com/facebook/flow/issues/4808
+
+
+    throw new Error(); // eslint-disable-line no-unreachable
+  }
+  /**
+   * Fully expand the given macro name and return the resulting list of
+   * tokens, or return `undefined` if no such macro is defined.
+   */
+  ;
+
+  _proto.expandMacro = function expandMacro(name) {
+    return this.macros.has(name) ? this.expandTokens([new Token_Token(name)]) : undefined;
+  }
+  /**
+   * Fully expand the given token stream and return the resulting list of tokens
+   */
+  ;
+
+  _proto.expandTokens = function expandTokens(tokens) {
+    var output = [];
+    var oldStackLength = this.stack.length;
+    this.pushTokens(tokens);
+
+    while (this.stack.length > oldStackLength) {
+      var expanded = this.expandOnce(true); // expand only expandable tokens
+      // expandOnce returns Token if and only if it's fully expanded.
+
+      if (expanded instanceof Token_Token) {
+        if (expanded.treatAsRelax) {
+          // the expansion of \noexpand is the token itself
+          expanded.noexpand = false;
+          expanded.treatAsRelax = false;
+        }
+
+        output.push(this.stack.pop());
+      }
+    }
+
+    return output;
+  }
+  /**
+   * Fully expand the given macro name and return the result as a string,
+   * or return `undefined` if no such macro is defined.
+   */
+  ;
+
+  _proto.expandMacroAsText = function expandMacroAsText(name) {
+    var tokens = this.expandMacro(name);
+
+    if (tokens) {
+      return tokens.map(function (token) {
+        return token.text;
+      }).join("");
+    } else {
+      return tokens;
+    }
+  }
+  /**
+   * Returns the expanded macro as a reversed array of tokens and a macro
+   * argument count.  Or returns `null` if no such macro.
+   */
+  ;
+
+  _proto._getExpansion = function _getExpansion(name) {
+    var definition = this.macros.get(name);
+
+    if (definition == null) {
+      // mainly checking for undefined here
+      return definition;
+    }
+
+    var expansion = typeof definition === "function" ? definition(this) : definition;
+
+    if (typeof expansion === "string") {
+      var numArgs = 0;
+
+      if (expansion.indexOf("#") !== -1) {
+        var stripped = expansion.replace(/##/g, "");
+
+        while (stripped.indexOf("#" + (numArgs + 1)) !== -1) {
+          ++numArgs;
+        }
+      }
+
+      var bodyLexer = new Lexer_Lexer(expansion, this.settings);
+      var tokens = [];
+      var tok = bodyLexer.lex();
+
+      while (tok.text !== "EOF") {
+        tokens.push(tok);
+        tok = bodyLexer.lex();
+      }
+
+      tokens.reverse(); // to fit in with stack using push and pop
+
+      var expanded = {
+        tokens: tokens,
+        numArgs: numArgs
+      };
+      return expanded;
+    }
+
+    return expansion;
+  }
+  /**
+   * Determine whether a command is currently "defined" (has some
+   * functionality), meaning that it's a macro (in the current group),
+   * a function, a symbol, or one of the special commands listed in
+   * `implicitCommands`.
+   */
+  ;
+
+  _proto.isDefined = function isDefined(name) {
+    return this.macros.has(name) || src_functions.hasOwnProperty(name) || src_symbols.math.hasOwnProperty(name) || src_symbols.text.hasOwnProperty(name) || implicitCommands.hasOwnProperty(name);
+  }
+  /**
+   * Determine whether a command is expandable.
+   */
+  ;
+
+  _proto.isExpandable = function isExpandable(name) {
+    var macro = this.macros.get(name);
+    return macro != null ? typeof macro === "string" || typeof macro === "function" || !macro.unexpandable // TODO(ylem): #2085
+    : src_functions.hasOwnProperty(name)
+    /* && !functions[name].primitive*/
+    ;
+  };
+
+  return MacroExpander;
+}();
+
+
+// CONCATENATED MODULE: ./src/Parser.js
+/* eslint no-constant-condition:0 */
+
+
+
+
+
+
+
+
+
+ // Pre-evaluate both modules as unicodeSymbols require String.normalize()
+
+var unicodeAccents = {
+  "́": {
+    "text": "\\'",
+    "math": "\\acute"
+  },
+  "̀": {
+    "text": "\\`",
+    "math": "\\grave"
+  },
+  "̈": {
+    "text": "\\\"",
+    "math": "\\ddot"
+  },
+  "̃": {
+    "text": "\\~",
+    "math": "\\tilde"
+  },
+  "̄": {
+    "text": "\\=",
+    "math": "\\bar"
+  },
+  "̆": {
+    "text": "\\u",
+    "math": "\\breve"
+  },
+  "̌": {
+    "text": "\\v",
+    "math": "\\check"
+  },
+  "̂": {
+    "text": "\\^",
+    "math": "\\hat"
+  },
+  "̇": {
+    "text": "\\.",
+    "math": "\\dot"
+  },
+  "̊": {
+    "text": "\\r",
+    "math": "\\mathring"
+  },
+  "̋": {
+    "text": "\\H"
+  }
+};
+var unicodeSymbols = {
+  "á": "á",
+  "à": "à",
+  "ä": "ä",
+  "ǟ": "ǟ",
+  "ã": "ã",
+  "ā": "ā",
+  "ă": "ă",
+  "ắ": "ắ",
+  "ằ": "ằ",
+  "ẵ": "ẵ",
+  "ǎ": "ǎ",
+  "â": "â",
+  "ấ": "ấ",
+  "ầ": "ầ",
+  "ẫ": "ẫ",
+  "ȧ": "ȧ",
+  "ǡ": "ǡ",
+  "å": "å",
+  "ǻ": "ǻ",
+  "ḃ": "ḃ",
+  "ć": "ć",
+  "č": "č",
+  "ĉ": "ĉ",
+  "ċ": "ċ",
+  "ď": "ď",
+  "ḋ": "ḋ",
+  "é": "é",
+  "è": "è",
+  "ë": "ë",
+  "ẽ": "ẽ",
+  "ē": "ē",
+  "ḗ": "ḗ",
+  "ḕ": "ḕ",
+  "ĕ": "ĕ",
+  "ě": "ě",
+  "ê": "ê",
+  "ế": "ế",
+  "ề": "ề",
+  "ễ": "ễ",
+  "ė": "ė",
+  "ḟ": "ḟ",
+  "ǵ": "ǵ",
+  "ḡ": "ḡ",
+  "ğ": "ğ",
+  "ǧ": "ǧ",
+  "ĝ": "ĝ",
+  "ġ": "ġ",
+  "ḧ": "ḧ",
+  "ȟ": "ȟ",
+  "ĥ": "ĥ",
+  "ḣ": "ḣ",
+  "í": "í",
+  "ì": "ì",
+  "ï": "ï",
+  "ḯ": "ḯ",
+  "ĩ": "ĩ",
+  "ī": "ī",
+  "ĭ": "ĭ",
+  "ǐ": "ǐ",
+  "î": "î",
+  "ǰ": "ǰ",
+  "ĵ": "ĵ",
+  "ḱ": "ḱ",
+  "ǩ": "ǩ",
+  "ĺ": "ĺ",
+  "ľ": "ľ",
+  "ḿ": "ḿ",
+  "ṁ": "ṁ",
+  "ń": "ń",
+  "ǹ": "ǹ",
+  "ñ": "ñ",
+  "ň": "ň",
+  "ṅ": "ṅ",
+  "ó": "ó",
+  "ò": "ò",
+  "ö": "ö",
+  "ȫ": "ȫ",
+  "õ": "õ",
+  "ṍ": "ṍ",
+  "ṏ": "ṏ",
+  "ȭ": "ȭ",
+  "ō": "ō",
+  "ṓ": "ṓ",
+  "ṑ": "ṑ",
+  "ŏ": "ŏ",
+  "ǒ": "ǒ",
+  "ô": "ô",
+  "ố": "ố",
+  "ồ": "ồ",
+  "ỗ": "ỗ",
+  "ȯ": "ȯ",
+  "ȱ": "ȱ",
+  "ő": "ő",
+  "ṕ": "ṕ",
+  "ṗ": "ṗ",
+  "ŕ": "ŕ",
+  "ř": "ř",
+  "ṙ": "ṙ",
+  "ś": "ś",
+  "ṥ": "ṥ",
+  "š": "š",
+  "ṧ": "ṧ",
+  "ŝ": "ŝ",
+  "ṡ": "ṡ",
+  "ẗ": "ẗ",
+  "ť": "ť",
+  "ṫ": "ṫ",
+  "ú": "ú",
+  "ù": "ù",
+  "ü": "ü",
+  "ǘ": "ǘ",
+  "ǜ": "ǜ",
+  "ǖ": "ǖ",
+  "ǚ": "ǚ",
+  "ũ": "ũ",
+  "ṹ": "ṹ",
+  "ū": "ū",
+  "ṻ": "ṻ",
+  "ŭ": "ŭ",
+  "ǔ": "ǔ",
+  "û": "û",
+  "ů": "ů",
+  "ű": "ű",
+  "ṽ": "ṽ",
+  "ẃ": "ẃ",
+  "ẁ": "ẁ",
+  "ẅ": "ẅ",
+  "ŵ": "ŵ",
+  "ẇ": "ẇ",
+  "ẘ": "ẘ",
+  "ẍ": "ẍ",
+  "ẋ": "ẋ",
+  "ý": "ý",
+  "ỳ": "ỳ",
+  "ÿ": "ÿ",
+  "ỹ": "ỹ",
+  "ȳ": "ȳ",
+  "ŷ": "ŷ",
+  "ẏ": "ẏ",
+  "ẙ": "ẙ",
+  "ź": "ź",
+  "ž": "ž",
+  "ẑ": "ẑ",
+  "ż": "ż",
+  "Á": "Á",
+  "À": "À",
+  "Ä": "Ä",
+  "Ǟ": "Ǟ",
+  "Ã": "Ã",
+  "Ā": "Ā",
+  "Ă": "Ă",
+  "Ắ": "Ắ",
+  "Ằ": "Ằ",
+  "Ẵ": "Ẵ",
+  "Ǎ": "Ǎ",
+  "Â": "Â",
+  "Ấ": "Ấ",
+  "Ầ": "Ầ",
+  "Ẫ": "Ẫ",
+  "Ȧ": "Ȧ",
+  "Ǡ": "Ǡ",
+  "Å": "Å",
+  "Ǻ": "Ǻ",
+  "Ḃ": "Ḃ",
+  "Ć": "Ć",
+  "Č": "Č",
+  "Ĉ": "Ĉ",
+  "Ċ": "Ċ",
+  "Ď": "Ď",
+  "Ḋ": "Ḋ",
+  "É": "É",
+  "È": "È",
+  "Ë": "Ë",
+  "Ẽ": "Ẽ",
+  "Ē": "Ē",
+  "Ḗ": "Ḗ",
+  "Ḕ": "Ḕ",
+  "Ĕ": "Ĕ",
+  "Ě": "Ě",
+  "Ê": "Ê",
+  "Ế": "Ế",
+  "Ề": "Ề",
+  "Ễ": "Ễ",
+  "Ė": "Ė",
+  "Ḟ": "Ḟ",
+  "Ǵ": "Ǵ",
+  "Ḡ": "Ḡ",
+  "Ğ": "Ğ",
+  "Ǧ": "Ǧ",
+  "Ĝ": "Ĝ",
+  "Ġ": "Ġ",
+  "Ḧ": "Ḧ",
+  "Ȟ": "Ȟ",
+  "Ĥ": "Ĥ",
+  "Ḣ": "Ḣ",
+  "Í": "Í",
+  "Ì": "Ì",
+  "Ï": "Ï",
+  "Ḯ": "Ḯ",
+  "Ĩ": "Ĩ",
+  "Ī": "Ī",
+  "Ĭ": "Ĭ",
+  "Ǐ": "Ǐ",
+  "Î": "Î",
+  "İ": "İ",
+  "Ĵ": "Ĵ",
+  "Ḱ": "Ḱ",
+  "Ǩ": "Ǩ",
+  "Ĺ": "Ĺ",
+  "Ľ": "Ľ",
+  "Ḿ": "Ḿ",
+  "Ṁ": "Ṁ",
+  "Ń": "Ń",
+  "Ǹ": "Ǹ",
+  "Ñ": "Ñ",
+  "Ň": "Ň",
+  "Ṅ": "Ṅ",
+  "Ó": "Ó",
+  "Ò": "Ò",
+  "Ö": "Ö",
+  "Ȫ": "Ȫ",
+  "Õ": "Õ",
+  "Ṍ": "Ṍ",
+  "Ṏ": "Ṏ",
+  "Ȭ": "Ȭ",
+  "Ō": "Ō",
+  "Ṓ": "Ṓ",
+  "Ṑ": "Ṑ",
+  "Ŏ": "Ŏ",
+  "Ǒ": "Ǒ",
+  "Ô": "Ô",
+  "Ố": "Ố",
+  "Ồ": "Ồ",
+  "Ỗ": "Ỗ",
+  "Ȯ": "Ȯ",
+  "Ȱ": "Ȱ",
+  "Ő": "Ő",
+  "Ṕ": "Ṕ",
+  "Ṗ": "Ṗ",
+  "Ŕ": "Ŕ",
+  "Ř": "Ř",
+  "Ṙ": "Ṙ",
+  "Ś": "Ś",
+  "Ṥ": "Ṥ",
+  "Š": "Š",
+  "Ṧ": "Ṧ",
+  "Ŝ": "Ŝ",
+  "Ṡ": "Ṡ",
+  "Ť": "Ť",
+  "Ṫ": "Ṫ",
+  "Ú": "Ú",
+  "Ù": "Ù",
+  "Ü": "Ü",
+  "Ǘ": "Ǘ",
+  "Ǜ": "Ǜ",
+  "Ǖ": "Ǖ",
+  "Ǚ": "Ǚ",
+  "Ũ": "Ũ",
+  "Ṹ": "Ṹ",
+  "Ū": "Ū",
+  "Ṻ": "Ṻ",
+  "Ŭ": "Ŭ",
+  "Ǔ": "Ǔ",
+  "Û": "Û",
+  "Ů": "Ů",
+  "Ű": "Ű",
+  "Ṽ": "Ṽ",
+  "Ẃ": "Ẃ",
+  "Ẁ": "Ẁ",
+  "Ẅ": "Ẅ",
+  "Ŵ": "Ŵ",
+  "Ẇ": "Ẇ",
+  "Ẍ": "Ẍ",
+  "Ẋ": "Ẋ",
+  "Ý": "Ý",
+  "Ỳ": "Ỳ",
+  "Ÿ": "Ÿ",
+  "Ỹ": "Ỹ",
+  "Ȳ": "Ȳ",
+  "Ŷ": "Ŷ",
+  "Ẏ": "Ẏ",
+  "Ź": "Ź",
+  "Ž": "Ž",
+  "Ẑ": "Ẑ",
+  "Ż": "Ż",
+  "ά": "ά",
+  "ὰ": "ὰ",
+  "ᾱ": "ᾱ",
+  "ᾰ": "ᾰ",
+  "έ": "έ",
+  "ὲ": "ὲ",
+  "ή": "ή",
+  "ὴ": "ὴ",
+  "ί": "ί",
+  "ὶ": "ὶ",
+  "ϊ": "ϊ",
+  "ΐ": "ΐ",
+  "ῒ": "ῒ",
+  "ῑ": "ῑ",
+  "ῐ": "ῐ",
+  "ό": "ό",
+  "ὸ": "ὸ",
+  "ύ": "ύ",
+  "ὺ": "ὺ",
+  "ϋ": "ϋ",
+  "ΰ": "ΰ",
+  "ῢ": "ῢ",
+  "ῡ": "ῡ",
+  "ῠ": "ῠ",
+  "ώ": "ώ",
+  "ὼ": "ὼ",
+  "Ύ": "Ύ",
+  "Ὺ": "Ὺ",
+  "Ϋ": "Ϋ",
+  "Ῡ": "Ῡ",
+  "Ῠ": "Ῠ",
+  "Ώ": "Ώ",
+  "Ὼ": "Ὼ"
+};
+
+/**
+ * This file contains the parser used to parse out a TeX expression from the
+ * input. Since TeX isn't context-free, standard parsers don't work particularly
+ * well.
+ *
+ * The strategy of this parser is as such:
+ *
+ * The main functions (the `.parse...` ones) take a position in the current
+ * parse string to parse tokens from. The lexer (found in Lexer.js, stored at
+ * this.gullet.lexer) also supports pulling out tokens at arbitrary places. When
+ * individual tokens are needed at a position, the lexer is called to pull out a
+ * token, which is then used.
+ *
+ * The parser has a property called "mode" indicating the mode that
+ * the parser is currently in. Currently it has to be one of "math" or
+ * "text", which denotes whether the current environment is a math-y
+ * one or a text-y one (e.g. inside \text). Currently, this serves to
+ * limit the functions which can be used in text mode.
+ *
+ * The main functions then return an object which contains the useful data that
+ * was parsed at its given point, and a new position at the end of the parsed
+ * data. The main functions can call each other and continue the parsing by
+ * using the returned position as a new starting point.
+ *
+ * There are also extra `.handle...` functions, which pull out some reused
+ * functionality into self-contained functions.
+ *
+ * The functions return ParseNodes.
+ */
+var Parser_Parser =
+/*#__PURE__*/
+function () {
+  function Parser(input, settings) {
+    this.mode = void 0;
+    this.gullet = void 0;
+    this.settings = void 0;
+    this.leftrightDepth = void 0;
+    this.nextToken = void 0;
+    // Start in math mode
+    this.mode = "math"; // Create a new macro expander (gullet) and (indirectly via that) also a
+    // new lexer (mouth) for this parser (stomach, in the language of TeX)
+
+    this.gullet = new MacroExpander_MacroExpander(input, settings, this.mode); // Store the settings for use in parsing
+
+    this.settings = settings; // Count leftright depth (for \middle errors)
+
+    this.leftrightDepth = 0;
+  }
+  /**
+   * Checks a result to make sure it has the right type, and throws an
+   * appropriate error otherwise.
+   */
+
+
+  var _proto = Parser.prototype;
+
+  _proto.expect = function expect(text, consume) {
+    if (consume === void 0) {
+      consume = true;
+    }
+
+    if (this.fetch().text !== text) {
+      throw new src_ParseError("Expected '" + text + "', got '" + this.fetch().text + "'", this.fetch());
+    }
+
+    if (consume) {
+      this.consume();
+    }
+  }
+  /**
+   * Discards the current lookahead token, considering it consumed.
+   */
+  ;
+
+  _proto.consume = function consume() {
+    this.nextToken = null;
+  }
+  /**
+   * Return the current lookahead token, or if there isn't one (at the
+   * beginning, or if the previous lookahead token was consume()d),
+   * fetch the next token as the new lookahead token and return it.
+   */
+  ;
+
+  _proto.fetch = function fetch() {
+    if (this.nextToken == null) {
+      this.nextToken = this.gullet.expandNextToken();
+    }
+
+    return this.nextToken;
+  }
+  /**
+   * Switches between "text" and "math" modes.
+   */
+  ;
+
+  _proto.switchMode = function switchMode(newMode) {
+    this.mode = newMode;
+    this.gullet.switchMode(newMode);
+  }
+  /**
+   * Main parsing function, which parses an entire input.
+   */
+  ;
+
+  _proto.parse = function parse() {
+    if (!this.settings.globalGroup) {
+      // Create a group namespace for the math expression.
+      // (LaTeX creates a new group for every $...$, $$...$$, \[...\].)
+      this.gullet.beginGroup();
+    } // Use old \color behavior (same as LaTeX's \textcolor) if requested.
+    // We do this within the group for the math expression, so it doesn't
+    // pollute settings.macros.
+
+
+    if (this.settings.colorIsTextColor) {
+      this.gullet.macros.set("\\color", "\\textcolor");
+    } // Try to parse the input
+
+
+    var parse = this.parseExpression(false); // If we succeeded, make sure there's an EOF at the end
+
+    this.expect("EOF"); // End the group namespace for the expression
+
+    if (!this.settings.globalGroup) {
+      this.gullet.endGroup();
+    }
+
+    return parse;
+  };
+
+  _proto.parseExpression = function parseExpression(breakOnInfix, breakOnTokenText) {
+    var body = []; // Keep adding atoms to the body until we can't parse any more atoms (either
+    // we reached the end, a }, or a \right)
+
+    while (true) {
+      // Ignore spaces in math mode
+      if (this.mode === "math") {
+        this.consumeSpaces();
+      }
+
+      var lex = this.fetch();
+
+      if (Parser.endOfExpression.indexOf(lex.text) !== -1) {
+        break;
+      }
+
+      if (breakOnTokenText && lex.text === breakOnTokenText) {
+        break;
+      }
+
+      if (breakOnInfix && src_functions[lex.text] && src_functions[lex.text].infix) {
+        break;
+      }
+
+      var atom = this.parseAtom(breakOnTokenText);
+
+      if (!atom) {
+        break;
+      } else if (atom.type === "internal") {
+        continue;
+      }
+
+      body.push(atom);
+    }
+
+    if (this.mode === "text") {
+      this.formLigatures(body);
+    }
+
+    return this.handleInfixNodes(body);
+  }
+  /**
+   * Rewrites infix operators such as \over with corresponding commands such
+   * as \frac.
+   *
+   * There can only be one infix operator per group.  If there's more than one
+   * then the expression is ambiguous.  This can be resolved by adding {}.
+   */
+  ;
+
+  _proto.handleInfixNodes = function handleInfixNodes(body) {
+    var overIndex = -1;
+    var funcName;
+
+    for (var i = 0; i < body.length; i++) {
+      if (body[i].type === "infix") {
+        if (overIndex !== -1) {
+          throw new src_ParseError("only one infix operator per group", body[i].token);
+        }
+
+        overIndex = i;
+        funcName = body[i].replaceWith;
+      }
+    }
+
+    if (overIndex !== -1 && funcName) {
+      var numerNode;
+      var denomNode;
+      var numerBody = body.slice(0, overIndex);
+      var denomBody = body.slice(overIndex + 1);
+
+      if (numerBody.length === 1 && numerBody[0].type === "ordgroup") {
+        numerNode = numerBody[0];
+      } else {
+        numerNode = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: numerBody
+        };
+      }
+
+      if (denomBody.length === 1 && denomBody[0].type === "ordgroup") {
+        denomNode = denomBody[0];
+      } else {
+        denomNode = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: denomBody
+        };
+      }
+
+      var node;
+
+      if (funcName === "\\\\abovefrac") {
+        node = this.callFunction(funcName, [numerNode, body[overIndex], denomNode], []);
+      } else {
+        node = this.callFunction(funcName, [numerNode, denomNode], []);
+      }
+
+      return [node];
+    } else {
+      return body;
+    }
+  } // The greediness of a superscript or subscript
+  ;
+
+  /**
+   * Handle a subscript or superscript with nice errors.
+   */
+  _proto.handleSupSubscript = function handleSupSubscript(name) {
+    var symbolToken = this.fetch();
+    var symbol = symbolToken.text;
+    this.consume();
+    var group = this.parseGroup(name, false, Parser.SUPSUB_GREEDINESS, undefined, undefined, true); // ignore spaces before sup/subscript argument
+
+    if (!group) {
+      throw new src_ParseError("Expected group after '" + symbol + "'", symbolToken);
+    }
+
+    return group;
+  }
+  /**
+   * Converts the textual input of an unsupported command into a text node
+   * contained within a color node whose color is determined by errorColor
+   */
+  ;
+
+  _proto.formatUnsupportedCmd = function formatUnsupportedCmd(text) {
+    var textordArray = [];
+
+    for (var i = 0; i < text.length; i++) {
+      textordArray.push({
+        type: "textord",
+        mode: "text",
+        text: text[i]
+      });
+    }
+
+    var textNode = {
+      type: "text",
+      mode: this.mode,
+      body: textordArray
+    };
+    var colorNode = {
+      type: "color",
+      mode: this.mode,
+      color: this.settings.errorColor,
+      body: [textNode]
+    };
+    return colorNode;
+  }
+  /**
+   * Parses a group with optional super/subscripts.
+   */
+  ;
+
+  _proto.parseAtom = function parseAtom(breakOnTokenText) {
+    // The body of an atom is an implicit group, so that things like
+    // \left(x\right)^2 work correctly.
+    var base = this.parseGroup("atom", false, null, breakOnTokenText); // In text mode, we don't have superscripts or subscripts
+
+    if (this.mode === "text") {
+      return base;
+    } // Note that base may be empty (i.e. null) at this point.
+
+
+    var superscript;
+    var subscript;
+
+    while (true) {
+      // Guaranteed in math mode, so eat any spaces first.
+      this.consumeSpaces(); // Lex the first token
+
+      var lex = this.fetch();
+
+      if (lex.text === "\\limits" || lex.text === "\\nolimits") {
+        // We got a limit control
+        if (base && base.type === "op") {
+          var limits = lex.text === "\\limits";
+          base.limits = limits;
+          base.alwaysHandleSupSub = true;
+        } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub) {
+          var _limits = lex.text === "\\limits";
+
+          base.limits = _limits;
+        } else {
+          throw new src_ParseError("Limit controls must follow a math operator", lex);
+        }
+
+        this.consume();
+      } else if (lex.text === "^") {
+        // We got a superscript start
+        if (superscript) {
+          throw new src_ParseError("Double superscript", lex);
+        }
+
+        superscript = this.handleSupSubscript("superscript");
+      } else if (lex.text === "_") {
+        // We got a subscript start
+        if (subscript) {
+          throw new src_ParseError("Double subscript", lex);
+        }
+
+        subscript = this.handleSupSubscript("subscript");
+      } else if (lex.text === "'") {
+        // We got a prime
+        if (superscript) {
+          throw new src_ParseError("Double superscript", lex);
+        }
+
+        var prime = {
+          type: "textord",
+          mode: this.mode,
+          text: "\\prime"
+        }; // Many primes can be grouped together, so we handle this here
+
+        var primes = [prime];
+        this.consume(); // Keep lexing tokens until we get something that's not a prime
+
+        while (this.fetch().text === "'") {
+          // For each one, add another prime to the list
+          primes.push(prime);
+          this.consume();
+        } // If there's a superscript following the primes, combine that
+        // superscript in with the primes.
+
+
+        if (this.fetch().text === "^") {
+          primes.push(this.handleSupSubscript("superscript"));
+        } // Put everything into an ordgroup as the superscript
+
+
+        superscript = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: primes
+        };
+      } else {
+        // If it wasn't ^, _, or ', stop parsing super/subscripts
+        break;
+      }
+    } // Base must be set if superscript or subscript are set per logic above,
+    // but need to check here for type check to pass.
+
+
+    if (superscript || subscript) {
+      // If we got either a superscript or subscript, create a supsub
+      return {
+        type: "supsub",
+        mode: this.mode,
+        base: base,
+        sup: superscript,
+        sub: subscript
+      };
+    } else {
+      // Otherwise return the original body
+      return base;
+    }
+  }
+  /**
+   * Parses an entire function, including its base and all of its arguments.
+   */
+  ;
+
+  _proto.parseFunction = function parseFunction(breakOnTokenText, name, // For error reporting.
+  greediness) {
+    var token = this.fetch();
+    var func = token.text;
+    var funcData = src_functions[func];
+
+    if (!funcData) {
+      return null;
+    }
+
+    this.consume(); // consume command token
+
+    if (greediness != null && funcData.greediness <= greediness) {
+      throw new src_ParseError("Got function '" + func + "' with no arguments" + (name ? " as " + name : ""), token);
+    } else if (this.mode === "text" && !funcData.allowedInText) {
+      throw new src_ParseError("Can't use function '" + func + "' in text mode", token);
+    } else if (this.mode === "math" && funcData.allowedInMath === false) {
+      throw new src_ParseError("Can't use function '" + func + "' in math mode", token);
+    }
+
+    var _this$parseArguments = this.parseArguments(func, funcData),
+        args = _this$parseArguments.args,
+        optArgs = _this$parseArguments.optArgs;
+
+    return this.callFunction(func, args, optArgs, token, breakOnTokenText);
+  }
+  /**
+   * Call a function handler with a suitable context and arguments.
+   */
+  ;
+
+  _proto.callFunction = function callFunction(name, args, optArgs, token, breakOnTokenText) {
+    var context = {
+      funcName: name,
+      parser: this,
+      token: token,
+      breakOnTokenText: breakOnTokenText
+    };
+    var func = src_functions[name];
+
+    if (func && func.handler) {
+      return func.handler(context, args, optArgs);
+    } else {
+      throw new src_ParseError("No function handler for " + name);
+    }
+  }
+  /**
+   * Parses the arguments of a function or environment
+   */
+  ;
+
+  _proto.parseArguments = function parseArguments(func, // Should look like "\name" or "\begin{name}".
+  funcData) {
+    var totalArgs = funcData.numArgs + funcData.numOptionalArgs;
+
+    if (totalArgs === 0) {
+      return {
+        args: [],
+        optArgs: []
+      };
+    }
+
+    var baseGreediness = funcData.greediness;
+    var args = [];
+    var optArgs = [];
+
+    for (var i = 0; i < totalArgs; i++) {
+      var argType = funcData.argTypes && funcData.argTypes[i];
+      var isOptional = i < funcData.numOptionalArgs; // Ignore spaces between arguments.  As the TeXbook says:
+      // "After you have said ‘\def\row#1#2{...}’, you are allowed to
+      //  put spaces between the arguments (e.g., ‘\row x n’), because
+      //  TeX doesn’t use single spaces as undelimited arguments."
+
+      var consumeSpaces = i > 0 && !isOptional || // Also consume leading spaces in math mode, as parseSymbol
+      // won't know what to do with them.  This can only happen with
+      // macros, e.g. \frac\foo\foo where \foo expands to a space symbol.
+      // In LaTeX, the \foo's get treated as (blank) arguments.
+      // In KaTeX, for now, both spaces will get consumed.
+      // TODO(edemaine)
+      i === 0 && !isOptional && this.mode === "math";
+      var arg = this.parseGroupOfType("argument to '" + func + "'", argType, isOptional, baseGreediness, consumeSpaces);
+
+      if (!arg) {
+        if (isOptional) {
+          optArgs.push(null);
+          continue;
+        }
+
+        throw new src_ParseError("Expected group after '" + func + "'", this.fetch());
+      }
+
+      (isOptional ? optArgs : args).push(arg);
+    }
+
+    return {
+      args: args,
+      optArgs: optArgs
+    };
+  }
+  /**
+   * Parses a group when the mode is changing.
+   */
+  ;
+
+  _proto.parseGroupOfType = function parseGroupOfType(name, type, optional, greediness, consumeSpaces) {
+    switch (type) {
+      case "color":
+        if (consumeSpaces) {
+          this.consumeSpaces();
+        }
+
+        return this.parseColorGroup(optional);
+
+      case "size":
+        if (consumeSpaces) {
+          this.consumeSpaces();
+        }
+
+        return this.parseSizeGroup(optional);
+
+      case "url":
+        return this.parseUrlGroup(optional, consumeSpaces);
+
+      case "math":
+      case "text":
+        return this.parseGroup(name, optional, greediness, undefined, type, consumeSpaces);
+
+      case "hbox":
+        {
+          // hbox argument type wraps the argument in the equivalent of
+          // \hbox, which is like \text but switching to \textstyle size.
+          var group = this.parseGroup(name, optional, greediness, undefined, "text", consumeSpaces);
+
+          if (!group) {
+            return group;
+          }
+
+          var styledGroup = {
+            type: "styling",
+            mode: group.mode,
+            body: [group],
+            style: "text" // simulate \textstyle
+
+          };
+          return styledGroup;
+        }
+
+      case "raw":
+        {
+          if (consumeSpaces) {
+            this.consumeSpaces();
+          }
+
+          if (optional && this.fetch().text === "{") {
+            return null;
+          }
+
+          var token = this.parseStringGroup("raw", optional, true);
+
+          if (token) {
+            return {
+              type: "raw",
+              mode: "text",
+              string: token.text
+            };
+          } else {
+            throw new src_ParseError("Expected raw group", this.fetch());
+          }
+        }
+
+      case "original":
+      case null:
+      case undefined:
+        return this.parseGroup(name, optional, greediness, undefined, undefined, consumeSpaces);
+
+      default:
+        throw new src_ParseError("Unknown group type as " + name, this.fetch());
+    }
+  }
+  /**
+   * Discard any space tokens, fetching the next non-space token.
+   */
+  ;
+
+  _proto.consumeSpaces = function consumeSpaces() {
+    while (this.fetch().text === " ") {
+      this.consume();
+    }
+  }
+  /**
+   * Parses a group, essentially returning the string formed by the
+   * brace-enclosed tokens plus some position information.
+   */
+  ;
+
+  _proto.parseStringGroup = function parseStringGroup(modeName, // Used to describe the mode in error messages.
+  optional, raw) {
+    var groupBegin = optional ? "[" : "{";
+    var groupEnd = optional ? "]" : "}";
+    var beginToken = this.fetch();
+
+    if (beginToken.text !== groupBegin) {
+      if (optional) {
+        return null;
+      } else if (raw && beginToken.text !== "EOF" && /[^{}[\]]/.test(beginToken.text)) {
+        this.consume();
+        return beginToken;
+      }
+    }
+
+    var outerMode = this.mode;
+    this.mode = "text";
+    this.expect(groupBegin);
+    var str = "";
+    var firstToken = this.fetch();
+    var nested = 0; // allow nested braces in raw string group
+
+    var lastToken = firstToken;
+    var nextToken;
+
+    while ((nextToken = this.fetch()).text !== groupEnd || raw && nested > 0) {
+      switch (nextToken.text) {
+        case "EOF":
+          throw new src_ParseError("Unexpected end of input in " + modeName, firstToken.range(lastToken, str));
+
+        case groupBegin:
+          nested++;
+          break;
+
+        case groupEnd:
+          nested--;
+          break;
+      }
+
+      lastToken = nextToken;
+      str += lastToken.text;
+      this.consume();
+    }
+
+    this.expect(groupEnd);
+    this.mode = outerMode;
+    return firstToken.range(lastToken, str);
+  }
+  /**
+   * Parses a regex-delimited group: the largest sequence of tokens
+   * whose concatenated strings match `regex`. Returns the string
+   * formed by the tokens plus some position information.
+   */
+  ;
+
+  _proto.parseRegexGroup = function parseRegexGroup(regex, modeName) {
+    var outerMode = this.mode;
+    this.mode = "text";
+    var firstToken = this.fetch();
+    var lastToken = firstToken;
+    var str = "";
+    var nextToken;
+
+    while ((nextToken = this.fetch()).text !== "EOF" && regex.test(str + nextToken.text)) {
+      lastToken = nextToken;
+      str += lastToken.text;
+      this.consume();
+    }
+
+    if (str === "") {
+      throw new src_ParseError("Invalid " + modeName + ": '" + firstToken.text + "'", firstToken);
+    }
+
+    this.mode = outerMode;
+    return firstToken.range(lastToken, str);
+  }
+  /**
+   * Parses a color description.
+   */
+  ;
+
+  _proto.parseColorGroup = function parseColorGroup(optional) {
+    var res = this.parseStringGroup("color", optional);
+
+    if (!res) {
+      return null;
+    }
+
+    var match = /^(#[a-f0-9]{3}|#?[a-f0-9]{6}|[a-z]+)$/i.exec(res.text);
+
+    if (!match) {
+      throw new src_ParseError("Invalid color: '" + res.text + "'", res);
+    }
+
+    var color = match[0];
+
+    if (/^[0-9a-f]{6}$/i.test(color)) {
+      // We allow a 6-digit HTML color spec without a leading "#".
+      // This follows the xcolor package's HTML color model.
+      // Predefined color names are all missed by this RegEx pattern.
+      color = "#" + color;
+    }
+
+    return {
+      type: "color-token",
+      mode: this.mode,
+      color: color
+    };
+  }
+  /**
+   * Parses a size specification, consisting of magnitude and unit.
+   */
+  ;
+
+  _proto.parseSizeGroup = function parseSizeGroup(optional) {
+    var res;
+    var isBlank = false;
+
+    if (!optional && this.fetch().text !== "{") {
+      res = this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2} *$/, "size");
+    } else {
+      res = this.parseStringGroup("size", optional);
+    }
+
+    if (!res) {
+      return null;
+    }
+
+    if (!optional && res.text.length === 0) {
+      // Because we've tested for what is !optional, this block won't
+      // affect \kern, \hspace, etc. It will capture the mandatory arguments
+      // to \genfrac and \above.
+      res.text = "0pt"; // Enable \above{}
+
+      isBlank = true; // This is here specifically for \genfrac
+    }
+
+    var match = /([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(res.text);
+
+    if (!match) {
+      throw new src_ParseError("Invalid size: '" + res.text + "'", res);
+    }
+
+    var data = {
+      number: +(match[1] + match[2]),
+      // sign + magnitude, cast to number
+      unit: match[3]
+    };
+
+    if (!validUnit(data)) {
+      throw new src_ParseError("Invalid unit: '" + data.unit + "'", res);
+    }
+
+    return {
+      type: "size",
+      mode: this.mode,
+      value: data,
+      isBlank: isBlank
+    };
+  }
+  /**
+   * Parses an URL, checking escaped letters and allowed protocols,
+   * and setting the catcode of % as an active character (as in \hyperref).
+   */
+  ;
+
+  _proto.parseUrlGroup = function parseUrlGroup(optional, consumeSpaces) {
+    this.gullet.lexer.setCatcode("%", 13); // active character
+
+    var res = this.parseStringGroup("url", optional, true); // get raw string
+
+    this.gullet.lexer.setCatcode("%", 14); // comment character
+
+    if (!res) {
+      return null;
+    } // hyperref package allows backslashes alone in href, but doesn't
+    // generate valid links in such cases; we interpret this as
+    // "undefined" behaviour, and keep them as-is. Some browser will
+    // replace backslashes with forward slashes.
+
+
+    var url = res.text.replace(/\\([#$%&~_^{}])/g, '$1');
+    return {
+      type: "url",
+      mode: this.mode,
+      url: url
+    };
+  }
+  /**
+   * If `optional` is false or absent, this parses an ordinary group,
+   * which is either a single nucleus (like "x") or an expression
+   * in braces (like "{x+y}") or an implicit group, a group that starts
+   * at the current position, and ends right before a higher explicit
+   * group ends, or at EOF.
+   * If `optional` is true, it parses either a bracket-delimited expression
+   * (like "[x+y]") or returns null to indicate the absence of a
+   * bracket-enclosed group.
+   * If `mode` is present, switches to that mode while parsing the group,
+   * and switches back after.
+   */
+  ;
+
+  _proto.parseGroup = function parseGroup(name, // For error reporting.
+  optional, greediness, breakOnTokenText, mode, consumeSpaces) {
+    // Switch to specified mode
+    var outerMode = this.mode;
+
+    if (mode) {
+      this.switchMode(mode);
+    } // Consume spaces if requested, crucially *after* we switch modes,
+    // so that the next non-space token is parsed in the correct mode.
+
+
+    if (consumeSpaces) {
+      this.consumeSpaces();
+    } // Get first token
+
+
+    var firstToken = this.fetch();
+    var text = firstToken.text;
+    var result; // Try to parse an open brace or \begingroup
+
+    if (optional ? text === "[" : text === "{" || text === "\\begingroup") {
+      this.consume();
+      var groupEnd = Parser.endOfGroup[text]; // Start a new group namespace
+
+      this.gullet.beginGroup(); // If we get a brace, parse an expression
+
+      var expression = this.parseExpression(false, groupEnd);
+      var lastToken = this.fetch(); // Check that we got a matching closing brace
+
+      this.expect(groupEnd); // End group namespace
+
+      this.gullet.endGroup();
+      result = {
+        type: "ordgroup",
+        mode: this.mode,
+        loc: SourceLocation.range(firstToken, lastToken),
+        body: expression,
+        // A group formed by \begingroup...\endgroup is a semi-simple group
+        // which doesn't affect spacing in math mode, i.e., is transparent.
+        // https://tex.stackexchange.com/questions/1930/when-should-one-
+        // use-begingroup-instead-of-bgroup
+        semisimple: text === "\\begingroup" || undefined
+      };
+    } else if (optional) {
+      // Return nothing for an optional group
+      result = null;
+    } else {
+      // If there exists a function with this name, parse the function.
+      // Otherwise, just return a nucleus
+      result = this.parseFunction(breakOnTokenText, name, greediness) || this.parseSymbol();
+
+      if (result == null && text[0] === "\\" && !implicitCommands.hasOwnProperty(text)) {
+        if (this.settings.throwOnError) {
+          throw new src_ParseError("Undefined control sequence: " + text, firstToken);
+        }
+
+        result = this.formatUnsupportedCmd(text);
+        this.consume();
+      }
+    } // Switch mode back
+
+
+    if (mode) {
+      this.switchMode(outerMode);
+    }
+
+    return result;
+  }
+  /**
+   * Form ligature-like combinations of characters for text mode.
+   * This includes inputs like "--", "---", "``" and "''".
+   * The result will simply replace multiple textord nodes with a single
+   * character in each value by a single textord node having multiple
+   * characters in its value.  The representation is still ASCII source.
+   * The group will be modified in place.
+   */
+  ;
+
+  _proto.formLigatures = function formLigatures(group) {
+    var n = group.length - 1;
+
+    for (var i = 0; i < n; ++i) {
+      var a = group[i]; // $FlowFixMe: Not every node type has a `text` property.
+
+      var v = a.text;
+
+      if (v === "-" && group[i + 1].text === "-") {
+        if (i + 1 < n && group[i + 2].text === "-") {
+          group.splice(i, 3, {
+            type: "textord",
+            mode: "text",
+            loc: SourceLocation.range(a, group[i + 2]),
+            text: "---"
+          });
+          n -= 2;
+        } else {
+          group.splice(i, 2, {
+            type: "textord",
+            mode: "text",
+            loc: SourceLocation.range(a, group[i + 1]),
+            text: "--"
+          });
+          n -= 1;
+        }
+      }
+
+      if ((v === "'" || v === "`") && group[i + 1].text === v) {
+        group.splice(i, 2, {
+          type: "textord",
+          mode: "text",
+          loc: SourceLocation.range(a, group[i + 1]),
+          text: v + v
+        });
+        n -= 1;
+      }
+    }
+  }
+  /**
+   * Parse a single symbol out of the string. Here, we handle single character
+   * symbols and special functions like \verb.
+   */
+  ;
+
+  _proto.parseSymbol = function parseSymbol() {
+    var nucleus = this.fetch();
+    var text = nucleus.text;
+
+    if (/^\\verb[^a-zA-Z]/.test(text)) {
+      this.consume();
+      var arg = text.slice(5);
+      var star = arg.charAt(0) === "*";
+
+      if (star) {
+        arg = arg.slice(1);
+      } // Lexer's tokenRegex is constructed to always have matching
+      // first/last characters.
+
+
+      if (arg.length < 2 || arg.charAt(0) !== arg.slice(-1)) {
+        throw new src_ParseError("\\verb assertion failed --\n                    please report what input caused this bug");
+      }
+
+      arg = arg.slice(1, -1); // remove first and last char
+
+      return {
+        type: "verb",
+        mode: "text",
+        body: arg,
+        star: star
+      };
+    } // At this point, we should have a symbol, possibly with accents.
+    // First expand any accented base symbol according to unicodeSymbols.
+
+
+    if (unicodeSymbols.hasOwnProperty(text[0]) && !src_symbols[this.mode][text[0]]) {
+      // This behavior is not strict (XeTeX-compatible) in math mode.
+      if (this.settings.strict && this.mode === "math") {
+        this.settings.reportNonstrict("unicodeTextInMathMode", "Accented Unicode text character \"" + text[0] + "\" used in " + "math mode", nucleus);
+      }
+
+      text = unicodeSymbols[text[0]] + text.substr(1);
+    } // Strip off any combining characters
+
+
+    var match = combiningDiacriticalMarksEndRegex.exec(text);
+
+    if (match) {
+      text = text.substring(0, match.index);
+
+      if (text === 'i') {
+        text = "\u0131"; // dotless i, in math and text mode
+      } else if (text === 'j') {
+        text = "\u0237"; // dotless j, in math and text mode
+      }
+    } // Recognize base symbol
+
+
+    var symbol;
+
+    if (src_symbols[this.mode][text]) {
+      if (this.settings.strict && this.mode === 'math' && extraLatin.indexOf(text) >= 0) {
+        this.settings.reportNonstrict("unicodeTextInMathMode", "Latin-1/Unicode text character \"" + text[0] + "\" used in " + "math mode", nucleus);
+      }
+
+      var group = src_symbols[this.mode][text].group;
+      var loc = SourceLocation.range(nucleus);
+      var s;
+
+      if (ATOMS.hasOwnProperty(group)) {
+        // $FlowFixMe
+        var family = group;
+        s = {
+          type: "atom",
+          mode: this.mode,
+          family: family,
+          loc: loc,
+          text: text
+        };
+      } else {
+        // $FlowFixMe
+        s = {
+          type: group,
+          mode: this.mode,
+          loc: loc,
+          text: text
+        };
+      }
+
+      symbol = s;
+    } else if (text.charCodeAt(0) >= 0x80) {
+      // no symbol for e.g. ^
+      if (this.settings.strict) {
+        if (!supportedCodepoint(text.charCodeAt(0))) {
+          this.settings.reportNonstrict("unknownSymbol", "Unrecognized Unicode character \"" + text[0] + "\"" + (" (" + text.charCodeAt(0) + ")"), nucleus);
+        } else if (this.mode === "math") {
+          this.settings.reportNonstrict("unicodeTextInMathMode", "Unicode text character \"" + text[0] + "\" used in math mode", nucleus);
+        }
+      } // All nonmathematical Unicode characters are rendered as if they
+      // are in text mode (wrapped in \text) because that's what it
+      // takes to render them in LaTeX.  Setting `mode: this.mode` is
+      // another natural choice (the user requested math mode), but
+      // this makes it more difficult for getCharacterMetrics() to
+      // distinguish Unicode characters without metrics and those for
+      // which we want to simulate the letter M.
+
+
+      symbol = {
+        type: "textord",
+        mode: "text",
+        loc: SourceLocation.range(nucleus),
+        text: text
+      };
+    } else {
+      return null; // EOF, ^, _, {, }, etc.
+    }
+
+    this.consume(); // Transform combining characters into accents
+
+    if (match) {
+      for (var i = 0; i < match[0].length; i++) {
+        var accent = match[0][i];
+
+        if (!unicodeAccents[accent]) {
+          throw new src_ParseError("Unknown accent ' " + accent + "'", nucleus);
+        }
+
+        var command = unicodeAccents[accent][this.mode];
+
+        if (!command) {
+          throw new src_ParseError("Accent " + accent + " unsupported in " + this.mode + " mode", nucleus);
+        }
+
+        symbol = {
+          type: "accent",
+          mode: this.mode,
+          loc: SourceLocation.range(nucleus),
+          label: command,
+          isStretchy: false,
+          isShifty: true,
+          base: symbol
+        };
+      }
+    }
+
+    return symbol;
+  };
+
+  return Parser;
+}();
+
+Parser_Parser.endOfExpression = ["}", "\\endgroup", "\\end", "\\right", "&"];
+Parser_Parser.endOfGroup = {
+  "[": "]",
+  "{": "}",
+  "\\begingroup": "\\endgroup"
+  /**
+   * Parses an "expression", which is a list of atoms.
+   *
+   * `breakOnInfix`: Should the parsing stop when we hit infix nodes? This
+   *                 happens when functions have higher precendence han infix
+   *                 nodes in implicit parses.
+   *
+   * `breakOnTokenText`: The text of the token that the expression should end
+   *                     with, or `null` if something else should end the
+   *                     expression.
+   */
+
+};
+Parser_Parser.SUPSUB_GREEDINESS = 1;
+
+// CONCATENATED MODULE: ./src/parseTree.js
+/**
+ * Provides a single function for parsing an expression using a Parser
+ * TODO(emily): Remove this
+ */
+
+
+
+/**
+ * Parses an expression using a Parser, then returns the parsed result.
+ */
+var parseTree_parseTree = function parseTree(toParse, settings) {
+  if (!(typeof toParse === 'string' || toParse instanceof String)) {
+    throw new TypeError('KaTeX can only parse string typed expression');
+  }
+
+  var parser = new Parser_Parser(toParse, settings); // Blank out any \df@tag to avoid spurious "Duplicate \tag" errors
+
+  delete parser.gullet.macros.current["\\df@tag"];
+  var tree = parser.parse(); // If the input used \tag, it will set the \df@tag macro to the tag.
+  // In this case, we separately parse the tag and wrap the tree.
+
+  if (parser.gullet.macros.get("\\df@tag")) {
+    if (!settings.displayMode) {
+      throw new src_ParseError("\\tag works only in display equations");
+    }
+
+    parser.gullet.feed("\\df@tag");
+    tree = [{
+      type: "tag",
+      mode: "text",
+      body: tree,
+      tag: parser.parse()
+    }];
+  }
+
+  return tree;
+};
+
+/* harmony default export */ var src_parseTree = (parseTree_parseTree);
+// CONCATENATED MODULE: ./katex.js
+/* eslint no-console:0 */
+
+/**
+ * This is the main entry point for KaTeX. Here, we expose functions for
+ * rendering expressions either to DOM nodes or to markup strings.
+ *
+ * We also expose the ParseError class to check if errors thrown from KaTeX are
+ * errors in the expression, or errors in javascript handling.
+ */
+
+
+
+
+
+
+
+
+
+
+/**
+ * Parse and build an expression, and place that expression in the DOM node
+ * given.
+ */
+var katex_render = function render(expression, baseNode, options) {
+  baseNode.textContent = "";
+  var node = katex_renderToDomTree(expression, options).toNode();
+  baseNode.appendChild(node);
+}; // KaTeX's styles don't work properly in quirks mode. Print out an error, and
+// disable rendering.
+
+
+if (typeof document !== "undefined") {
+  if (document.compatMode !== "CSS1Compat") {
+    typeof console !== "undefined" && console.warn("Warning: KaTeX doesn't work in quirks mode. Make sure your " + "website has a suitable doctype.");
+
+    katex_render = function render() {
+      throw new src_ParseError("KaTeX doesn't work in quirks mode.");
+    };
+  }
+}
+/**
+ * Parse and build an expression, and return the markup for that.
+ */
+
+
+var renderToString = function renderToString(expression, options) {
+  var markup = katex_renderToDomTree(expression, options).toMarkup();
+  return markup;
+};
+/**
+ * Parse an expression and return the parse tree.
+ */
+
+
+var katex_generateParseTree = function generateParseTree(expression, options) {
+  var settings = new Settings_Settings(options);
+  return src_parseTree(expression, settings);
+};
+/**
+ * If the given error is a KaTeX ParseError and options.throwOnError is false,
+ * renders the invalid LaTeX as a span with hover title giving the KaTeX
+ * error message.  Otherwise, simply throws the error.
+ */
+
+
+var katex_renderError = function renderError(error, expression, options) {
+  if (options.throwOnError || !(error instanceof src_ParseError)) {
+    throw error;
+  }
+
+  var node = buildCommon.makeSpan(["katex-error"], [new domTree_SymbolNode(expression)]);
+  node.setAttribute("title", error.toString());
+  node.setAttribute("style", "color:" + options.errorColor);
+  return node;
+};
+/**
+ * Generates and returns the katex build tree. This is used for advanced
+ * use cases (like rendering to custom output).
+ */
+
+
+var katex_renderToDomTree = function renderToDomTree(expression, options) {
+  var settings = new Settings_Settings(options);
+
+  try {
+    var tree = src_parseTree(expression, settings);
+    return buildTree_buildTree(tree, expression, settings);
+  } catch (error) {
+    return katex_renderError(error, expression, settings);
+  }
+};
+/**
+ * Generates and returns the katex build tree, with just HTML (no MathML).
+ * This is used for advanced use cases (like rendering to custom output).
+ */
+
+
+var katex_renderToHTMLTree = function renderToHTMLTree(expression, options) {
+  var settings = new Settings_Settings(options);
+
+  try {
+    var tree = src_parseTree(expression, settings);
+    return buildTree_buildHTMLTree(tree, expression, settings);
+  } catch (error) {
+    return katex_renderError(error, expression, settings);
+  }
+};
+
+/* harmony default export */ var katex_0 = ({
+  /**
+   * Current KaTeX version
+   */
+  version: "0.12.0",
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML combination, and adds
+   * it as a child to the specified DOM node.
+   */
+  render: katex_render,
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML combination string,
+   * for sending to the client.
+   */
+  renderToString: renderToString,
+
+  /**
+   * KaTeX error, usually during parsing.
+   */
+  ParseError: src_ParseError,
+
+  /**
+   * Parses the given LaTeX into KaTeX's internal parse tree structure,
+   * without rendering to HTML or MathML.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __parse: katex_generateParseTree,
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML internal DOM tree
+   * representation, without flattening that representation to a string.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __renderToDomTree: katex_renderToDomTree,
+
+  /**
+   * Renders the given LaTeX into an HTML internal DOM tree representation,
+   * without MathML and without flattening that representation to a string.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __renderToHTMLTree: katex_renderToHTMLTree,
+
+  /**
+   * extends internal font metrics object with a new object
+   * each key in the new object represents a font name
+  */
+  __setFontMetrics: setFontMetrics,
+
+  /**
+   * adds a new symbol to builtin symbols table
+   */
+  __defineSymbol: defineSymbol,
+
+  /**
+   * adds a new macro to builtin macro list
+   */
+  __defineMacro: defineMacro,
+
+  /**
+   * Expose the dom tree node types, which can be useful for type checking nodes.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __domTree: {
+    Span: domTree_Span,
+    Anchor: domTree_Anchor,
+    SymbolNode: domTree_SymbolNode,
+    SvgNode: SvgNode,
+    PathNode: domTree_PathNode,
+    LineNode: LineNode
+  }
+});
+// CONCATENATED MODULE: ./katex.webpack.js
+/**
+ * This is the webpack entry point for KaTeX. As ECMAScript, flow[1] and jest[2]
+ * doesn't support CSS modules natively, a separate entry point is used and
+ * it is not flowtyped.
+ *
+ * [1] https://gist.github.com/lambdahands/d19e0da96285b749f0ef
+ * [2] https://facebook.github.io/jest/docs/en/webpack.html
+ */
+
+
+/* harmony default export */ var katex_webpack = __webpack_exports__["default"] = (katex_0);
+
+/***/ })
+/******/ ])["default"];
+});
\ No newline at end of file
diff --git a/katex.bk/katex.min.css b/katex.bk/katex.min.css
new file mode 100644
index 00000000..98b7c7bd
--- /dev/null
+++ b/katex.bk/katex.min.css
@@ -0,0 +1 @@
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n=this.fetch();if(-1!==t.endOfExpression.indexOf(n.text))break;if(r&&n.text===r)break;if(e&&ia[n.text]&&ia[n.text].infix)break;var i=this.parseAtom(r);if(!i)break;"internal"!==i.type&&a.push(i)}return"text"===this.mode&&this.formLigatures(a),this.handleInfixNodes(a)},e.handleInfixNodes=function(t){for(var e,r=-1,a=0;a<t.length;a++)if("infix"===t[a].type){if(-1!==r)throw new o("only one infix operator per group",t[a].token);r=a,e=t[a].replaceWith}if(-1!==r&&e){var n,i,s=t.slice(0,r),l=t.slice(r+1);return n=1===s.length&&"ordgroup"===s[0].type?s[0]:{type:"ordgroup",mode:this.mode,body:s},i=1===l.length&&"ordgroup"===l[0].type?l[0]:{type:"ordgroup",mode:this.mode,body:l},["\\\\abovefrac"===e?this.callFunction(e,[n,t[r],i],[]):this.callFunction(e,[n,i],[])]}return t},e.handleSupSubscript=function(e){var r=this.fetch(),a=r.text;this.consume();var n=this.parseGroup(e,!1,t.SUPSUB_GREEDINESS,void 0,void 0,!0);if(!n)throw new o("Expected group after '"+a+"'",r);return n},e.formatUnsupportedCmd=function(t){for(var e=[],r=0;r<t.length;r++)e.push({type:"textord",mode:"text",text:t[r]});var a={type:"text",mode:this.mode,body:e};return{type:"color",mode:this.mode,color:this.settings.errorColor,body:[a]}},e.parseAtom=function(t){var e,r,a=this.parseGroup("atom",!1,null,t);if("text"===this.mode)return a;for(;;){this.consumeSpaces();var n=this.fetch();if("\\limits"===n.text||"\\nolimits"===n.text){if(a&&"op"===a.type){var i="\\limits"===n.text;a.limits=i,a.alwaysHandleSupSub=!0}else{if(!a||"operatorname"!==a.type||!a.alwaysHandleSupSub)throw new o("Limit controls must follow a math operator",n);var s="\\limits"===n.text;a.limits=s}this.consume()}else if("^"===n.text){if(e)throw new o("Double superscript",n);e=this.handleSupSubscript("superscript")}else if("_"===n.text){if(r)throw new o("Double subscript",n);r=this.handleSupSubscript("subscript")}else{if("'"!==n.text)break;if(e)throw new o("Double superscript",n);var l={type:"textord",mode:this.mode,text:"\\prime"},h=[l];for(this.consume();"'"===this.fetch().text;)h.push(l),this.consume();"^"===this.fetch().text&&h.push(this.handleSupSubscript("superscript")),e={type:"ordgroup",mode:this.mode,body:h}}}return e||r?{type:"supsub",mode:this.mode,base:a,sup:e,sub:r}:a},e.parseFunction=function(t,e,r){var a=this.fetch(),n=a.text,i=ia[n];if(!i)return null;if(this.consume(),null!=r&&i.greediness<=r)throw new o("Got function '"+n+"' with no arguments"+(e?" as "+e:""),a);if("text"===this.mode&&!i.allowedInText)throw new o("Can't use function '"+n+"' in text mode",a);if("math"===this.mode&&!1===i.allowedInMath)throw new o("Can't use function '"+n+"' in math mode",a);var s=this.parseArguments(n,i),l=s.args,h=s.optArgs;return this.callFunction(n,l,h,a,t)},e.callFunction=function(t,e,r,a,n){var i={funcName:t,parser:this,token:a,breakOnTokenText:n},s=ia[t];if(s&&s.handler)return s.handler(i,e,r);throw new o("No function handler for "+t)},e.parseArguments=function(t,e){var r=e.numArgs+e.numOptionalArgs;if(0===r)return{args:[],optArgs:[]};for(var a=e.greediness,n=[],i=[],s=0;s<r;s++){var l=e.argTypes&&e.argTypes[s],h=s<e.numOptionalArgs,m=s>0&&!h||0===s&&!h&&"math"===this.mode,c=this.parseGroupOfType("argument to '"+t+"'",l,h,a,m);if(!c){if(h){i.push(null);continue}throw new o("Expected group after '"+t+"'",this.fetch())}(h?i:n).push(c)}return{args:n,optArgs:i}},e.parseGroupOfType=function(t,e,r,a,n){switch(e){case"color":return n&&this.consumeSpaces(),this.parseColorGroup(r);case"size":return n&&this.consumeSpaces(),this.parseSizeGroup(r);case"url":return this.parseUrlGroup(r,n);case"math":case"text":return this.parseGroup(t,r,a,void 0,e,n);case"hbox":var i=this.parseGroup(t,r,a,void 0,"text",n);return i?{type:"styling",mode:i.mode,body:[i],style:"text"}:i;case"raw":if(n&&this.consumeSpaces(),r&&"{"===this.fetch().text)return null;var s=this.parseStringGroup("raw",r,!0);if(s)return{type:"raw",mode:"text",string:s.text};throw new o("Expected raw group",this.fetch());case"original":case null:case void 0:return this.parseGroup(t,r,a,void 0,void 0,n);default:throw new o("Unknown group type as "+t,this.fetch())}},e.consumeSpaces=function(){for(;" "===this.fetch().text;)this.consume()},e.parseStringGroup=function(t,e,r){var a=e?"[":"{",n=e?"]":"}",i=this.fetch();if(i.text!==a){if(e)return null;if(r&&"EOF"!==i.text&&/[^{}[\]]/.test(i.text))return this.consume(),i}var s=this.mode;this.mode="text",this.expect(a);for(var l,h="",m=this.fetch(),c=0,u=m;(l=this.fetch()).text!==n||r&&c>0;){switch(l.text){case"EOF":throw new o("Unexpected end of input in "+t,m.range(u,h));case a:c++;break;case n:c--}h+=(u=l).text,this.consume()}return this.expect(n),this.mode=s,m.range(u,h)},e.parseRegexGroup=function(t,e){var r=this.mode;this.mode="text";for(var a,n=this.fetch(),i=n,s="";"EOF"!==(a=this.fetch()).text&&t.test(s+a.text);)s+=(i=a).text,this.consume();if(""===s)throw new o("Invalid "+e+": '"+n.text+"'",n);return this.mode=r,n.range(i,s)},e.parseColorGroup=function(t){var e=this.parseStringGroup("color",t);if(!e)return null;var r=/^(#[a-f0-9]{3}|#?[a-f0-9]{6}|[a-z]+)$/i.exec(e.text);if(!r)throw new o("Invalid color: '"+e.text+"'",e);var a=r[0];return/^[0-9a-f]{6}$/i.test(a)&&(a="#"+a),{type:"color-token",mode:this.mode,color:a}},e.parseSizeGroup=function(t){var e,r=!1;if(!(e=t||"{"===this.fetch().text?this.parseStringGroup("size",t):this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2} *$/,"size")))return null;t||0!==e.text.length||(e.text="0pt",r=!0);var a=/([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(e.text);if(!a)throw new o("Invalid size: '"+e.text+"'",e);var n={number:+(a[1]+a[2]),unit:a[3]};if(!At(n))throw new o("Invalid unit: '"+n.unit+"'",e);return{type:"size",mode:this.mode,value:n,isBlank:r}},e.parseUrlGroup=function(t,e){this.gullet.lexer.setCatcode("%",13);var r=this.parseStringGroup("url",t,!0);if(this.gullet.lexer.setCatcode("%",14),!r)return null;var a=r.text.replace(/\\([#$%&~_^{}])/g,"$1");return{type:"url",mode:this.mode,url:a}},e.parseGroup=function(e,r,n,i,s,l){var h=this.mode;s&&this.switchMode(s),l&&this.consumeSpaces();var m,c=this.fetch(),u=c.text;if(r?"["===u:"{"===u||"\\begingroup"===u){this.consume();var p=t.endOfGroup[u];this.gullet.beginGroup();var d=this.parseExpression(!1,p),f=this.fetch();this.expect(p),this.gullet.endGroup(),m={type:"ordgroup",mode:this.mode,loc:a.range(c,f),body:d,semisimple:"\\begingroup"===u||void 0}}else if(r)m=null;else if(null==(m=this.parseFunction(i,e,n)||this.parseSymbol())&&"\\"===u[0]&&!ba.hasOwnProperty(u)){if(this.settings.throwOnError)throw new o("Undefined control sequence: "+u,c);m=this.formatUnsupportedCmd(u),this.consume()}return s&&this.switchMode(h),m},e.formLigatures=function(t){for(var e=t.length-1,r=0;r<e;++r){var n=t[r],i=n.text;"-"===i&&"-"===t[r+1].text&&(r+1<e&&"-"===t[r+2].text?(t.splice(r,3,{type:"textord",mode:"text",loc:a.range(n,t[r+2]),text:"---"}),e-=2):(t.splice(r,2,{type:"textord",mode:"text",loc:a.range(n,t[r+1]),text:"--"}),e-=1)),"'"!==i&&"`"!==i||t[r+1].text!==i||(t.splice(r,2,{type:"textord",mode:"text",loc:a.range(n,t[r+1]),text:i+i}),e-=1)}},e.parseSymbol=function(){var t=this.fetch(),e=t.text;if(/^\\verb[^a-zA-Z]/.test(e)){this.consume();var r=e.slice(5),n="*"===r.charAt(0);if(n&&(r=r.slice(1)),r.length<2||r.charAt(0)!==r.slice(-1))throw new o("\\verb assertion failed --\n                    please report what input caused this bug");return{type:"verb",mode:"text",body:r=r.slice(1,-1),star:n}}ka.hasOwnProperty(e[0])&&!j[this.mode][e[0]]&&(this.settings.strict&&"math"===this.mode&&this.settings.reportNonstrict("unicodeTextInMathMode",'Accented Unicode text character "'+e[0]+'" used in math mode',t),e=ka[e[0]]+e.substr(1));var i,s=sa.exec(e);if(s&&("i"===(e=e.substring(0,s.index))?e="\u0131":"j"===e&&(e="\u0237")),j[this.mode][e]){this.settings.strict&&"math"===this.mode&&"\xc7\xd0\xde\xe7\xfe".indexOf(e)>=0&&this.settings.reportNonstrict("unicodeTextInMathMode",'Latin-1/Unicode text character "'+e[0]+'" used in math mode',t);var l,h=j[this.mode][e].group,m=a.range(t);if(W.hasOwnProperty(h)){var c=h;l={type:"atom",mode:this.mode,family:c,loc:m,text:e}}else l={type:h,mode:this.mode,loc:m,text:e};i=l}else{if(!(e.charCodeAt(0)>=128))return null;this.settings.strict&&(M(e.charCodeAt(0))?"math"===this.mode&&this.settings.reportNonstrict("unicodeTextInMathMode",'Unicode text character "'+e[0]+'" used in math mode',t):this.settings.reportNonstrict("unknownSymbol",'Unrecognized Unicode character "'+e[0]+'" ('+e.charCodeAt(0)+")",t)),i={type:"textord",mode:"text",loc:a.range(t),text:e}}if(this.consume(),s)for(var u=0;u<s[0].length;u++){var p=s[0][u];if(!wa[p])throw new o("Unknown accent ' "+p+"'",t);var d=wa[p][this.mode];if(!d)throw new o("Accent "+p+" unsupported in "+this.mode+" mode",t);i={type:"accent",mode:this.mode,loc:a.range(t),label:d,isStretchy:!1,isShifty:!0,base:i}}return i},t}();Sa.endOfExpression=["}","\\endgroup","\\end","\\right","&"],Sa.endOfGroup={"[":"]","{":"}","\\begingroup":"\\endgroup"},Sa.SUPSUB_GREEDINESS=1;var Ma=function(t,e){if(!("string"==typeof t||t instanceof String))throw new TypeError("KaTeX can only parse string typed expression");var r=new Sa(t,e);delete r.gullet.macros.current["\\df@tag"];var a=r.parse();if(r.gullet.macros.get("\\df@tag")){if(!e.displayMode)throw new o("\\tag works only in display equations");r.gullet.feed("\\df@tag"),a=[{type:"tag",mode:"text",body:a,tag:r.parse()}]}return a},za=function(t,e,r){e.textContent="";var a=Ta(t,r).toNode();e.appendChild(a)};"undefined"!=typeof document&&"CSS1Compat"!==document.compatMode&&("undefined"!=typeof console&&console.warn("Warning: KaTeX doesn't work in quirks mode. Make sure your website has a suitable doctype."),za=function(){throw new o("KaTeX doesn't work in quirks mode.")});var Aa=function(t,e,r){if(r.throwOnError||!(t instanceof o))throw t;var a=Dt.makeSpan(["katex-error"],[new E(e)]);return a.setAttribute("title",t.toString()),a.setAttribute("style","color:"+r.errorColor),a},Ta=function(t,e){var r=new u(e);try{var a=Ma(t,r);return ke(a,t,r)}catch(e){return Aa(e,t,r)}},Ba={version:"0.12.0",render:za,renderToString:function(t,e){return Ta(t,e).toMarkup()},ParseError:o,__parse:function(t,e){var r=new u(e);return Ma(t,r)},__renderToDomTree:Ta,__renderToHTMLTree:function(t,e){var r=new u(e);try{return function(t,e,r){var a=le(t,ye(r)),n=Dt.makeSpan(["katex"],[a]);return we(n,r)}(Ma(t,r),0,r)}catch(e){return Aa(e,t,r)}},__setFontMetrics:function(t,e){F[t]=e},__defineSymbol:$,__defineMacro:pa,__domTree:{Span:N,Anchor:I,SymbolNode:E,SvgNode:L,PathNode:P,LineNode:D}};e.default=Ba}]).default});
\ No newline at end of file
diff --git a/katex.bk/katex.mjs b/katex.bk/katex.mjs
new file mode 100644
index 00000000..76938229
--- /dev/null
+++ b/katex.bk/katex.mjs
@@ -0,0 +1,16911 @@
+/**
+ * Lexing or parsing positional information for error reporting.
+ * This object is immutable.
+ */
+class SourceLocation {
+  // The + prefix indicates that these fields aren't writeable
+  // Lexer holding the input string.
+  // Start offset, zero-based inclusive.
+  // End offset, zero-based exclusive.
+  constructor(lexer, start, end) {
+    this.lexer = void 0;
+    this.start = void 0;
+    this.end = void 0;
+    this.lexer = lexer;
+    this.start = start;
+    this.end = end;
+  }
+  /**
+   * Merges two `SourceLocation`s from location providers, given they are
+   * provided in order of appearance.
+   * - Returns the first one's location if only the first is provided.
+   * - Returns a merged range of the first and the last if both are provided
+   *   and their lexers match.
+   * - Otherwise, returns null.
+   */
+
+
+  static range(first, second) {
+    if (!second) {
+      return first && first.loc;
+    } else if (!first || !first.loc || !second.loc || first.loc.lexer !== second.loc.lexer) {
+      return null;
+    } else {
+      return new SourceLocation(first.loc.lexer, first.loc.start, second.loc.end);
+    }
+  }
+
+}
+
+/**
+ * Interface required to break circular dependency between Token, Lexer, and
+ * ParseError.
+ */
+
+/**
+ * The resulting token returned from `lex`.
+ *
+ * It consists of the token text plus some position information.
+ * The position information is essentially a range in an input string,
+ * but instead of referencing the bare input string, we refer to the lexer.
+ * That way it is possible to attach extra metadata to the input string,
+ * like for example a file name or similar.
+ *
+ * The position information is optional, so it is OK to construct synthetic
+ * tokens if appropriate. Not providing available position information may
+ * lead to degraded error reporting, though.
+ */
+class Token {
+  // don't expand the token
+  // used in \noexpand
+  constructor(text, // the text of this token
+  loc) {
+    this.text = void 0;
+    this.loc = void 0;
+    this.noexpand = void 0;
+    this.treatAsRelax = void 0;
+    this.text = text;
+    this.loc = loc;
+  }
+  /**
+   * Given a pair of tokens (this and endToken), compute a `Token` encompassing
+   * the whole input range enclosed by these two.
+   */
+
+
+  range(endToken, // last token of the range, inclusive
+  text) // the text of the newly constructed token
+  {
+    return new Token(text, SourceLocation.range(this, endToken));
+  }
+
+}
+
+/**
+ * This is the ParseError class, which is the main error thrown by KaTeX
+ * functions when something has gone wrong. This is used to distinguish internal
+ * errors from errors in the expression that the user provided.
+ *
+ * If possible, a caller should provide a Token or ParseNode with information
+ * about where in the source string the problem occurred.
+ */
+class ParseError {
+  // Error position based on passed-in Token or ParseNode.
+  constructor(message, // The error message
+  token) // An object providing position information
+  {
+    this.position = void 0;
+    let error = "KaTeX parse error: " + message;
+    let start;
+    const loc = token && token.loc;
+
+    if (loc && loc.start <= loc.end) {
+      // If we have the input and a position, make the error a bit fancier
+      // Get the input
+      const input = loc.lexer.input; // Prepend some information
+
+      start = loc.start;
+      const end = loc.end;
+
+      if (start === input.length) {
+        error += " at end of input: ";
+      } else {
+        error += " at position " + (start + 1) + ": ";
+      } // Underline token in question using combining underscores
+
+
+      const underlined = input.slice(start, end).replace(/[^]/g, "$&\u0332"); // Extract some context from the input and add it to the error
+
+      let left;
+
+      if (start > 15) {
+        left = "…" + input.slice(start - 15, start);
+      } else {
+        left = input.slice(0, start);
+      }
+
+      let right;
+
+      if (end + 15 < input.length) {
+        right = input.slice(end, end + 15) + "…";
+      } else {
+        right = input.slice(end);
+      }
+
+      error += left + underlined + right;
+    } // Some hackery to make ParseError a prototype of Error
+    // See http://stackoverflow.com/a/8460753
+
+
+    const self = new Error(error);
+    self.name = "ParseError"; // $FlowFixMe
+
+    self.__proto__ = ParseError.prototype; // $FlowFixMe
+
+    self.position = start;
+    return self;
+  }
+
+} // $FlowFixMe More hackery
+
+
+ParseError.prototype.__proto__ = Error.prototype;
+
+/**
+ * This file contains a list of utility functions which are useful in other
+ * files.
+ */
+
+/**
+ * Return whether an element is contained in a list
+ */
+const contains = function contains(list, elem) {
+  return list.indexOf(elem) !== -1;
+};
+/**
+ * Provide a default value if a setting is undefined
+ * NOTE: Couldn't use `T` as the output type due to facebook/flow#5022.
+ */
+
+
+const deflt = function deflt(setting, defaultIfUndefined) {
+  return setting === undefined ? defaultIfUndefined : setting;
+}; // hyphenate and escape adapted from Facebook's React under Apache 2 license
+
+
+const uppercase = /([A-Z])/g;
+
+const hyphenate = function hyphenate(str) {
+  return str.replace(uppercase, "-$1").toLowerCase();
+};
+
+const ESCAPE_LOOKUP = {
+  "&": "&amp;",
+  ">": "&gt;",
+  "<": "&lt;",
+  "\"": "&quot;",
+  "'": "&#x27;"
+};
+const ESCAPE_REGEX = /[&><"']/g;
+/**
+ * Escapes text to prevent scripting attacks.
+ */
+
+function escape(text) {
+  return String(text).replace(ESCAPE_REGEX, match => ESCAPE_LOOKUP[match]);
+}
+/**
+ * Sometimes we want to pull out the innermost element of a group. In most
+ * cases, this will just be the group itself, but when ordgroups and colors have
+ * a single element, we want to pull that out.
+ */
+
+
+const getBaseElem = function getBaseElem(group) {
+  if (group.type === "ordgroup") {
+    if (group.body.length === 1) {
+      return getBaseElem(group.body[0]);
+    } else {
+      return group;
+    }
+  } else if (group.type === "color") {
+    if (group.body.length === 1) {
+      return getBaseElem(group.body[0]);
+    } else {
+      return group;
+    }
+  } else if (group.type === "font") {
+    return getBaseElem(group.body);
+  } else {
+    return group;
+  }
+};
+/**
+ * TeXbook algorithms often reference "character boxes", which are simply groups
+ * with a single character in them. To decide if something is a character box,
+ * we find its innermost group, and see if it is a single character.
+ */
+
+
+const isCharacterBox = function isCharacterBox(group) {
+  const baseElem = getBaseElem(group); // These are all they types of groups which hold single characters
+
+  return baseElem.type === "mathord" || baseElem.type === "textord" || baseElem.type === "atom";
+};
+
+const assert = function assert(value) {
+  if (!value) {
+    throw new Error('Expected non-null, but got ' + String(value));
+  }
+
+  return value;
+};
+/**
+ * Return the protocol of a URL, or "_relative" if the URL does not specify a
+ * protocol (and thus is relative).
+ */
+
+const protocolFromUrl = function protocolFromUrl(url) {
+  const protocol = /^\s*([^\\/#]*?)(?::|&#0*58|&#x0*3a)/i.exec(url);
+  return protocol != null ? protocol[1] : "_relative";
+};
+var utils = {
+  contains,
+  deflt,
+  escape,
+  hyphenate,
+  getBaseElem,
+  isCharacterBox,
+  protocolFromUrl
+};
+
+/* eslint no-console:0 */
+
+/**
+ * The main Settings object
+ *
+ * The current options stored are:
+ *  - displayMode: Whether the expression should be typeset as inline math
+ *                 (false, the default), meaning that the math starts in
+ *                 \textstyle and is placed in an inline-block); or as display
+ *                 math (true), meaning that the math starts in \displaystyle
+ *                 and is placed in a block with vertical margin.
+ */
+class Settings {
+  constructor(options) {
+    this.displayMode = void 0;
+    this.output = void 0;
+    this.leqno = void 0;
+    this.fleqn = void 0;
+    this.throwOnError = void 0;
+    this.errorColor = void 0;
+    this.macros = void 0;
+    this.minRuleThickness = void 0;
+    this.colorIsTextColor = void 0;
+    this.strict = void 0;
+    this.trust = void 0;
+    this.maxSize = void 0;
+    this.maxExpand = void 0;
+    this.globalGroup = void 0;
+    // allow null options
+    options = options || {};
+    this.displayMode = utils.deflt(options.displayMode, false);
+    this.output = utils.deflt(options.output, "htmlAndMathml");
+    this.leqno = utils.deflt(options.leqno, false);
+    this.fleqn = utils.deflt(options.fleqn, false);
+    this.throwOnError = utils.deflt(options.throwOnError, true);
+    this.errorColor = utils.deflt(options.errorColor, "#cc0000");
+    this.macros = options.macros || {};
+    this.minRuleThickness = Math.max(0, utils.deflt(options.minRuleThickness, 0));
+    this.colorIsTextColor = utils.deflt(options.colorIsTextColor, false);
+    this.strict = utils.deflt(options.strict, "warn");
+    this.trust = utils.deflt(options.trust, false);
+    this.maxSize = Math.max(0, utils.deflt(options.maxSize, Infinity));
+    this.maxExpand = Math.max(0, utils.deflt(options.maxExpand, 1000));
+    this.globalGroup = utils.deflt(options.globalGroup, false);
+  }
+  /**
+   * Report nonstrict (non-LaTeX-compatible) input.
+   * Can safely not be called if `this.strict` is false in JavaScript.
+   */
+
+
+  reportNonstrict(errorCode, errorMsg, token) {
+    let strict = this.strict;
+
+    if (typeof strict === "function") {
+      // Allow return value of strict function to be boolean or string
+      // (or null/undefined, meaning no further processing).
+      strict = strict(errorCode, errorMsg, token);
+    }
+
+    if (!strict || strict === "ignore") {
+      return;
+    } else if (strict === true || strict === "error") {
+      throw new ParseError("LaTeX-incompatible input and strict mode is set to 'error': " + `${errorMsg} [${errorCode}]`, token);
+    } else if (strict === "warn") {
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to 'warn': " + `${errorMsg} [${errorCode}]`);
+    } else {
+      // won't happen in type-safe code
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to " + `unrecognized '${strict}': ${errorMsg} [${errorCode}]`);
+    }
+  }
+  /**
+   * Check whether to apply strict (LaTeX-adhering) behavior for unusual
+   * input (like `\\`).  Unlike `nonstrict`, will not throw an error;
+   * instead, "error" translates to a return value of `true`, while "ignore"
+   * translates to a return value of `false`.  May still print a warning:
+   * "warn" prints a warning and returns `false`.
+   * This is for the second category of `errorCode`s listed in the README.
+   */
+
+
+  useStrictBehavior(errorCode, errorMsg, token) {
+    let strict = this.strict;
+
+    if (typeof strict === "function") {
+      // Allow return value of strict function to be boolean or string
+      // (or null/undefined, meaning no further processing).
+      // But catch any exceptions thrown by function, treating them
+      // like "error".
+      try {
+        strict = strict(errorCode, errorMsg, token);
+      } catch (error) {
+        strict = "error";
+      }
+    }
+
+    if (!strict || strict === "ignore") {
+      return false;
+    } else if (strict === true || strict === "error") {
+      return true;
+    } else if (strict === "warn") {
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to 'warn': " + `${errorMsg} [${errorCode}]`);
+      return false;
+    } else {
+      // won't happen in type-safe code
+      typeof console !== "undefined" && console.warn("LaTeX-incompatible input and strict mode is set to " + `unrecognized '${strict}': ${errorMsg} [${errorCode}]`);
+      return false;
+    }
+  }
+  /**
+   * Check whether to test potentially dangerous input, and return
+   * `true` (trusted) or `false` (untrusted).  The sole argument `context`
+   * should be an object with `command` field specifying the relevant LaTeX
+   * command (as a string starting with `\`), and any other arguments, etc.
+   * If `context` has a `url` field, a `protocol` field will automatically
+   * get added by this function (changing the specified object).
+   */
+
+
+  isTrusted(context) {
+    if (context.url && !context.protocol) {
+      context.protocol = utils.protocolFromUrl(context.url);
+    }
+
+    const trust = typeof this.trust === "function" ? this.trust(context) : this.trust;
+    return Boolean(trust);
+  }
+
+}
+
+/**
+ * This file contains information and classes for the various kinds of styles
+ * used in TeX. It provides a generic `Style` class, which holds information
+ * about a specific style. It then provides instances of all the different kinds
+ * of styles possible, and provides functions to move between them and get
+ * information about them.
+ */
+
+/**
+ * The main style class. Contains a unique id for the style, a size (which is
+ * the same for cramped and uncramped version of a style), and a cramped flag.
+ */
+class Style {
+  constructor(id, size, cramped) {
+    this.id = void 0;
+    this.size = void 0;
+    this.cramped = void 0;
+    this.id = id;
+    this.size = size;
+    this.cramped = cramped;
+  }
+  /**
+   * Get the style of a superscript given a base in the current style.
+   */
+
+
+  sup() {
+    return styles[sup[this.id]];
+  }
+  /**
+   * Get the style of a subscript given a base in the current style.
+   */
+
+
+  sub() {
+    return styles[sub[this.id]];
+  }
+  /**
+   * Get the style of a fraction numerator given the fraction in the current
+   * style.
+   */
+
+
+  fracNum() {
+    return styles[fracNum[this.id]];
+  }
+  /**
+   * Get the style of a fraction denominator given the fraction in the current
+   * style.
+   */
+
+
+  fracDen() {
+    return styles[fracDen[this.id]];
+  }
+  /**
+   * Get the cramped version of a style (in particular, cramping a cramped style
+   * doesn't change the style).
+   */
+
+
+  cramp() {
+    return styles[cramp[this.id]];
+  }
+  /**
+   * Get a text or display version of this style.
+   */
+
+
+  text() {
+    return styles[text[this.id]];
+  }
+  /**
+   * Return true if this style is tightly spaced (scriptstyle/scriptscriptstyle)
+   */
+
+
+  isTight() {
+    return this.size >= 2;
+  }
+
+} // Export an interface for type checking, but don't expose the implementation.
+// This way, no more styles can be generated.
+
+
+// IDs of the different styles
+const D = 0;
+const Dc = 1;
+const T = 2;
+const Tc = 3;
+const S = 4;
+const Sc = 5;
+const SS = 6;
+const SSc = 7; // Instances of the different styles
+
+const styles = [new Style(D, 0, false), new Style(Dc, 0, true), new Style(T, 1, false), new Style(Tc, 1, true), new Style(S, 2, false), new Style(Sc, 2, true), new Style(SS, 3, false), new Style(SSc, 3, true)]; // Lookup tables for switching from one style to another
+
+const sup = [S, Sc, S, Sc, SS, SSc, SS, SSc];
+const sub = [Sc, Sc, Sc, Sc, SSc, SSc, SSc, SSc];
+const fracNum = [T, Tc, S, Sc, SS, SSc, SS, SSc];
+const fracDen = [Tc, Tc, Sc, Sc, SSc, SSc, SSc, SSc];
+const cramp = [Dc, Dc, Tc, Tc, Sc, Sc, SSc, SSc];
+const text = [D, Dc, T, Tc, T, Tc, T, Tc]; // We only export some of the styles.
+
+var Style$1 = {
+  DISPLAY: styles[D],
+  TEXT: styles[T],
+  SCRIPT: styles[S],
+  SCRIPTSCRIPT: styles[SS]
+};
+
+/*
+ * This file defines the Unicode scripts and script families that we
+ * support. To add new scripts or families, just add a new entry to the
+ * scriptData array below. Adding scripts to the scriptData array allows
+ * characters from that script to appear in \text{} environments.
+ */
+
+/**
+ * Each script or script family has a name and an array of blocks.
+ * Each block is an array of two numbers which specify the start and
+ * end points (inclusive) of a block of Unicode codepoints.
+ */
+
+/**
+ * Unicode block data for the families of scripts we support in \text{}.
+ * Scripts only need to appear here if they do not have font metrics.
+ */
+const scriptData = [{
+  // Latin characters beyond the Latin-1 characters we have metrics for.
+  // Needed for Czech, Hungarian and Turkish text, for example.
+  name: 'latin',
+  blocks: [[0x0100, 0x024f], // Latin Extended-A and Latin Extended-B
+  [0x0300, 0x036f]]
+}, {
+  // The Cyrillic script used by Russian and related languages.
+  // A Cyrillic subset used to be supported as explicitly defined
+  // symbols in symbols.js
+  name: 'cyrillic',
+  blocks: [[0x0400, 0x04ff]]
+}, {
+  // The Brahmic scripts of South and Southeast Asia
+  // Devanagari (0900–097F)
+  // Bengali (0980–09FF)
+  // Gurmukhi (0A00–0A7F)
+  // Gujarati (0A80–0AFF)
+  // Oriya (0B00–0B7F)
+  // Tamil (0B80–0BFF)
+  // Telugu (0C00–0C7F)
+  // Kannada (0C80–0CFF)
+  // Malayalam (0D00–0D7F)
+  // Sinhala (0D80–0DFF)
+  // Thai (0E00–0E7F)
+  // Lao (0E80–0EFF)
+  // Tibetan (0F00–0FFF)
+  // Myanmar (1000–109F)
+  name: 'brahmic',
+  blocks: [[0x0900, 0x109F]]
+}, {
+  name: 'georgian',
+  blocks: [[0x10A0, 0x10ff]]
+}, {
+  // Chinese and Japanese.
+  // The "k" in cjk is for Korean, but we've separated Korean out
+  name: "cjk",
+  blocks: [[0x3000, 0x30FF], // CJK symbols and punctuation, Hiragana, Katakana
+  [0x4E00, 0x9FAF], // CJK ideograms
+  [0xFF00, 0xFF60]]
+}, {
+  // Korean
+  name: 'hangul',
+  blocks: [[0xAC00, 0xD7AF]]
+}];
+/**
+ * Given a codepoint, return the name of the script or script family
+ * it is from, or null if it is not part of a known block
+ */
+
+function scriptFromCodepoint(codepoint) {
+  for (let i = 0; i < scriptData.length; i++) {
+    const script = scriptData[i];
+
+    for (let i = 0; i < script.blocks.length; i++) {
+      const block = script.blocks[i];
+
+      if (codepoint >= block[0] && codepoint <= block[1]) {
+        return script.name;
+      }
+    }
+  }
+
+  return null;
+}
+/**
+ * A flattened version of all the supported blocks in a single array.
+ * This is an optimization to make supportedCodepoint() fast.
+ */
+
+const allBlocks = [];
+scriptData.forEach(s => s.blocks.forEach(b => allBlocks.push(...b)));
+/**
+ * Given a codepoint, return true if it falls within one of the
+ * scripts or script families defined above and false otherwise.
+ *
+ * Micro benchmarks shows that this is faster than
+ * /[\u3000-\u30FF\u4E00-\u9FAF\uFF00-\uFF60\uAC00-\uD7AF\u0900-\u109F]/.test()
+ * in Firefox, Chrome and Node.
+ */
+
+function supportedCodepoint(codepoint) {
+  for (let i = 0; i < allBlocks.length; i += 2) {
+    if (codepoint >= allBlocks[i] && codepoint <= allBlocks[i + 1]) {
+      return true;
+    }
+  }
+
+  return false;
+}
+
+/**
+ * This file provides support to domTree.js and delimiter.js.
+ * It's a storehouse of path geometry for SVG images.
+ */
+// In all paths below, the viewBox-to-em scale is 1000:1.
+const hLinePad = 80; // padding above a sqrt viniculum. Prevents image cropping.
+// The viniculum of a \sqrt can be made thicker by a KaTeX rendering option.
+// Think of variable extraViniculum as two detours in the SVG path.
+// The detour begins at the lower left of the area labeled extraViniculum below.
+// The detour proceeds one extraViniculum distance up and slightly to the right,
+// displacing the radiused corner between surd and viniculum. The radius is
+// traversed as usual, then the detour resumes. It goes right, to the end of
+// the very long viniculumn, then down one extraViniculum distance,
+// after which it resumes regular path geometry for the radical.
+
+/*                                                  viniculum
+                                                   /
+         /▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒←extraViniculum
+        / █████████████████████←0.04em (40 unit) std viniculum thickness
+       / /
+      / /
+     / /\
+    / / surd
+*/
+
+const sqrtMain = function sqrtMain(extraViniculum, hLinePad) {
+  // sqrtMain path geometry is from glyph U221A in the font KaTeX Main
+  return `M95,${622 + extraViniculum + hLinePad}
+c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
+c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
+c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
+s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
+c69,-144,104.5,-217.7,106.5,-221
+l${extraViniculum / 2.075} -${extraViniculum}
+c5.3,-9.3,12,-14,20,-14
+H400000v${40 + extraViniculum}H845.2724
+s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
+c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
+M${834 + extraViniculum} ${hLinePad}h400000v${40 + extraViniculum}h-400000z`;
+};
+
+const sqrtSize1 = function sqrtSize1(extraViniculum, hLinePad) {
+  // size1 is from glyph U221A in the font KaTeX_Size1-Regular
+  return `M263,${601 + extraViniculum + hLinePad}c0.7,0,18,39.7,52,119
+c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120
+c340,-704.7,510.7,-1060.3,512,-1067
+l${extraViniculum / 2.084} -${extraViniculum}
+c4.7,-7.3,11,-11,19,-11
+H40000v${40 + extraViniculum}H1012.3
+s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232
+c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1
+s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26
+c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z
+M${1001 + extraViniculum} ${hLinePad}h400000v${40 + extraViniculum}h-400000z`;
+};
+
+const sqrtSize2 = function sqrtSize2(extraViniculum, hLinePad) {
+  // size2 is from glyph U221A in the font KaTeX_Size2-Regular
+  return `M983 ${10 + extraViniculum + hLinePad}
+l${extraViniculum / 3.13} -${extraViniculum}
+c4,-6.7,10,-10,18,-10 H400000v${40 + extraViniculum}
+H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
+s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744
+c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30
+c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722
+c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5
+c53.7,-170.3,84.5,-266.8,92.5,-289.5z
+M${1001 + extraViniculum} ${hLinePad}h400000v${40 + extraViniculum}h-400000z`;
+};
+
+const sqrtSize3 = function sqrtSize3(extraViniculum, hLinePad) {
+  // size3 is from glyph U221A in the font KaTeX_Size3-Regular
+  return `M424,${2398 + extraViniculum + hLinePad}
+c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514
+c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20
+s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121
+s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081
+l${extraViniculum / 4.223} -${extraViniculum}c4,-6.7,10,-10,18,-10 H400000
+v${40 + extraViniculum}H1014.6
+s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185
+c-2,6,-10,9,-24,9
+c-8,0,-12,-0.7,-12,-2z M${1001 + extraViniculum} ${hLinePad}
+h400000v${40 + extraViniculum}h-400000z`;
+};
+
+const sqrtSize4 = function sqrtSize4(extraViniculum, hLinePad) {
+  // size4 is from glyph U221A in the font KaTeX_Size4-Regular
+  return `M473,${2713 + extraViniculum + hLinePad}
+c339.3,-1799.3,509.3,-2700,510,-2702 l${extraViniculum / 5.298} -${extraViniculum}
+c3.3,-7.3,9.3,-11,18,-11 H400000v${40 + extraViniculum}H1017.7
+s-90.5,478,-276.2,1466c-185.7,988,-279.5,1483,-281.5,1485c-2,6,-10,9,-24,9
+c-8,0,-12,-0.7,-12,-2c0,-1.3,-5.3,-32,-16,-92c-50.7,-293.3,-119.7,-693.3,-207,-1200
+c0,-1.3,-5.3,8.7,-16,30c-10.7,21.3,-21.3,42.7,-32,64s-16,33,-16,33s-26,-26,-26,-26
+s76,-153,76,-153s77,-151,77,-151c0.7,0.7,35.7,202,105,604c67.3,400.7,102,602.7,104,
+606zM${1001 + extraViniculum} ${hLinePad}h400000v${40 + extraViniculum}H1017.7z`;
+};
+
+const sqrtTall = function sqrtTall(extraViniculum, hLinePad, viewBoxHeight) {
+  // sqrtTall is from glyph U23B7 in the font KaTeX_Size4-Regular
+  // One path edge has a variable length. It runs vertically from the viniculumn
+  // to a point near (14 units) the bottom of the surd. The viniculum
+  // is normally 40 units thick. So the length of the line in question is:
+  const vertSegment = viewBoxHeight - 54 - hLinePad - extraViniculum;
+  return `M702 ${extraViniculum + hLinePad}H400000${40 + extraViniculum}
+H742v${vertSegment}l-4 4-4 4c-.667.7 -2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1
+h-12l-28-84c-16.667-52-96.667 -294.333-240-727l-212 -643 -85 170
+c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667
+219 661 l218 661zM702 ${hLinePad}H400000v${40 + extraViniculum}H742z`;
+};
+
+const sqrtPath = function sqrtPath(size, extraViniculum, viewBoxHeight) {
+  extraViniculum = 1000 * extraViniculum; // Convert from document ems to viewBox.
+
+  let path = "";
+
+  switch (size) {
+    case "sqrtMain":
+      path = sqrtMain(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize1":
+      path = sqrtSize1(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize2":
+      path = sqrtSize2(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize3":
+      path = sqrtSize3(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtSize4":
+      path = sqrtSize4(extraViniculum, hLinePad);
+      break;
+
+    case "sqrtTall":
+      path = sqrtTall(extraViniculum, hLinePad, viewBoxHeight);
+  }
+
+  return path;
+};
+const path = {
+  // Two paths that cover gaps in built-up parentheses.
+  leftParenInner: `M291 0 H417 V300 H291 z`,
+  rightParenInner: `M457 0 H583 V300 H457 z`,
+  // The doubleleftarrow geometry is from glyph U+21D0 in the font KaTeX Main
+  doubleleftarrow: `M262 157
+l10-10c34-36 62.7-77 86-123 3.3-8 5-13.3 5-16 0-5.3-6.7-8-20-8-7.3
+ 0-12.2.5-14.5 1.5-2.3 1-4.8 4.5-7.5 10.5-49.3 97.3-121.7 169.3-217 216-28
+ 14-57.3 25-88 33-6.7 2-11 3.8-13 5.5-2 1.7-3 4.2-3 7.5s1 5.8 3 7.5
+c2 1.7 6.3 3.5 13 5.5 68 17.3 128.2 47.8 180.5 91.5 52.3 43.7 93.8 96.2 124.5
+ 157.5 9.3 8 15.3 12.3 18 13h6c12-.7 18-4 18-10 0-2-1.7-7-5-15-23.3-46-52-87
+-86-123l-10-10h399738v-40H218c328 0 0 0 0 0l-10-8c-26.7-20-65.7-43-117-69 2.7
+-2 6-3.7 10-5 36.7-16 72.3-37.3 107-64l10-8h399782v-40z
+m8 0v40h399730v-40zm0 194v40h399730v-40z`,
+  // doublerightarrow is from glyph U+21D2 in font KaTeX Main
+  doublerightarrow: `M399738 392l
+-10 10c-34 36-62.7 77-86 123-3.3 8-5 13.3-5 16 0 5.3 6.7 8 20 8 7.3 0 12.2-.5
+ 14.5-1.5 2.3-1 4.8-4.5 7.5-10.5 49.3-97.3 121.7-169.3 217-216 28-14 57.3-25 88
+-33 6.7-2 11-3.8 13-5.5 2-1.7 3-4.2 3-7.5s-1-5.8-3-7.5c-2-1.7-6.3-3.5-13-5.5-68
+-17.3-128.2-47.8-180.5-91.5-52.3-43.7-93.8-96.2-124.5-157.5-9.3-8-15.3-12.3-18
+-13h-6c-12 .7-18 4-18 10 0 2 1.7 7 5 15 23.3 46 52 87 86 123l10 10H0v40h399782
+c-328 0 0 0 0 0l10 8c26.7 20 65.7 43 117 69-2.7 2-6 3.7-10 5-36.7 16-72.3 37.3
+-107 64l-10 8H0v40zM0 157v40h399730v-40zm0 194v40h399730v-40z`,
+  // leftarrow is from glyph U+2190 in font KaTeX Main
+  leftarrow: `M400000 241H110l3-3c68.7-52.7 113.7-120
+ 135-202 4-14.7 6-23 6-25 0-7.3-7-11-21-11-8 0-13.2.8-15.5 2.5-2.3 1.7-4.2 5.8
+-5.5 12.5-1.3 4.7-2.7 10.3-4 17-12 48.7-34.8 92-68.5 130S65.3 228.3 18 247
+c-10 4-16 7.7-18 11 0 8.7 6 14.3 18 17 47.3 18.7 87.8 47 121.5 85S196 441.3 208
+ 490c.7 2 1.3 5 2 9s1.2 6.7 1.5 8c.3 1.3 1 3.3 2 6s2.2 4.5 3.5 5.5c1.3 1 3.3
+ 1.8 6 2.5s6 1 10 1c14 0 21-3.7 21-11 0-2-2-10.3-6-25-20-79.3-65-146.7-135-202
+ l-3-3h399890zM100 241v40h399900v-40z`,
+  // overbrace is from glyphs U+23A9/23A8/23A7 in font KaTeX_Size4-Regular
+  leftbrace: `M6 548l-6-6v-35l6-11c56-104 135.3-181.3 238-232 57.3-28.7 117
+-45 179-50h399577v120H403c-43.3 7-81 15-113 26-100.7 33-179.7 91-237 174-2.7
+ 5-6 9-10 13-.7 1-7.3 1-20 1H6z`,
+  leftbraceunder: `M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13
+ 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688
+ 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7
+-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z`,
+  // overgroup is from the MnSymbol package (public domain)
+  leftgroup: `M400000 80
+H435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0
+ 435 0h399565z`,
+  leftgroupunder: `M400000 262
+H435C64 262 168.3 112.6 21 82c-5.9-1.2-18 0-18 0-2 0-3 1-3 3v38c76 158 257 219
+ 435 219h399565z`,
+  // Harpoons are from glyph U+21BD in font KaTeX Main
+  leftharpoon: `M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3
+-3.3 10.2-9.5 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5
+-18.3 3-21-1.3-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7
+-196 228-6.7 4.7-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40z`,
+  leftharpoonplus: `M0 267c.7 5.3 3 10 7 14h399993v-40H93c3.3-3.3 10.2-9.5
+ 20.5-18.5s17.8-15.8 22.5-20.5c50.7-52 88-110.3 112-175 4-11.3 5-18.3 3-21-1.3
+-4-7.3-6-18-6-8 0-13 .7-15 2s-4.7 6.7-8 16c-42 98.7-107.3 174.7-196 228-6.7 4.7
+-10.7 8-12 10-1.3 2-2 5.7-2 11zm100-26v40h399900v-40zM0 435v40h400000v-40z
+m0 0v40h400000v-40z`,
+  leftharpoondown: `M7 241c-4 4-6.333 8.667-7 14 0 5.333.667 9 2 11s5.333
+ 5.333 12 10c90.667 54 156 130 196 228 3.333 10.667 6.333 16.333 9 17 2 .667 5
+ 1 9 1h5c10.667 0 16.667-2 18-6 2-2.667 1-9.667-3-21-32-87.333-82.667-157.667
+-152-211l-3-3h399907v-40zM93 281 H400000 v-40L7 241z`,
+  leftharpoondownplus: `M7 435c-4 4-6.3 8.7-7 14 0 5.3.7 9 2 11s5.3 5.3 12
+ 10c90.7 54 156 130 196 228 3.3 10.7 6.3 16.3 9 17 2 .7 5 1 9 1h5c10.7 0 16.7
+-2 18-6 2-2.7 1-9.7-3-21-32-87.3-82.7-157.7-152-211l-3-3h399907v-40H7zm93 0
+v40h399900v-40zM0 241v40h399900v-40zm0 0v40h399900v-40z`,
+  // hook is from glyph U+21A9 in font KaTeX Main
+  lefthook: `M400000 281 H103s-33-11.2-61-33.5S0 197.3 0 164s14.2-61.2 42.5
+-83.5C70.8 58.2 104 47 142 47 c16.7 0 25 6.7 25 20 0 12-8.7 18.7-26 20-40 3.3
+-68.7 15.7-86 37-10 12-15 25.3-15 40 0 22.7 9.8 40.7 29.5 54 19.7 13.3 43.5 21
+ 71.5 23h399859zM103 281v-40h399897v40z`,
+  leftlinesegment: `M40 281 V428 H0 V94 H40 V241 H400000 v40z
+M40 281 V428 H0 V94 H40 V241 H400000 v40z`,
+  leftmapsto: `M40 281 V448H0V74H40V241H400000v40z
+M40 281 V448H0V74H40V241H400000v40z`,
+  // tofrom is from glyph U+21C4 in font KaTeX AMS Regular
+  leftToFrom: `M0 147h400000v40H0zm0 214c68 40 115.7 95.7 143 167h22c15.3 0 23
+-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69-70-101l-7-8h399905v-40H95l7-8
+c28.7-32 52-65.7 70-101 10.7-23.3 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 265.3
+ 68 321 0 361zm0-174v-40h399900v40zm100 154v40h399900v-40z`,
+  longequal: `M0 50 h400000 v40H0z m0 194h40000v40H0z
+M0 50 h400000 v40H0z m0 194h40000v40H0z`,
+  midbrace: `M200428 334
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+ 311.7-78.3 403-201 6-8 9.7-12 11-12 .7-.7 6.7-1 18-1s17.3.3 18 1c1.3 0 5 4 11
+ 12 44.7 59.3 101.3 106.3 170 141s145.3 54.3 229 60h199572v120z`,
+  midbraceunder: `M199572 214
+c100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14
+ 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3
+ 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0
+-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z`,
+  oiintSize1: `M512.6 71.6c272.6 0 320.3 106.8 320.3 178.2 0 70.8-47.7 177.6
+-320.3 177.6S193.1 320.6 193.1 249.8c0-71.4 46.9-178.2 319.5-178.2z
+m368.1 178.2c0-86.4-60.9-215.4-368.1-215.4-306.4 0-367.3 129-367.3 215.4 0 85.8
+60.9 214.8 367.3 214.8 307.2 0 368.1-129 368.1-214.8z`,
+  oiintSize2: `M757.8 100.1c384.7 0 451.1 137.6 451.1 230 0 91.3-66.4 228.8
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+m502.4 230c0-111.2-82.4-277.2-502.4-277.2s-504 166-504 277.2
+c0 110 84 276 504 276s502.4-166 502.4-276z`,
+  oiiintSize1: `M681.4 71.6c408.9 0 480.5 106.8 480.5 178.2 0 70.8-71.6 177.6
+-480.5 177.6S202.1 320.6 202.1 249.8c0-71.4 70.5-178.2 479.3-178.2z
+m525.8 178.2c0-86.4-86.8-215.4-525.7-215.4-437.9 0-524.7 129-524.7 215.4 0
+85.8 86.8 214.8 524.7 214.8 438.9 0 525.7-129 525.7-214.8z`,
+  oiiintSize2: `M1021.2 53c603.6 0 707.8 165.8 707.8 277.2 0 110-104.2 275.8
+-707.8 275.8-606 0-710.2-165.8-710.2-275.8C311 218.8 415.2 53 1021.2 53z
+m770.4 277.1c0-131.2-126.4-327.6-770.5-327.6S248.4 198.9 248.4 330.1
+c0 130 128.8 326.4 772.7 326.4s770.5-196.4 770.5-326.4z`,
+  rightarrow: `M0 241v40h399891c-47.3 35.3-84 78-110 128
+-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20
+ 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7
+ 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85
+-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+ 151.7 139 205zm0 0v40h399900v-40z`,
+  rightbrace: `M400000 542l
+-6 6h-17c-12.7 0-19.3-.3-20-1-4-4-7.3-8.3-10-13-35.3-51.3-80.8-93.8-136.5-127.5
+s-117.2-55.8-184.5-66.5c-.7 0-2-.3-4-1-18.7-2.7-76-4.3-172-5H0V214h399571l6 1
+c124.7 8 235 61.7 331 161 31.3 33.3 59.7 72.7 85 118l7 13v35z`,
+  rightbraceunder: `M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3
+ 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237
+-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z`,
+  rightgroup: `M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0
+ 3-1 3-3v-38c-76-158-257-219-435-219H0z`,
+  rightgroupunder: `M0 262h399565c371 0 266.7-149.4 414-180 5.9-1.2 18 0 18
+ 0 2 0 3 1 3 3v38c-76 158-257 219-435 219H0z`,
+  rightharpoon: `M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3
+-3.7-15.3-11-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2
+-10.7 0-16.7 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58
+ 69.2 92 94.5zm0 0v40h399900v-40z`,
+  rightharpoonplus: `M0 241v40h399993c4.7-4.7 7-9.3 7-14 0-9.3-3.7-15.3-11
+-18-92.7-56.7-159-133.7-199-231-3.3-9.3-6-14.7-8-16-2-1.3-7-2-15-2-10.7 0-16.7
+ 2-18 6-2 2.7-1 9.7 3 21 15.3 42 36.7 81.8 64 119.5 27.3 37.7 58 69.2 92 94.5z
+m0 0v40h399900v-40z m100 194v40h399900v-40zm0 0v40h399900v-40z`,
+  rightharpoondown: `M399747 511c0 7.3 6.7 11 20 11 8 0 13-.8 15-2.5s4.7-6.8
+ 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3 8.5-5.8 9.5
+-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3-64.7 57-92 95
+-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 241v40h399900v-40z`,
+  rightharpoondownplus: `M399747 705c0 7.3 6.7 11 20 11 8 0 13-.8
+ 15-2.5s4.7-6.8 8-15.5c40-94 99.3-166.3 178-217 13.3-8 20.3-12.3 21-13 5.3-3.3
+ 8.5-5.8 9.5-7.5 1-1.7 1.5-5.2 1.5-10.5s-2.3-10.3-7-15H0v40h399908c-34 25.3
+-64.7 57-92 95-27.3 38-48.7 77.7-64 119-3.3 8.7-5 14-5 16zM0 435v40h399900v-40z
+m0-194v40h400000v-40zm0 0v40h400000v-40z`,
+  righthook: `M399859 241c-764 0 0 0 0 0 40-3.3 68.7-15.7 86-37 10-12 15-25.3
+ 15-40 0-22.7-9.8-40.7-29.5-54-19.7-13.3-43.5-21-71.5-23-17.3-1.3-26-8-26-20 0
+-13.3 8.7-20 26-20 38 0 71 11.2 99 33.5 0 0 7 5.6 21 16.7 14 11.2 21 33.5 21
+ 66.8s-14 61.2-42 83.5c-28 22.3-61 33.5-99 33.5L0 241z M0 281v-40h399859v40z`,
+  rightlinesegment: `M399960 241 V94 h40 V428 h-40 V281 H0 v-40z
+M399960 241 V94 h40 V428 h-40 V281 H0 v-40z`,
+  rightToFrom: `M400000 167c-70.7-42-118-97.7-142-167h-23c-15.3 0-23 .3-23
+ 1 0 1.3 5.3 13.7 16 37 18 35.3 41.3 69 70 101l7 8H0v40h399905l-7 8c-28.7 32
+-52 65.7-70 101-10.7 23.3-16 35.7-16 37 0 .7 7.7 1 23 1h23c24-69.3 71.3-125 142
+-167z M100 147v40h399900v-40zM0 341v40h399900v-40z`,
+  // twoheadleftarrow is from glyph U+219E in font KaTeX AMS Regular
+  twoheadleftarrow: `M0 167c68 40
+ 115.7 95.7 143 167h22c15.3 0 23-.3 23-1 0-1.3-5.3-13.7-16-37-18-35.3-41.3-69
+-70-101l-7-8h125l9 7c50.7 39.3 85 86 103 140h46c0-4.7-6.3-18.7-19-42-18-35.3
+-40-67.3-66-96l-9-9h399716v-40H284l9-9c26-28.7 48-60.7 66-96 12.7-23.333 19
+-37.333 19-42h-46c-18 54-52.3 100.7-103 140l-9 7H95l7-8c28.7-32 52-65.7 70-101
+ 10.7-23.333 16-35.7 16-37 0-.7-7.7-1-23-1h-22C115.7 71.3 68 127 0 167z`,
+  twoheadrightarrow: `M400000 167
+c-68-40-115.7-95.7-143-167h-22c-15.3 0-23 .3-23 1 0 1.3 5.3 13.7 16 37 18 35.3
+ 41.3 69 70 101l7 8h-125l-9-7c-50.7-39.3-85-86-103-140h-46c0 4.7 6.3 18.7 19 42
+ 18 35.3 40 67.3 66 96l9 9H0v40h399716l-9 9c-26 28.7-48 60.7-66 96-12.7 23.333
+-19 37.333-19 42h46c18-54 52.3-100.7 103-140l9-7h125l-7 8c-28.7 32-52 65.7-70
+ 101-10.7 23.333-16 35.7-16 37 0 .7 7.7 1 23 1h22c27.3-71.3 75-127 143-167z`,
+  // tilde1 is a modified version of a glyph from the MnSymbol package
+  tilde1: `M200 55.538c-77 0-168 73.953-177 73.953-3 0-7
+-2.175-9-5.437L2 97c-1-2-2-4-2-6 0-4 2-7 5-9l20-12C116 12 171 0 207 0c86 0
+ 114 68 191 68 78 0 168-68 177-68 4 0 7 2 9 5l12 19c1 2.175 2 4.35 2 6.525 0
+ 4.35-2 7.613-5 9.788l-19 13.05c-92 63.077-116.937 75.308-183 76.128
+-68.267.847-113-73.952-191-73.952z`,
+  // ditto tilde2, tilde3, & tilde4
+  tilde2: `M344 55.266c-142 0-300.638 81.316-311.5 86.418
+-8.01 3.762-22.5 10.91-23.5 5.562L1 120c-1-2-1-3-1-4 0-5 3-9 8-10l18.4-9C160.9
+ 31.9 283 0 358 0c148 0 188 122 331 122s314-97 326-97c4 0 8 2 10 7l7 21.114
+c1 2.14 1 3.21 1 4.28 0 5.347-3 9.626-7 10.696l-22.3 12.622C852.6 158.372 751
+ 181.476 676 181.476c-149 0-189-126.21-332-126.21z`,
+  tilde3: `M786 59C457 59 32 175.242 13 175.242c-6 0-10-3.457
+-11-10.37L.15 138c-1-7 3-12 10-13l19.2-6.4C378.4 40.7 634.3 0 804.3 0c337 0
+ 411.8 157 746.8 157 328 0 754-112 773-112 5 0 10 3 11 9l1 14.075c1 8.066-.697
+ 16.595-6.697 17.492l-21.052 7.31c-367.9 98.146-609.15 122.696-778.15 122.696
+ -338 0-409-156.573-744-156.573z`,
+  tilde4: `M786 58C457 58 32 177.487 13 177.487c-6 0-10-3.345
+-11-10.035L.15 143c-1-7 3-12 10-13l22-6.7C381.2 35 637.15 0 807.15 0c337 0 409
+ 177 744 177 328 0 754-127 773-127 5 0 10 3 11 9l1 14.794c1 7.805-3 13.38-9
+ 14.495l-20.7 5.574c-366.85 99.79-607.3 139.372-776.3 139.372-338 0-409
+ -175.236-744-175.236z`,
+  // vec is from glyph U+20D7 in font KaTeX Main
+  vec: `M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
+3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
+10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
+-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
+-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
+H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
+c-16-25.333-24-45-24-59z`,
+  // widehat1 is a modified version of a glyph from the MnSymbol package
+  widehat1: `M529 0h5l519 115c5 1 9 5 9 10 0 1-1 2-1 3l-4 22
+c-1 5-5 9-11 9h-2L532 67 19 159h-2c-5 0-9-4-11-9l-5-22c-1-6 2-12 8-13z`,
+  // ditto widehat2, widehat3, & widehat4
+  widehat2: `M1181 0h2l1171 176c6 0 10 5 10 11l-2 23c-1 6-5 10
+-11 10h-1L1182 67 15 220h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z`,
+  widehat3: `M1181 0h2l1171 236c6 0 10 5 10 11l-2 23c-1 6-5 10
+-11 10h-1L1182 67 15 280h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z`,
+  widehat4: `M1181 0h2l1171 296c6 0 10 5 10 11l-2 23c-1 6-5 10
+-11 10h-1L1182 67 15 340h-1c-6 0-10-4-11-10l-2-23c-1-6 4-11 10-11z`,
+  // widecheck paths are all inverted versions of widehat
+  widecheck1: `M529,159h5l519,-115c5,-1,9,-5,9,-10c0,-1,-1,-2,-1,-3l-4,-22c-1,
+-5,-5,-9,-11,-9h-2l-512,92l-513,-92h-2c-5,0,-9,4,-11,9l-5,22c-1,6,2,12,8,13z`,
+  widecheck2: `M1181,220h2l1171,-176c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
+-11,-10h-1l-1168,153l-1167,-153h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z`,
+  widecheck3: `M1181,280h2l1171,-236c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
+-11,-10h-1l-1168,213l-1167,-213h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z`,
+  widecheck4: `M1181,340h2l1171,-296c6,0,10,-5,10,-11l-2,-23c-1,-6,-5,-10,
+-11,-10h-1l-1168,273l-1167,-273h-1c-6,0,-10,4,-11,10l-2,23c-1,6,4,11,10,11z`,
+  // The next ten paths support reaction arrows from the mhchem package.
+  // Arrows for \ce{<-->} are offset from xAxis by 0.22ex, per mhchem in LaTeX
+  // baraboveleftarrow is mostly from from glyph U+2190 in font KaTeX Main
+  baraboveleftarrow: `M400000 620h-399890l3 -3c68.7 -52.7 113.7 -120 135 -202
+c4 -14.7 6 -23 6 -25c0 -7.3 -7 -11 -21 -11c-8 0 -13.2 0.8 -15.5 2.5
+c-2.3 1.7 -4.2 5.8 -5.5 12.5c-1.3 4.7 -2.7 10.3 -4 17c-12 48.7 -34.8 92 -68.5 130
+s-74.2 66.3 -121.5 85c-10 4 -16 7.7 -18 11c0 8.7 6 14.3 18 17c47.3 18.7 87.8 47
+121.5 85s56.5 81.3 68.5 130c0.7 2 1.3 5 2 9s1.2 6.7 1.5 8c0.3 1.3 1 3.3 2 6
+s2.2 4.5 3.5 5.5c1.3 1 3.3 1.8 6 2.5s6 1 10 1c14 0 21 -3.7 21 -11
+c0 -2 -2 -10.3 -6 -25c-20 -79.3 -65 -146.7 -135 -202l-3 -3h399890z
+M100 620v40h399900v-40z M0 241v40h399900v-40zM0 241v40h399900v-40z`,
+  // rightarrowabovebar is mostly from glyph U+2192, KaTeX Main
+  rightarrowabovebar: `M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32
+-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0
+13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39
+-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5
+-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5
+-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67
+151.7 139 205zm96 379h399894v40H0zm0 0h399904v40H0z`,
+  // The short left harpoon has 0.5em (i.e. 500 units) kern on the left end.
+  // Ref from mhchem.sty: \rlap{\raisebox{-.22ex}{$\kern0.5em
+  baraboveshortleftharpoon: `M507,435c-4,4,-6.3,8.7,-7,14c0,5.3,0.7,9,2,11
+c1.3,2,5.3,5.3,12,10c90.7,54,156,130,196,228c3.3,10.7,6.3,16.3,9,17
+c2,0.7,5,1,9,1c0,0,5,0,5,0c10.7,0,16.7,-2,18,-6c2,-2.7,1,-9.7,-3,-21
+c-32,-87.3,-82.7,-157.7,-152,-211c0,0,-3,-3,-3,-3l399351,0l0,-40
+c-398570,0,-399437,0,-399437,0z M593 435 v40 H399500 v-40z
+M0 281 v-40 H399908 v40z M0 281 v-40 H399908 v40z`,
+  rightharpoonaboveshortbar: `M0,241 l0,40c399126,0,399993,0,399993,0
+c4.7,-4.7,7,-9.3,7,-14c0,-9.3,-3.7,-15.3,-11,-18c-92.7,-56.7,-159,-133.7,-199,
+-231c-3.3,-9.3,-6,-14.7,-8,-16c-2,-1.3,-7,-2,-15,-2c-10.7,0,-16.7,2,-18,6
+c-2,2.7,-1,9.7,3,21c15.3,42,36.7,81.8,64,119.5c27.3,37.7,58,69.2,92,94.5z
+M0 241 v40 H399908 v-40z M0 475 v-40 H399500 v40z M0 475 v-40 H399500 v40z`,
+  shortbaraboveleftharpoon: `M7,435c-4,4,-6.3,8.7,-7,14c0,5.3,0.7,9,2,11
+c1.3,2,5.3,5.3,12,10c90.7,54,156,130,196,228c3.3,10.7,6.3,16.3,9,17c2,0.7,5,1,9,
+1c0,0,5,0,5,0c10.7,0,16.7,-2,18,-6c2,-2.7,1,-9.7,-3,-21c-32,-87.3,-82.7,-157.7,
+-152,-211c0,0,-3,-3,-3,-3l399907,0l0,-40c-399126,0,-399993,0,-399993,0z
+M93 435 v40 H400000 v-40z M500 241 v40 H400000 v-40z M500 241 v40 H400000 v-40z`,
+  shortrightharpoonabovebar: `M53,241l0,40c398570,0,399437,0,399437,0
+c4.7,-4.7,7,-9.3,7,-14c0,-9.3,-3.7,-15.3,-11,-18c-92.7,-56.7,-159,-133.7,-199,
+-231c-3.3,-9.3,-6,-14.7,-8,-16c-2,-1.3,-7,-2,-15,-2c-10.7,0,-16.7,2,-18,6
+c-2,2.7,-1,9.7,3,21c15.3,42,36.7,81.8,64,119.5c27.3,37.7,58,69.2,92,94.5z
+M500 241 v40 H399408 v-40z M500 435 v40 H400000 v-40z`
+};
+
+/**
+ * This node represents a document fragment, which contains elements, but when
+ * placed into the DOM doesn't have any representation itself. It only contains
+ * children and doesn't have any DOM node properties.
+ */
+class DocumentFragment {
+  // HtmlDomNode
+  // Never used; needed for satisfying interface.
+  constructor(children) {
+    this.children = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    this.children = children;
+    this.classes = [];
+    this.height = 0;
+    this.depth = 0;
+    this.maxFontSize = 0;
+    this.style = {};
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+  /** Convert the fragment into a node. */
+
+
+  toNode() {
+    const frag = document.createDocumentFragment();
+
+    for (let i = 0; i < this.children.length; i++) {
+      frag.appendChild(this.children[i].toNode());
+    }
+
+    return frag;
+  }
+  /** Convert the fragment into HTML markup. */
+
+
+  toMarkup() {
+    let markup = ""; // Simply concatenate the markup for the children together.
+
+    for (let i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    return markup;
+  }
+  /**
+   * Converts the math node into a string, similar to innerText. Applies to
+   * MathDomNode's only.
+   */
+
+
+  toText() {
+    // To avoid this, we would subclass documentFragment separately for
+    // MathML, but polyfills for subclassing is expensive per PR 1469.
+    // $FlowFixMe: Only works for ChildType = MathDomNode.
+    const toText = child => child.toText();
+
+    return this.children.map(toText).join("");
+  }
+
+}
+
+/**
+ * These objects store the data about the DOM nodes we create, as well as some
+ * extra data. They can then be transformed into real DOM nodes with the
+ * `toNode` function or HTML markup using `toMarkup`. They are useful for both
+ * storing extra properties on the nodes, as well as providing a way to easily
+ * work with the DOM.
+ *
+ * Similar functions for working with MathML nodes exist in mathMLTree.js.
+ *
+ * TODO: refactor `span` and `anchor` into common superclass when
+ * target environments support class inheritance
+ */
+
+/**
+ * Create an HTML className based on a list of classes. In addition to joining
+ * with spaces, we also remove empty classes.
+ */
+const createClass = function createClass(classes) {
+  return classes.filter(cls => cls).join(" ");
+};
+
+const initNode = function initNode(classes, options, style) {
+  this.classes = classes || [];
+  this.attributes = {};
+  this.height = 0;
+  this.depth = 0;
+  this.maxFontSize = 0;
+  this.style = style || {};
+
+  if (options) {
+    if (options.style.isTight()) {
+      this.classes.push("mtight");
+    }
+
+    const color = options.getColor();
+
+    if (color) {
+      this.style.color = color;
+    }
+  }
+};
+/**
+ * Convert into an HTML node
+ */
+
+
+const toNode = function toNode(tagName) {
+  const node = document.createElement(tagName); // Apply the class
+
+  node.className = createClass(this.classes); // Apply inline styles
+
+  for (const style in this.style) {
+    if (this.style.hasOwnProperty(style)) {
+      // $FlowFixMe Flow doesn't seem to understand span.style's type.
+      node.style[style] = this.style[style];
+    }
+  } // Apply attributes
+
+
+  for (const attr in this.attributes) {
+    if (this.attributes.hasOwnProperty(attr)) {
+      node.setAttribute(attr, this.attributes[attr]);
+    }
+  } // Append the children, also as HTML nodes
+
+
+  for (let i = 0; i < this.children.length; i++) {
+    node.appendChild(this.children[i].toNode());
+  }
+
+  return node;
+};
+/**
+ * Convert into an HTML markup string
+ */
+
+
+const toMarkup = function toMarkup(tagName) {
+  let markup = `<${tagName}`; // Add the class
+
+  if (this.classes.length) {
+    markup += ` class="${utils.escape(createClass(this.classes))}"`;
+  }
+
+  let styles = ""; // Add the styles, after hyphenation
+
+  for (const style in this.style) {
+    if (this.style.hasOwnProperty(style)) {
+      styles += `${utils.hyphenate(style)}:${this.style[style]};`;
+    }
+  }
+
+  if (styles) {
+    markup += ` style="${utils.escape(styles)}"`;
+  } // Add the attributes
+
+
+  for (const attr in this.attributes) {
+    if (this.attributes.hasOwnProperty(attr)) {
+      markup += ` ${attr}="${utils.escape(this.attributes[attr])}"`;
+    }
+  }
+
+  markup += ">"; // Add the markup of the children, also as markup
+
+  for (let i = 0; i < this.children.length; i++) {
+    markup += this.children[i].toMarkup();
+  }
+
+  markup += `</${tagName}>`;
+  return markup;
+}; // Making the type below exact with all optional fields doesn't work due to
+// - https://github.com/facebook/flow/issues/4582
+// - https://github.com/facebook/flow/issues/5688
+// However, since *all* fields are optional, $Shape<> works as suggested in 5688
+// above.
+// This type does not include all CSS properties. Additional properties should
+// be added as needed.
+
+
+/**
+ * This node represents a span node, with a className, a list of children, and
+ * an inline style. It also contains information about its height, depth, and
+ * maxFontSize.
+ *
+ * Represents two types with different uses: SvgSpan to wrap an SVG and DomSpan
+ * otherwise. This typesafety is important when HTML builders access a span's
+ * children.
+ */
+class Span {
+  constructor(classes, children, options, style) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.width = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    initNode.call(this, classes, options, style);
+    this.children = children || [];
+  }
+  /**
+   * Sets an arbitrary attribute on the span. Warning: use this wisely. Not
+   * all browsers support attributes the same, and having too many custom
+   * attributes is probably bad.
+   */
+
+
+  setAttribute(attribute, value) {
+    this.attributes[attribute] = value;
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+
+  toNode() {
+    return toNode.call(this, "span");
+  }
+
+  toMarkup() {
+    return toMarkup.call(this, "span");
+  }
+
+}
+/**
+ * This node represents an anchor (<a>) element with a hyperlink.  See `span`
+ * for further details.
+ */
+
+class Anchor {
+  constructor(href, classes, children, options) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    initNode.call(this, classes, options);
+    this.children = children || [];
+    this.setAttribute('href', href);
+  }
+
+  setAttribute(attribute, value) {
+    this.attributes[attribute] = value;
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+
+  toNode() {
+    return toNode.call(this, "a");
+  }
+
+  toMarkup() {
+    return toMarkup.call(this, "a");
+  }
+
+}
+/**
+ * This node represents an image embed (<img>) element.
+ */
+
+class Img {
+  constructor(src, alt, style) {
+    this.src = void 0;
+    this.alt = void 0;
+    this.classes = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.maxFontSize = void 0;
+    this.style = void 0;
+    this.alt = alt;
+    this.src = src;
+    this.classes = ["mord"];
+    this.style = style;
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+
+  toNode() {
+    const node = document.createElement("img");
+    node.src = this.src;
+    node.alt = this.alt;
+    node.className = "mord"; // Apply inline styles
+
+    for (const style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        // $FlowFixMe
+        node.style[style] = this.style[style];
+      }
+    }
+
+    return node;
+  }
+
+  toMarkup() {
+    let markup = `<img  src='${this.src} 'alt='${this.alt}' `; // Add the styles, after hyphenation
+
+    let styles = "";
+
+    for (const style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        styles += `${utils.hyphenate(style)}:${this.style[style]};`;
+      }
+    }
+
+    if (styles) {
+      markup += ` style="${utils.escape(styles)}"`;
+    }
+
+    markup += "'/>";
+    return markup;
+  }
+
+}
+const iCombinations = {
+  'î': '\u0131\u0302',
+  'ï': '\u0131\u0308',
+  'í': '\u0131\u0301',
+  // 'ī': '\u0131\u0304', // enable when we add Extended Latin
+  'ì': '\u0131\u0300'
+};
+/**
+ * A symbol node contains information about a single symbol. It either renders
+ * to a single text node, or a span with a single text node in it, depending on
+ * whether it has CSS classes, styles, or needs italic correction.
+ */
+
+class SymbolNode {
+  constructor(text, height, depth, italic, skew, width, classes, style) {
+    this.text = void 0;
+    this.height = void 0;
+    this.depth = void 0;
+    this.italic = void 0;
+    this.skew = void 0;
+    this.width = void 0;
+    this.maxFontSize = void 0;
+    this.classes = void 0;
+    this.style = void 0;
+    this.text = text;
+    this.height = height || 0;
+    this.depth = depth || 0;
+    this.italic = italic || 0;
+    this.skew = skew || 0;
+    this.width = width || 0;
+    this.classes = classes || [];
+    this.style = style || {};
+    this.maxFontSize = 0; // Mark text from non-Latin scripts with specific classes so that we
+    // can specify which fonts to use.  This allows us to render these
+    // characters with a serif font in situations where the browser would
+    // either default to a sans serif or render a placeholder character.
+    // We use CSS class names like cjk_fallback, hangul_fallback and
+    // brahmic_fallback. See ./unicodeScripts.js for the set of possible
+    // script names
+
+    const script = scriptFromCodepoint(this.text.charCodeAt(0));
+
+    if (script) {
+      this.classes.push(script + "_fallback");
+    }
+
+    if (/[îïíì]/.test(this.text)) {
+      // add ī when we add Extended Latin
+      this.text = iCombinations[this.text];
+    }
+  }
+
+  hasClass(className) {
+    return utils.contains(this.classes, className);
+  }
+  /**
+   * Creates a text node or span from a symbol node. Note that a span is only
+   * created if it is needed.
+   */
+
+
+  toNode() {
+    const node = document.createTextNode(this.text);
+    let span = null;
+
+    if (this.italic > 0) {
+      span = document.createElement("span");
+      span.style.marginRight = this.italic + "em";
+    }
+
+    if (this.classes.length > 0) {
+      span = span || document.createElement("span");
+      span.className = createClass(this.classes);
+    }
+
+    for (const style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        span = span || document.createElement("span"); // $FlowFixMe Flow doesn't seem to understand span.style's type.
+
+        span.style[style] = this.style[style];
+      }
+    }
+
+    if (span) {
+      span.appendChild(node);
+      return span;
+    } else {
+      return node;
+    }
+  }
+  /**
+   * Creates markup for a symbol node.
+   */
+
+
+  toMarkup() {
+    // TODO(alpert): More duplication than I'd like from
+    // span.prototype.toMarkup and symbolNode.prototype.toNode...
+    let needsSpan = false;
+    let markup = "<span";
+
+    if (this.classes.length) {
+      needsSpan = true;
+      markup += " class=\"";
+      markup += utils.escape(createClass(this.classes));
+      markup += "\"";
+    }
+
+    let styles = "";
+
+    if (this.italic > 0) {
+      styles += "margin-right:" + this.italic + "em;";
+    }
+
+    for (const style in this.style) {
+      if (this.style.hasOwnProperty(style)) {
+        styles += utils.hyphenate(style) + ":" + this.style[style] + ";";
+      }
+    }
+
+    if (styles) {
+      needsSpan = true;
+      markup += " style=\"" + utils.escape(styles) + "\"";
+    }
+
+    const escaped = utils.escape(this.text);
+
+    if (needsSpan) {
+      markup += ">";
+      markup += escaped;
+      markup += "</span>";
+      return markup;
+    } else {
+      return escaped;
+    }
+  }
+
+}
+/**
+ * SVG nodes are used to render stretchy wide elements.
+ */
+
+class SvgNode {
+  constructor(children, attributes) {
+    this.children = void 0;
+    this.attributes = void 0;
+    this.children = children || [];
+    this.attributes = attributes || {};
+  }
+
+  toNode() {
+    const svgNS = "http://www.w3.org/2000/svg";
+    const node = document.createElementNS(svgNS, "svg"); // Apply attributes
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    for (let i = 0; i < this.children.length; i++) {
+      node.appendChild(this.children[i].toNode());
+    }
+
+    return node;
+  }
+
+  toMarkup() {
+    let markup = "<svg"; // Apply attributes
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += ` ${attr}='${this.attributes[attr]}'`;
+      }
+    }
+
+    markup += ">";
+
+    for (let i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    markup += "</svg>";
+    return markup;
+  }
+
+}
+class PathNode {
+  constructor(pathName, alternate) {
+    this.pathName = void 0;
+    this.alternate = void 0;
+    this.pathName = pathName;
+    this.alternate = alternate; // Used only for \sqrt
+  }
+
+  toNode() {
+    const svgNS = "http://www.w3.org/2000/svg";
+    const node = document.createElementNS(svgNS, "path");
+
+    if (this.alternate) {
+      node.setAttribute("d", this.alternate);
+    } else {
+      node.setAttribute("d", path[this.pathName]);
+    }
+
+    return node;
+  }
+
+  toMarkup() {
+    if (this.alternate) {
+      return `<path d='${this.alternate}'/>`;
+    } else {
+      return `<path d='${path[this.pathName]}'/>`;
+    }
+  }
+
+}
+class LineNode {
+  constructor(attributes) {
+    this.attributes = void 0;
+    this.attributes = attributes || {};
+  }
+
+  toNode() {
+    const svgNS = "http://www.w3.org/2000/svg";
+    const node = document.createElementNS(svgNS, "line"); // Apply attributes
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    return node;
+  }
+
+  toMarkup() {
+    let markup = "<line";
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += ` ${attr}='${this.attributes[attr]}'`;
+      }
+    }
+
+    markup += "/>";
+    return markup;
+  }
+
+}
+function assertSymbolDomNode(group) {
+  if (group instanceof SymbolNode) {
+    return group;
+  } else {
+    throw new Error(`Expected symbolNode but got ${String(group)}.`);
+  }
+}
+function assertSpan(group) {
+  if (group instanceof Span) {
+    return group;
+  } else {
+    throw new Error(`Expected span<HtmlDomNode> but got ${String(group)}.`);
+  }
+}
+
+// This file is GENERATED by buildMetrics.sh. DO NOT MODIFY.
+var metricMap = {
+  "AMS-Regular": {
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+    "8739": [0.08167, 0.58167, 0, 0, 0.22222],
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+    "73": [0, 0.61111, 0, 0, 0.525],
+    "74": [0, 0.61111, 0, 0, 0.525],
+    "75": [0, 0.61111, 0, 0, 0.525],
+    "76": [0, 0.61111, 0, 0, 0.525],
+    "77": [0, 0.61111, 0, 0, 0.525],
+    "78": [0, 0.61111, 0, 0, 0.525],
+    "79": [0, 0.61111, 0, 0, 0.525],
+    "80": [0, 0.61111, 0, 0, 0.525],
+    "81": [0.13889, 0.61111, 0, 0, 0.525],
+    "82": [0, 0.61111, 0, 0, 0.525],
+    "83": [0, 0.61111, 0, 0, 0.525],
+    "84": [0, 0.61111, 0, 0, 0.525],
+    "85": [0, 0.61111, 0, 0, 0.525],
+    "86": [0, 0.61111, 0, 0, 0.525],
+    "87": [0, 0.61111, 0, 0, 0.525],
+    "88": [0, 0.61111, 0, 0, 0.525],
+    "89": [0, 0.61111, 0, 0, 0.525],
+    "90": [0, 0.61111, 0, 0, 0.525],
+    "91": [0.08333, 0.69444, 0, 0, 0.525],
+    "92": [0.08333, 0.69444, 0, 0, 0.525],
+    "93": [0.08333, 0.69444, 0, 0, 0.525],
+    "94": [0, 0.61111, 0, 0, 0.525],
+    "95": [0.09514, 0, 0, 0, 0.525],
+    "96": [0, 0.61111, 0, 0, 0.525],
+    "97": [0, 0.43056, 0, 0, 0.525],
+    "98": [0, 0.61111, 0, 0, 0.525],
+    "99": [0, 0.43056, 0, 0, 0.525],
+    "100": [0, 0.61111, 0, 0, 0.525],
+    "101": [0, 0.43056, 0, 0, 0.525],
+    "102": [0, 0.61111, 0, 0, 0.525],
+    "103": [0.22222, 0.43056, 0, 0, 0.525],
+    "104": [0, 0.61111, 0, 0, 0.525],
+    "105": [0, 0.61111, 0, 0, 0.525],
+    "106": [0.22222, 0.61111, 0, 0, 0.525],
+    "107": [0, 0.61111, 0, 0, 0.525],
+    "108": [0, 0.61111, 0, 0, 0.525],
+    "109": [0, 0.43056, 0, 0, 0.525],
+    "110": [0, 0.43056, 0, 0, 0.525],
+    "111": [0, 0.43056, 0, 0, 0.525],
+    "112": [0.22222, 0.43056, 0, 0, 0.525],
+    "113": [0.22222, 0.43056, 0, 0, 0.525],
+    "114": [0, 0.43056, 0, 0, 0.525],
+    "115": [0, 0.43056, 0, 0, 0.525],
+    "116": [0, 0.55358, 0, 0, 0.525],
+    "117": [0, 0.43056, 0, 0, 0.525],
+    "118": [0, 0.43056, 0, 0, 0.525],
+    "119": [0, 0.43056, 0, 0, 0.525],
+    "120": [0, 0.43056, 0, 0, 0.525],
+    "121": [0.22222, 0.43056, 0, 0, 0.525],
+    "122": [0, 0.43056, 0, 0, 0.525],
+    "123": [0.08333, 0.69444, 0, 0, 0.525],
+    "124": [0.08333, 0.69444, 0, 0, 0.525],
+    "125": [0.08333, 0.69444, 0, 0, 0.525],
+    "126": [0, 0.61111, 0, 0, 0.525],
+    "127": [0, 0.61111, 0, 0, 0.525],
+    "160": [0, 0, 0, 0, 0.525],
+    "176": [0, 0.61111, 0, 0, 0.525],
+    "184": [0.19445, 0, 0, 0, 0.525],
+    "305": [0, 0.43056, 0, 0, 0.525],
+    "567": [0.22222, 0.43056, 0, 0, 0.525],
+    "711": [0, 0.56597, 0, 0, 0.525],
+    "713": [0, 0.56555, 0, 0, 0.525],
+    "714": [0, 0.61111, 0, 0, 0.525],
+    "715": [0, 0.61111, 0, 0, 0.525],
+    "728": [0, 0.61111, 0, 0, 0.525],
+    "730": [0, 0.61111, 0, 0, 0.525],
+    "770": [0, 0.61111, 0, 0, 0.525],
+    "771": [0, 0.61111, 0, 0, 0.525],
+    "776": [0, 0.61111, 0, 0, 0.525],
+    "915": [0, 0.61111, 0, 0, 0.525],
+    "916": [0, 0.61111, 0, 0, 0.525],
+    "920": [0, 0.61111, 0, 0, 0.525],
+    "923": [0, 0.61111, 0, 0, 0.525],
+    "926": [0, 0.61111, 0, 0, 0.525],
+    "928": [0, 0.61111, 0, 0, 0.525],
+    "931": [0, 0.61111, 0, 0, 0.525],
+    "933": [0, 0.61111, 0, 0, 0.525],
+    "934": [0, 0.61111, 0, 0, 0.525],
+    "936": [0, 0.61111, 0, 0, 0.525],
+    "937": [0, 0.61111, 0, 0, 0.525],
+    "8216": [0, 0.61111, 0, 0, 0.525],
+    "8217": [0, 0.61111, 0, 0, 0.525],
+    "8242": [0, 0.61111, 0, 0, 0.525],
+    "9251": [0.11111, 0.21944, 0, 0, 0.525]
+  }
+};
+
+/**
+ * This file contains metrics regarding fonts and individual symbols. The sigma
+ * and xi variables, as well as the metricMap map contain data extracted from
+ * TeX, TeX font metrics, and the TTF files. These data are then exposed via the
+ * `metrics` variable and the getCharacterMetrics function.
+ */
+// In TeX, there are actually three sets of dimensions, one for each of
+// textstyle (size index 5 and higher: >=9pt), scriptstyle (size index 3 and 4:
+// 7-8pt), and scriptscriptstyle (size index 1 and 2: 5-6pt).  These are
+// provided in the the arrays below, in that order.
+//
+// The font metrics are stored in fonts cmsy10, cmsy7, and cmsy5 respsectively.
+// This was determined by running the following script:
+//
+//     latex -interaction=nonstopmode \
+//     '\documentclass{article}\usepackage{amsmath}\begin{document}' \
+//     '$a$ \expandafter\show\the\textfont2' \
+//     '\expandafter\show\the\scriptfont2' \
+//     '\expandafter\show\the\scriptscriptfont2' \
+//     '\stop'
+//
+// The metrics themselves were retreived using the following commands:
+//
+//     tftopl cmsy10
+//     tftopl cmsy7
+//     tftopl cmsy5
+//
+// The output of each of these commands is quite lengthy.  The only part we
+// care about is the FONTDIMEN section. Each value is measured in EMs.
+const sigmasAndXis = {
+  slant: [0.250, 0.250, 0.250],
+  // sigma1
+  space: [0.000, 0.000, 0.000],
+  // sigma2
+  stretch: [0.000, 0.000, 0.000],
+  // sigma3
+  shrink: [0.000, 0.000, 0.000],
+  // sigma4
+  xHeight: [0.431, 0.431, 0.431],
+  // sigma5
+  quad: [1.000, 1.171, 1.472],
+  // sigma6
+  extraSpace: [0.000, 0.000, 0.000],
+  // sigma7
+  num1: [0.677, 0.732, 0.925],
+  // sigma8
+  num2: [0.394, 0.384, 0.387],
+  // sigma9
+  num3: [0.444, 0.471, 0.504],
+  // sigma10
+  denom1: [0.686, 0.752, 1.025],
+  // sigma11
+  denom2: [0.345, 0.344, 0.532],
+  // sigma12
+  sup1: [0.413, 0.503, 0.504],
+  // sigma13
+  sup2: [0.363, 0.431, 0.404],
+  // sigma14
+  sup3: [0.289, 0.286, 0.294],
+  // sigma15
+  sub1: [0.150, 0.143, 0.200],
+  // sigma16
+  sub2: [0.247, 0.286, 0.400],
+  // sigma17
+  supDrop: [0.386, 0.353, 0.494],
+  // sigma18
+  subDrop: [0.050, 0.071, 0.100],
+  // sigma19
+  delim1: [2.390, 1.700, 1.980],
+  // sigma20
+  delim2: [1.010, 1.157, 1.420],
+  // sigma21
+  axisHeight: [0.250, 0.250, 0.250],
+  // sigma22
+  // These font metrics are extracted from TeX by using tftopl on cmex10.tfm;
+  // they correspond to the font parameters of the extension fonts (family 3).
+  // See the TeXbook, page 441. In AMSTeX, the extension fonts scale; to
+  // match cmex7, we'd use cmex7.tfm values for script and scriptscript
+  // values.
+  defaultRuleThickness: [0.04, 0.049, 0.049],
+  // xi8; cmex7: 0.049
+  bigOpSpacing1: [0.111, 0.111, 0.111],
+  // xi9
+  bigOpSpacing2: [0.166, 0.166, 0.166],
+  // xi10
+  bigOpSpacing3: [0.2, 0.2, 0.2],
+  // xi11
+  bigOpSpacing4: [0.6, 0.611, 0.611],
+  // xi12; cmex7: 0.611
+  bigOpSpacing5: [0.1, 0.143, 0.143],
+  // xi13; cmex7: 0.143
+  // The \sqrt rule width is taken from the height of the surd character.
+  // Since we use the same font at all sizes, this thickness doesn't scale.
+  sqrtRuleThickness: [0.04, 0.04, 0.04],
+  // This value determines how large a pt is, for metrics which are defined
+  // in terms of pts.
+  // This value is also used in katex.less; if you change it make sure the
+  // values match.
+  ptPerEm: [10.0, 10.0, 10.0],
+  // The space between adjacent `|` columns in an array definition. From
+  // `\showthe\doublerulesep` in LaTeX. Equals 2.0 / ptPerEm.
+  doubleRuleSep: [0.2, 0.2, 0.2],
+  // The width of separator lines in {array} environments. From
+  // `\showthe\arrayrulewidth` in LaTeX. Equals 0.4 / ptPerEm.
+  arrayRuleWidth: [0.04, 0.04, 0.04],
+  // Two values from LaTeX source2e:
+  fboxsep: [0.3, 0.3, 0.3],
+  //        3 pt / ptPerEm
+  fboxrule: [0.04, 0.04, 0.04] // 0.4 pt / ptPerEm
+
+}; // This map contains a mapping from font name and character code to character
+// should have Latin-1 and Cyrillic characters, but may not depending on the
+// operating system.  The metrics do not account for extra height from the
+// accents.  In the case of Cyrillic characters which have both ascenders and
+// descenders we prefer approximations with ascenders, primarily to prevent
+// the fraction bar or root line from intersecting the glyph.
+// TODO(kevinb) allow union of multiple glyph metrics for better accuracy.
+
+const extraCharacterMap = {
+  // Latin-1
+  'Å': 'A',
+  'Ç': 'C',
+  'Ð': 'D',
+  'Þ': 'o',
+  'å': 'a',
+  'ç': 'c',
+  'ð': 'd',
+  'þ': 'o',
+  // Cyrillic
+  'А': 'A',
+  'Б': 'B',
+  'В': 'B',
+  'Г': 'F',
+  'Д': 'A',
+  'Е': 'E',
+  'Ж': 'K',
+  'З': '3',
+  'И': 'N',
+  'Й': 'N',
+  'К': 'K',
+  'Л': 'N',
+  'М': 'M',
+  'Н': 'H',
+  'О': 'O',
+  'П': 'N',
+  'Р': 'P',
+  'С': 'C',
+  'Т': 'T',
+  'У': 'y',
+  'Ф': 'O',
+  'Х': 'X',
+  'Ц': 'U',
+  'Ч': 'h',
+  'Ш': 'W',
+  'Щ': 'W',
+  'Ъ': 'B',
+  'Ы': 'X',
+  'Ь': 'B',
+  'Э': '3',
+  'Ю': 'X',
+  'Я': 'R',
+  'а': 'a',
+  'б': 'b',
+  'в': 'a',
+  'г': 'r',
+  'д': 'y',
+  'е': 'e',
+  'ж': 'm',
+  'з': 'e',
+  'и': 'n',
+  'й': 'n',
+  'к': 'n',
+  'л': 'n',
+  'м': 'm',
+  'н': 'n',
+  'о': 'o',
+  'п': 'n',
+  'р': 'p',
+  'с': 'c',
+  'т': 'o',
+  'у': 'y',
+  'ф': 'b',
+  'х': 'x',
+  'ц': 'n',
+  'ч': 'n',
+  'ш': 'w',
+  'щ': 'w',
+  'ъ': 'a',
+  'ы': 'm',
+  'ь': 'a',
+  'э': 'e',
+  'ю': 'm',
+  'я': 'r'
+};
+
+/**
+ * This function adds new font metrics to default metricMap
+ * It can also override existing metrics
+ */
+function setFontMetrics(fontName, metrics) {
+  metricMap[fontName] = metrics;
+}
+/**
+ * This function is a convenience function for looking up information in the
+ * metricMap table. It takes a character as a string, and a font.
+ *
+ * Note: the `width` property may be undefined if fontMetricsData.js wasn't
+ * built using `Make extended_metrics`.
+ */
+
+function getCharacterMetrics(character, font, mode) {
+  if (!metricMap[font]) {
+    throw new Error(`Font metrics not found for font: ${font}.`);
+  }
+
+  let ch = character.charCodeAt(0);
+  let metrics = metricMap[font][ch];
+
+  if (!metrics && character[0] in extraCharacterMap) {
+    ch = extraCharacterMap[character[0]].charCodeAt(0);
+    metrics = metricMap[font][ch];
+  }
+
+  if (!metrics && mode === 'text') {
+    // We don't typically have font metrics for Asian scripts.
+    // But since we support them in text mode, we need to return
+    // some sort of metrics.
+    // So if the character is in a script we support but we
+    // don't have metrics for it, just use the metrics for
+    // the Latin capital letter M. This is close enough because
+    // we (currently) only care about the height of the glpyh
+    // not its width.
+    if (supportedCodepoint(ch)) {
+      metrics = metricMap[font][77]; // 77 is the charcode for 'M'
+    }
+  }
+
+  if (metrics) {
+    return {
+      depth: metrics[0],
+      height: metrics[1],
+      italic: metrics[2],
+      skew: metrics[3],
+      width: metrics[4]
+    };
+  }
+}
+const fontMetricsBySizeIndex = {};
+/**
+ * Get the font metrics for a given size.
+ */
+
+function getGlobalMetrics(size) {
+  let sizeIndex;
+
+  if (size >= 5) {
+    sizeIndex = 0;
+  } else if (size >= 3) {
+    sizeIndex = 1;
+  } else {
+    sizeIndex = 2;
+  }
+
+  if (!fontMetricsBySizeIndex[sizeIndex]) {
+    const metrics = fontMetricsBySizeIndex[sizeIndex] = {
+      cssEmPerMu: sigmasAndXis.quad[sizeIndex] / 18
+    };
+
+    for (const key in sigmasAndXis) {
+      if (sigmasAndXis.hasOwnProperty(key)) {
+        metrics[key] = sigmasAndXis[key][sizeIndex];
+      }
+    }
+  }
+
+  return fontMetricsBySizeIndex[sizeIndex];
+}
+
+/**
+ * This file holds a list of all no-argument functions and single-character
+ * symbols (like 'a' or ';').
+ *
+ * For each of the symbols, there are three properties they can have:
+ * - font (required): the font to be used for this symbol. Either "main" (the
+     normal font), or "ams" (the ams fonts).
+ * - group (required): the ParseNode group type the symbol should have (i.e.
+     "textord", "mathord", etc).
+     See https://github.com/KaTeX/KaTeX/wiki/Examining-TeX#group-types
+ * - replace: the character that this symbol or function should be
+ *   replaced with (i.e. "\phi" has a replace value of "\u03d5", the phi
+ *   character in the main font).
+ *
+ * The outermost map in the table indicates what mode the symbols should be
+ * accepted in (e.g. "math" or "text").
+ */
+// Some of these have a "-token" suffix since these are also used as `ParseNode`
+// types for raw text tokens, and we want to avoid conflicts with higher-level
+// `ParseNode` types. These `ParseNode`s are constructed within `Parser` by
+// looking up the `symbols` map.
+const ATOMS = {
+  "bin": 1,
+  "close": 1,
+  "inner": 1,
+  "open": 1,
+  "punct": 1,
+  "rel": 1
+};
+const NON_ATOMS = {
+  "accent-token": 1,
+  "mathord": 1,
+  "op-token": 1,
+  "spacing": 1,
+  "textord": 1
+};
+const symbols = {
+  "math": {},
+  "text": {}
+};
+/** `acceptUnicodeChar = true` is only applicable if `replace` is set. */
+
+function defineSymbol(mode, font, group, replace, name, acceptUnicodeChar) {
+  symbols[mode][name] = {
+    font,
+    group,
+    replace
+  };
+
+  if (acceptUnicodeChar && replace) {
+    symbols[mode][replace] = symbols[mode][name];
+  }
+} // Some abbreviations for commonly used strings.
+// This helps minify the code, and also spotting typos using jshint.
+// modes:
+
+const math = "math";
+const text$1 = "text"; // fonts:
+
+const main = "main";
+const ams = "ams"; // groups:
+
+const accent = "accent-token";
+const bin = "bin";
+const close = "close";
+const inner = "inner";
+const mathord = "mathord";
+const op = "op-token";
+const open = "open";
+const punct = "punct";
+const rel = "rel";
+const spacing = "spacing";
+const textord = "textord"; // Now comes the symbol table
+// Relation Symbols
+
+defineSymbol(math, main, rel, "\u2261", "\\equiv", true);
+defineSymbol(math, main, rel, "\u227a", "\\prec", true);
+defineSymbol(math, main, rel, "\u227b", "\\succ", true);
+defineSymbol(math, main, rel, "\u223c", "\\sim", true);
+defineSymbol(math, main, rel, "\u22a5", "\\perp");
+defineSymbol(math, main, rel, "\u2aaf", "\\preceq", true);
+defineSymbol(math, main, rel, "\u2ab0", "\\succeq", true);
+defineSymbol(math, main, rel, "\u2243", "\\simeq", true);
+defineSymbol(math, main, rel, "\u2223", "\\mid", true);
+defineSymbol(math, main, rel, "\u226a", "\\ll", true);
+defineSymbol(math, main, rel, "\u226b", "\\gg", true);
+defineSymbol(math, main, rel, "\u224d", "\\asymp", true);
+defineSymbol(math, main, rel, "\u2225", "\\parallel");
+defineSymbol(math, main, rel, "\u22c8", "\\bowtie", true);
+defineSymbol(math, main, rel, "\u2323", "\\smile", true);
+defineSymbol(math, main, rel, "\u2291", "\\sqsubseteq", true);
+defineSymbol(math, main, rel, "\u2292", "\\sqsupseteq", true);
+defineSymbol(math, main, rel, "\u2250", "\\doteq", true);
+defineSymbol(math, main, rel, "\u2322", "\\frown", true);
+defineSymbol(math, main, rel, "\u220b", "\\ni", true);
+defineSymbol(math, main, rel, "\u221d", "\\propto", true);
+defineSymbol(math, main, rel, "\u22a2", "\\vdash", true);
+defineSymbol(math, main, rel, "\u22a3", "\\dashv", true);
+defineSymbol(math, main, rel, "\u220b", "\\owns"); // Punctuation
+
+defineSymbol(math, main, punct, "\u002e", "\\ldotp");
+defineSymbol(math, main, punct, "\u22c5", "\\cdotp"); // Misc Symbols
+
+defineSymbol(math, main, textord, "\u0023", "\\#");
+defineSymbol(text$1, main, textord, "\u0023", "\\#");
+defineSymbol(math, main, textord, "\u0026", "\\&");
+defineSymbol(text$1, main, textord, "\u0026", "\\&");
+defineSymbol(math, main, textord, "\u2135", "\\aleph", true);
+defineSymbol(math, main, textord, "\u2200", "\\forall", true);
+defineSymbol(math, main, textord, "\u210f", "\\hbar", true);
+defineSymbol(math, main, textord, "\u2203", "\\exists", true);
+defineSymbol(math, main, textord, "\u2207", "\\nabla", true);
+defineSymbol(math, main, textord, "\u266d", "\\flat", true);
+defineSymbol(math, main, textord, "\u2113", "\\ell", true);
+defineSymbol(math, main, textord, "\u266e", "\\natural", true);
+defineSymbol(math, main, textord, "\u2663", "\\clubsuit", true);
+defineSymbol(math, main, textord, "\u2118", "\\wp", true);
+defineSymbol(math, main, textord, "\u266f", "\\sharp", true);
+defineSymbol(math, main, textord, "\u2662", "\\diamondsuit", true);
+defineSymbol(math, main, textord, "\u211c", "\\Re", true);
+defineSymbol(math, main, textord, "\u2661", "\\heartsuit", true);
+defineSymbol(math, main, textord, "\u2111", "\\Im", true);
+defineSymbol(math, main, textord, "\u2660", "\\spadesuit", true);
+defineSymbol(text$1, main, textord, "\u00a7", "\\S", true);
+defineSymbol(text$1, main, textord, "\u00b6", "\\P", true); // Math and Text
+
+defineSymbol(math, main, textord, "\u2020", "\\dag");
+defineSymbol(text$1, main, textord, "\u2020", "\\dag");
+defineSymbol(text$1, main, textord, "\u2020", "\\textdagger");
+defineSymbol(math, main, textord, "\u2021", "\\ddag");
+defineSymbol(text$1, main, textord, "\u2021", "\\ddag");
+defineSymbol(text$1, main, textord, "\u2021", "\\textdaggerdbl"); // Large Delimiters
+
+defineSymbol(math, main, close, "\u23b1", "\\rmoustache", true);
+defineSymbol(math, main, open, "\u23b0", "\\lmoustache", true);
+defineSymbol(math, main, close, "\u27ef", "\\rgroup", true);
+defineSymbol(math, main, open, "\u27ee", "\\lgroup", true); // Binary Operators
+
+defineSymbol(math, main, bin, "\u2213", "\\mp", true);
+defineSymbol(math, main, bin, "\u2296", "\\ominus", true);
+defineSymbol(math, main, bin, "\u228e", "\\uplus", true);
+defineSymbol(math, main, bin, "\u2293", "\\sqcap", true);
+defineSymbol(math, main, bin, "\u2217", "\\ast");
+defineSymbol(math, main, bin, "\u2294", "\\sqcup", true);
+defineSymbol(math, main, bin, "\u25ef", "\\bigcirc");
+defineSymbol(math, main, bin, "\u2219", "\\bullet");
+defineSymbol(math, main, bin, "\u2021", "\\ddagger");
+defineSymbol(math, main, bin, "\u2240", "\\wr", true);
+defineSymbol(math, main, bin, "\u2a3f", "\\amalg");
+defineSymbol(math, main, bin, "\u0026", "\\And"); // from amsmath
+// Arrow Symbols
+
+defineSymbol(math, main, rel, "\u27f5", "\\longleftarrow", true);
+defineSymbol(math, main, rel, "\u21d0", "\\Leftarrow", true);
+defineSymbol(math, main, rel, "\u27f8", "\\Longleftarrow", true);
+defineSymbol(math, main, rel, "\u27f6", "\\longrightarrow", true);
+defineSymbol(math, main, rel, "\u21d2", "\\Rightarrow", true);
+defineSymbol(math, main, rel, "\u27f9", "\\Longrightarrow", true);
+defineSymbol(math, main, rel, "\u2194", "\\leftrightarrow", true);
+defineSymbol(math, main, rel, "\u27f7", "\\longleftrightarrow", true);
+defineSymbol(math, main, rel, "\u21d4", "\\Leftrightarrow", true);
+defineSymbol(math, main, rel, "\u27fa", "\\Longleftrightarrow", true);
+defineSymbol(math, main, rel, "\u21a6", "\\mapsto", true);
+defineSymbol(math, main, rel, "\u27fc", "\\longmapsto", true);
+defineSymbol(math, main, rel, "\u2197", "\\nearrow", true);
+defineSymbol(math, main, rel, "\u21a9", "\\hookleftarrow", true);
+defineSymbol(math, main, rel, "\u21aa", "\\hookrightarrow", true);
+defineSymbol(math, main, rel, "\u2198", "\\searrow", true);
+defineSymbol(math, main, rel, "\u21bc", "\\leftharpoonup", true);
+defineSymbol(math, main, rel, "\u21c0", "\\rightharpoonup", true);
+defineSymbol(math, main, rel, "\u2199", "\\swarrow", true);
+defineSymbol(math, main, rel, "\u21bd", "\\leftharpoondown", true);
+defineSymbol(math, main, rel, "\u21c1", "\\rightharpoondown", true);
+defineSymbol(math, main, rel, "\u2196", "\\nwarrow", true);
+defineSymbol(math, main, rel, "\u21cc", "\\rightleftharpoons", true); // AMS Negated Binary Relations
+
+defineSymbol(math, ams, rel, "\u226e", "\\nless", true); // Symbol names preceeded by "@" each have a corresponding macro.
+
+defineSymbol(math, ams, rel, "\ue010", "\\@nleqslant");
+defineSymbol(math, ams, rel, "\ue011", "\\@nleqq");
+defineSymbol(math, ams, rel, "\u2a87", "\\lneq", true);
+defineSymbol(math, ams, rel, "\u2268", "\\lneqq", true);
+defineSymbol(math, ams, rel, "\ue00c", "\\@lvertneqq");
+defineSymbol(math, ams, rel, "\u22e6", "\\lnsim", true);
+defineSymbol(math, ams, rel, "\u2a89", "\\lnapprox", true);
+defineSymbol(math, ams, rel, "\u2280", "\\nprec", true); // unicode-math maps \u22e0 to \npreccurlyeq. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u22e0", "\\npreceq", true);
+defineSymbol(math, ams, rel, "\u22e8", "\\precnsim", true);
+defineSymbol(math, ams, rel, "\u2ab9", "\\precnapprox", true);
+defineSymbol(math, ams, rel, "\u2241", "\\nsim", true);
+defineSymbol(math, ams, rel, "\ue006", "\\@nshortmid");
+defineSymbol(math, ams, rel, "\u2224", "\\nmid", true);
+defineSymbol(math, ams, rel, "\u22ac", "\\nvdash", true);
+defineSymbol(math, ams, rel, "\u22ad", "\\nvDash", true);
+defineSymbol(math, ams, rel, "\u22ea", "\\ntriangleleft");
+defineSymbol(math, ams, rel, "\u22ec", "\\ntrianglelefteq", true);
+defineSymbol(math, ams, rel, "\u228a", "\\subsetneq", true);
+defineSymbol(math, ams, rel, "\ue01a", "\\@varsubsetneq");
+defineSymbol(math, ams, rel, "\u2acb", "\\subsetneqq", true);
+defineSymbol(math, ams, rel, "\ue017", "\\@varsubsetneqq");
+defineSymbol(math, ams, rel, "\u226f", "\\ngtr", true);
+defineSymbol(math, ams, rel, "\ue00f", "\\@ngeqslant");
+defineSymbol(math, ams, rel, "\ue00e", "\\@ngeqq");
+defineSymbol(math, ams, rel, "\u2a88", "\\gneq", true);
+defineSymbol(math, ams, rel, "\u2269", "\\gneqq", true);
+defineSymbol(math, ams, rel, "\ue00d", "\\@gvertneqq");
+defineSymbol(math, ams, rel, "\u22e7", "\\gnsim", true);
+defineSymbol(math, ams, rel, "\u2a8a", "\\gnapprox", true);
+defineSymbol(math, ams, rel, "\u2281", "\\nsucc", true); // unicode-math maps \u22e1 to \nsucccurlyeq. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u22e1", "\\nsucceq", true);
+defineSymbol(math, ams, rel, "\u22e9", "\\succnsim", true);
+defineSymbol(math, ams, rel, "\u2aba", "\\succnapprox", true); // unicode-math maps \u2246 to \simneqq. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u2246", "\\ncong", true);
+defineSymbol(math, ams, rel, "\ue007", "\\@nshortparallel");
+defineSymbol(math, ams, rel, "\u2226", "\\nparallel", true);
+defineSymbol(math, ams, rel, "\u22af", "\\nVDash", true);
+defineSymbol(math, ams, rel, "\u22eb", "\\ntriangleright");
+defineSymbol(math, ams, rel, "\u22ed", "\\ntrianglerighteq", true);
+defineSymbol(math, ams, rel, "\ue018", "\\@nsupseteqq");
+defineSymbol(math, ams, rel, "\u228b", "\\supsetneq", true);
+defineSymbol(math, ams, rel, "\ue01b", "\\@varsupsetneq");
+defineSymbol(math, ams, rel, "\u2acc", "\\supsetneqq", true);
+defineSymbol(math, ams, rel, "\ue019", "\\@varsupsetneqq");
+defineSymbol(math, ams, rel, "\u22ae", "\\nVdash", true);
+defineSymbol(math, ams, rel, "\u2ab5", "\\precneqq", true);
+defineSymbol(math, ams, rel, "\u2ab6", "\\succneqq", true);
+defineSymbol(math, ams, rel, "\ue016", "\\@nsubseteqq");
+defineSymbol(math, ams, bin, "\u22b4", "\\unlhd");
+defineSymbol(math, ams, bin, "\u22b5", "\\unrhd"); // AMS Negated Arrows
+
+defineSymbol(math, ams, rel, "\u219a", "\\nleftarrow", true);
+defineSymbol(math, ams, rel, "\u219b", "\\nrightarrow", true);
+defineSymbol(math, ams, rel, "\u21cd", "\\nLeftarrow", true);
+defineSymbol(math, ams, rel, "\u21cf", "\\nRightarrow", true);
+defineSymbol(math, ams, rel, "\u21ae", "\\nleftrightarrow", true);
+defineSymbol(math, ams, rel, "\u21ce", "\\nLeftrightarrow", true); // AMS Misc
+
+defineSymbol(math, ams, rel, "\u25b3", "\\vartriangle");
+defineSymbol(math, ams, textord, "\u210f", "\\hslash");
+defineSymbol(math, ams, textord, "\u25bd", "\\triangledown");
+defineSymbol(math, ams, textord, "\u25ca", "\\lozenge");
+defineSymbol(math, ams, textord, "\u24c8", "\\circledS");
+defineSymbol(math, ams, textord, "\u00ae", "\\circledR");
+defineSymbol(text$1, ams, textord, "\u00ae", "\\circledR");
+defineSymbol(math, ams, textord, "\u2221", "\\measuredangle", true);
+defineSymbol(math, ams, textord, "\u2204", "\\nexists");
+defineSymbol(math, ams, textord, "\u2127", "\\mho");
+defineSymbol(math, ams, textord, "\u2132", "\\Finv", true);
+defineSymbol(math, ams, textord, "\u2141", "\\Game", true);
+defineSymbol(math, ams, textord, "\u2035", "\\backprime");
+defineSymbol(math, ams, textord, "\u25b2", "\\blacktriangle");
+defineSymbol(math, ams, textord, "\u25bc", "\\blacktriangledown");
+defineSymbol(math, ams, textord, "\u25a0", "\\blacksquare");
+defineSymbol(math, ams, textord, "\u29eb", "\\blacklozenge");
+defineSymbol(math, ams, textord, "\u2605", "\\bigstar");
+defineSymbol(math, ams, textord, "\u2222", "\\sphericalangle", true);
+defineSymbol(math, ams, textord, "\u2201", "\\complement", true); // unicode-math maps U+F0 to \matheth. We map to AMS function \eth
+
+defineSymbol(math, ams, textord, "\u00f0", "\\eth", true);
+defineSymbol(text$1, main, textord, "\u00f0", "\u00f0");
+defineSymbol(math, ams, textord, "\u2571", "\\diagup");
+defineSymbol(math, ams, textord, "\u2572", "\\diagdown");
+defineSymbol(math, ams, textord, "\u25a1", "\\square");
+defineSymbol(math, ams, textord, "\u25a1", "\\Box");
+defineSymbol(math, ams, textord, "\u25ca", "\\Diamond"); // unicode-math maps U+A5 to \mathyen. We map to AMS function \yen
+
+defineSymbol(math, ams, textord, "\u00a5", "\\yen", true);
+defineSymbol(text$1, ams, textord, "\u00a5", "\\yen", true);
+defineSymbol(math, ams, textord, "\u2713", "\\checkmark", true);
+defineSymbol(text$1, ams, textord, "\u2713", "\\checkmark"); // AMS Hebrew
+
+defineSymbol(math, ams, textord, "\u2136", "\\beth", true);
+defineSymbol(math, ams, textord, "\u2138", "\\daleth", true);
+defineSymbol(math, ams, textord, "\u2137", "\\gimel", true); // AMS Greek
+
+defineSymbol(math, ams, textord, "\u03dd", "\\digamma", true);
+defineSymbol(math, ams, textord, "\u03f0", "\\varkappa"); // AMS Delimiters
+
+defineSymbol(math, ams, open, "\u250c", "\\@ulcorner", true);
+defineSymbol(math, ams, close, "\u2510", "\\@urcorner", true);
+defineSymbol(math, ams, open, "\u2514", "\\@llcorner", true);
+defineSymbol(math, ams, close, "\u2518", "\\@lrcorner", true); // AMS Binary Relations
+
+defineSymbol(math, ams, rel, "\u2266", "\\leqq", true);
+defineSymbol(math, ams, rel, "\u2a7d", "\\leqslant", true);
+defineSymbol(math, ams, rel, "\u2a95", "\\eqslantless", true);
+defineSymbol(math, ams, rel, "\u2272", "\\lesssim", true);
+defineSymbol(math, ams, rel, "\u2a85", "\\lessapprox", true);
+defineSymbol(math, ams, rel, "\u224a", "\\approxeq", true);
+defineSymbol(math, ams, bin, "\u22d6", "\\lessdot");
+defineSymbol(math, ams, rel, "\u22d8", "\\lll", true);
+defineSymbol(math, ams, rel, "\u2276", "\\lessgtr", true);
+defineSymbol(math, ams, rel, "\u22da", "\\lesseqgtr", true);
+defineSymbol(math, ams, rel, "\u2a8b", "\\lesseqqgtr", true);
+defineSymbol(math, ams, rel, "\u2251", "\\doteqdot");
+defineSymbol(math, ams, rel, "\u2253", "\\risingdotseq", true);
+defineSymbol(math, ams, rel, "\u2252", "\\fallingdotseq", true);
+defineSymbol(math, ams, rel, "\u223d", "\\backsim", true);
+defineSymbol(math, ams, rel, "\u22cd", "\\backsimeq", true);
+defineSymbol(math, ams, rel, "\u2ac5", "\\subseteqq", true);
+defineSymbol(math, ams, rel, "\u22d0", "\\Subset", true);
+defineSymbol(math, ams, rel, "\u228f", "\\sqsubset", true);
+defineSymbol(math, ams, rel, "\u227c", "\\preccurlyeq", true);
+defineSymbol(math, ams, rel, "\u22de", "\\curlyeqprec", true);
+defineSymbol(math, ams, rel, "\u227e", "\\precsim", true);
+defineSymbol(math, ams, rel, "\u2ab7", "\\precapprox", true);
+defineSymbol(math, ams, rel, "\u22b2", "\\vartriangleleft");
+defineSymbol(math, ams, rel, "\u22b4", "\\trianglelefteq");
+defineSymbol(math, ams, rel, "\u22a8", "\\vDash", true);
+defineSymbol(math, ams, rel, "\u22aa", "\\Vvdash", true);
+defineSymbol(math, ams, rel, "\u2323", "\\smallsmile");
+defineSymbol(math, ams, rel, "\u2322", "\\smallfrown");
+defineSymbol(math, ams, rel, "\u224f", "\\bumpeq", true);
+defineSymbol(math, ams, rel, "\u224e", "\\Bumpeq", true);
+defineSymbol(math, ams, rel, "\u2267", "\\geqq", true);
+defineSymbol(math, ams, rel, "\u2a7e", "\\geqslant", true);
+defineSymbol(math, ams, rel, "\u2a96", "\\eqslantgtr", true);
+defineSymbol(math, ams, rel, "\u2273", "\\gtrsim", true);
+defineSymbol(math, ams, rel, "\u2a86", "\\gtrapprox", true);
+defineSymbol(math, ams, bin, "\u22d7", "\\gtrdot");
+defineSymbol(math, ams, rel, "\u22d9", "\\ggg", true);
+defineSymbol(math, ams, rel, "\u2277", "\\gtrless", true);
+defineSymbol(math, ams, rel, "\u22db", "\\gtreqless", true);
+defineSymbol(math, ams, rel, "\u2a8c", "\\gtreqqless", true);
+defineSymbol(math, ams, rel, "\u2256", "\\eqcirc", true);
+defineSymbol(math, ams, rel, "\u2257", "\\circeq", true);
+defineSymbol(math, ams, rel, "\u225c", "\\triangleq", true);
+defineSymbol(math, ams, rel, "\u223c", "\\thicksim");
+defineSymbol(math, ams, rel, "\u2248", "\\thickapprox");
+defineSymbol(math, ams, rel, "\u2ac6", "\\supseteqq", true);
+defineSymbol(math, ams, rel, "\u22d1", "\\Supset", true);
+defineSymbol(math, ams, rel, "\u2290", "\\sqsupset", true);
+defineSymbol(math, ams, rel, "\u227d", "\\succcurlyeq", true);
+defineSymbol(math, ams, rel, "\u22df", "\\curlyeqsucc", true);
+defineSymbol(math, ams, rel, "\u227f", "\\succsim", true);
+defineSymbol(math, ams, rel, "\u2ab8", "\\succapprox", true);
+defineSymbol(math, ams, rel, "\u22b3", "\\vartriangleright");
+defineSymbol(math, ams, rel, "\u22b5", "\\trianglerighteq");
+defineSymbol(math, ams, rel, "\u22a9", "\\Vdash", true);
+defineSymbol(math, ams, rel, "\u2223", "\\shortmid");
+defineSymbol(math, ams, rel, "\u2225", "\\shortparallel");
+defineSymbol(math, ams, rel, "\u226c", "\\between", true);
+defineSymbol(math, ams, rel, "\u22d4", "\\pitchfork", true);
+defineSymbol(math, ams, rel, "\u221d", "\\varpropto");
+defineSymbol(math, ams, rel, "\u25c0", "\\blacktriangleleft"); // unicode-math says that \therefore is a mathord atom.
+// We kept the amssymb atom type, which is rel.
+
+defineSymbol(math, ams, rel, "\u2234", "\\therefore", true);
+defineSymbol(math, ams, rel, "\u220d", "\\backepsilon");
+defineSymbol(math, ams, rel, "\u25b6", "\\blacktriangleright"); // unicode-math says that \because is a mathord atom.
+// We kept the amssymb atom type, which is rel.
+
+defineSymbol(math, ams, rel, "\u2235", "\\because", true);
+defineSymbol(math, ams, rel, "\u22d8", "\\llless");
+defineSymbol(math, ams, rel, "\u22d9", "\\gggtr");
+defineSymbol(math, ams, bin, "\u22b2", "\\lhd");
+defineSymbol(math, ams, bin, "\u22b3", "\\rhd");
+defineSymbol(math, ams, rel, "\u2242", "\\eqsim", true);
+defineSymbol(math, main, rel, "\u22c8", "\\Join");
+defineSymbol(math, ams, rel, "\u2251", "\\Doteq", true); // AMS Binary Operators
+
+defineSymbol(math, ams, bin, "\u2214", "\\dotplus", true);
+defineSymbol(math, ams, bin, "\u2216", "\\smallsetminus");
+defineSymbol(math, ams, bin, "\u22d2", "\\Cap", true);
+defineSymbol(math, ams, bin, "\u22d3", "\\Cup", true);
+defineSymbol(math, ams, bin, "\u2a5e", "\\doublebarwedge", true);
+defineSymbol(math, ams, bin, "\u229f", "\\boxminus", true);
+defineSymbol(math, ams, bin, "\u229e", "\\boxplus", true);
+defineSymbol(math, ams, bin, "\u22c7", "\\divideontimes", true);
+defineSymbol(math, ams, bin, "\u22c9", "\\ltimes", true);
+defineSymbol(math, ams, bin, "\u22ca", "\\rtimes", true);
+defineSymbol(math, ams, bin, "\u22cb", "\\leftthreetimes", true);
+defineSymbol(math, ams, bin, "\u22cc", "\\rightthreetimes", true);
+defineSymbol(math, ams, bin, "\u22cf", "\\curlywedge", true);
+defineSymbol(math, ams, bin, "\u22ce", "\\curlyvee", true);
+defineSymbol(math, ams, bin, "\u229d", "\\circleddash", true);
+defineSymbol(math, ams, bin, "\u229b", "\\circledast", true);
+defineSymbol(math, ams, bin, "\u22c5", "\\centerdot");
+defineSymbol(math, ams, bin, "\u22ba", "\\intercal", true);
+defineSymbol(math, ams, bin, "\u22d2", "\\doublecap");
+defineSymbol(math, ams, bin, "\u22d3", "\\doublecup");
+defineSymbol(math, ams, bin, "\u22a0", "\\boxtimes", true); // AMS Arrows
+// Note: unicode-math maps \u21e2 to their own function \rightdasharrow.
+// We'll map it to AMS function \dashrightarrow. It produces the same atom.
+
+defineSymbol(math, ams, rel, "\u21e2", "\\dashrightarrow", true); // unicode-math maps \u21e0 to \leftdasharrow. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u21e0", "\\dashleftarrow", true);
+defineSymbol(math, ams, rel, "\u21c7", "\\leftleftarrows", true);
+defineSymbol(math, ams, rel, "\u21c6", "\\leftrightarrows", true);
+defineSymbol(math, ams, rel, "\u21da", "\\Lleftarrow", true);
+defineSymbol(math, ams, rel, "\u219e", "\\twoheadleftarrow", true);
+defineSymbol(math, ams, rel, "\u21a2", "\\leftarrowtail", true);
+defineSymbol(math, ams, rel, "\u21ab", "\\looparrowleft", true);
+defineSymbol(math, ams, rel, "\u21cb", "\\leftrightharpoons", true);
+defineSymbol(math, ams, rel, "\u21b6", "\\curvearrowleft", true); // unicode-math maps \u21ba to \acwopencirclearrow. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u21ba", "\\circlearrowleft", true);
+defineSymbol(math, ams, rel, "\u21b0", "\\Lsh", true);
+defineSymbol(math, ams, rel, "\u21c8", "\\upuparrows", true);
+defineSymbol(math, ams, rel, "\u21bf", "\\upharpoonleft", true);
+defineSymbol(math, ams, rel, "\u21c3", "\\downharpoonleft", true);
+defineSymbol(math, ams, rel, "\u22b8", "\\multimap", true);
+defineSymbol(math, ams, rel, "\u21ad", "\\leftrightsquigarrow", true);
+defineSymbol(math, ams, rel, "\u21c9", "\\rightrightarrows", true);
+defineSymbol(math, ams, rel, "\u21c4", "\\rightleftarrows", true);
+defineSymbol(math, ams, rel, "\u21a0", "\\twoheadrightarrow", true);
+defineSymbol(math, ams, rel, "\u21a3", "\\rightarrowtail", true);
+defineSymbol(math, ams, rel, "\u21ac", "\\looparrowright", true);
+defineSymbol(math, ams, rel, "\u21b7", "\\curvearrowright", true); // unicode-math maps \u21bb to \cwopencirclearrow. We'll use the AMS synonym.
+
+defineSymbol(math, ams, rel, "\u21bb", "\\circlearrowright", true);
+defineSymbol(math, ams, rel, "\u21b1", "\\Rsh", true);
+defineSymbol(math, ams, rel, "\u21ca", "\\downdownarrows", true);
+defineSymbol(math, ams, rel, "\u21be", "\\upharpoonright", true);
+defineSymbol(math, ams, rel, "\u21c2", "\\downharpoonright", true);
+defineSymbol(math, ams, rel, "\u21dd", "\\rightsquigarrow", true);
+defineSymbol(math, ams, rel, "\u21dd", "\\leadsto");
+defineSymbol(math, ams, rel, "\u21db", "\\Rrightarrow", true);
+defineSymbol(math, ams, rel, "\u21be", "\\restriction");
+defineSymbol(math, main, textord, "\u2018", "`");
+defineSymbol(math, main, textord, "$", "\\$");
+defineSymbol(text$1, main, textord, "$", "\\$");
+defineSymbol(text$1, main, textord, "$", "\\textdollar");
+defineSymbol(math, main, textord, "%", "\\%");
+defineSymbol(text$1, main, textord, "%", "\\%");
+defineSymbol(math, main, textord, "_", "\\_");
+defineSymbol(text$1, main, textord, "_", "\\_");
+defineSymbol(text$1, main, textord, "_", "\\textunderscore");
+defineSymbol(math, main, textord, "\u2220", "\\angle", true);
+defineSymbol(math, main, textord, "\u221e", "\\infty", true);
+defineSymbol(math, main, textord, "\u2032", "\\prime");
+defineSymbol(math, main, textord, "\u25b3", "\\triangle");
+defineSymbol(math, main, textord, "\u0393", "\\Gamma", true);
+defineSymbol(math, main, textord, "\u0394", "\\Delta", true);
+defineSymbol(math, main, textord, "\u0398", "\\Theta", true);
+defineSymbol(math, main, textord, "\u039b", "\\Lambda", true);
+defineSymbol(math, main, textord, "\u039e", "\\Xi", true);
+defineSymbol(math, main, textord, "\u03a0", "\\Pi", true);
+defineSymbol(math, main, textord, "\u03a3", "\\Sigma", true);
+defineSymbol(math, main, textord, "\u03a5", "\\Upsilon", true);
+defineSymbol(math, main, textord, "\u03a6", "\\Phi", true);
+defineSymbol(math, main, textord, "\u03a8", "\\Psi", true);
+defineSymbol(math, main, textord, "\u03a9", "\\Omega", true);
+defineSymbol(math, main, textord, "A", "\u0391");
+defineSymbol(math, main, textord, "B", "\u0392");
+defineSymbol(math, main, textord, "E", "\u0395");
+defineSymbol(math, main, textord, "Z", "\u0396");
+defineSymbol(math, main, textord, "H", "\u0397");
+defineSymbol(math, main, textord, "I", "\u0399");
+defineSymbol(math, main, textord, "K", "\u039A");
+defineSymbol(math, main, textord, "M", "\u039C");
+defineSymbol(math, main, textord, "N", "\u039D");
+defineSymbol(math, main, textord, "O", "\u039F");
+defineSymbol(math, main, textord, "P", "\u03A1");
+defineSymbol(math, main, textord, "T", "\u03A4");
+defineSymbol(math, main, textord, "X", "\u03A7");
+defineSymbol(math, main, textord, "\u00ac", "\\neg", true);
+defineSymbol(math, main, textord, "\u00ac", "\\lnot");
+defineSymbol(math, main, textord, "\u22a4", "\\top");
+defineSymbol(math, main, textord, "\u22a5", "\\bot");
+defineSymbol(math, main, textord, "\u2205", "\\emptyset");
+defineSymbol(math, ams, textord, "\u2205", "\\varnothing");
+defineSymbol(math, main, mathord, "\u03b1", "\\alpha", true);
+defineSymbol(math, main, mathord, "\u03b2", "\\beta", true);
+defineSymbol(math, main, mathord, "\u03b3", "\\gamma", true);
+defineSymbol(math, main, mathord, "\u03b4", "\\delta", true);
+defineSymbol(math, main, mathord, "\u03f5", "\\epsilon", true);
+defineSymbol(math, main, mathord, "\u03b6", "\\zeta", true);
+defineSymbol(math, main, mathord, "\u03b7", "\\eta", true);
+defineSymbol(math, main, mathord, "\u03b8", "\\theta", true);
+defineSymbol(math, main, mathord, "\u03b9", "\\iota", true);
+defineSymbol(math, main, mathord, "\u03ba", "\\kappa", true);
+defineSymbol(math, main, mathord, "\u03bb", "\\lambda", true);
+defineSymbol(math, main, mathord, "\u03bc", "\\mu", true);
+defineSymbol(math, main, mathord, "\u03bd", "\\nu", true);
+defineSymbol(math, main, mathord, "\u03be", "\\xi", true);
+defineSymbol(math, main, mathord, "\u03bf", "\\omicron", true);
+defineSymbol(math, main, mathord, "\u03c0", "\\pi", true);
+defineSymbol(math, main, mathord, "\u03c1", "\\rho", true);
+defineSymbol(math, main, mathord, "\u03c3", "\\sigma", true);
+defineSymbol(math, main, mathord, "\u03c4", "\\tau", true);
+defineSymbol(math, main, mathord, "\u03c5", "\\upsilon", true);
+defineSymbol(math, main, mathord, "\u03d5", "\\phi", true);
+defineSymbol(math, main, mathord, "\u03c7", "\\chi", true);
+defineSymbol(math, main, mathord, "\u03c8", "\\psi", true);
+defineSymbol(math, main, mathord, "\u03c9", "\\omega", true);
+defineSymbol(math, main, mathord, "\u03b5", "\\varepsilon", true);
+defineSymbol(math, main, mathord, "\u03d1", "\\vartheta", true);
+defineSymbol(math, main, mathord, "\u03d6", "\\varpi", true);
+defineSymbol(math, main, mathord, "\u03f1", "\\varrho", true);
+defineSymbol(math, main, mathord, "\u03c2", "\\varsigma", true);
+defineSymbol(math, main, mathord, "\u03c6", "\\varphi", true);
+defineSymbol(math, main, bin, "\u2217", "*");
+defineSymbol(math, main, bin, "+", "+");
+defineSymbol(math, main, bin, "\u2212", "-");
+defineSymbol(math, main, bin, "\u22c5", "\\cdot", true);
+defineSymbol(math, main, bin, "\u2218", "\\circ");
+defineSymbol(math, main, bin, "\u00f7", "\\div", true);
+defineSymbol(math, main, bin, "\u00b1", "\\pm", true);
+defineSymbol(math, main, bin, "\u00d7", "\\times", true);
+defineSymbol(math, main, bin, "\u2229", "\\cap", true);
+defineSymbol(math, main, bin, "\u222a", "\\cup", true);
+defineSymbol(math, main, bin, "\u2216", "\\setminus");
+defineSymbol(math, main, bin, "\u2227", "\\land");
+defineSymbol(math, main, bin, "\u2228", "\\lor");
+defineSymbol(math, main, bin, "\u2227", "\\wedge", true);
+defineSymbol(math, main, bin, "\u2228", "\\vee", true);
+defineSymbol(math, main, textord, "\u221a", "\\surd");
+defineSymbol(math, main, open, "\u27e8", "\\langle", true);
+defineSymbol(math, main, open, "\u2223", "\\lvert");
+defineSymbol(math, main, open, "\u2225", "\\lVert");
+defineSymbol(math, main, close, "?", "?");
+defineSymbol(math, main, close, "!", "!");
+defineSymbol(math, main, close, "\u27e9", "\\rangle", true);
+defineSymbol(math, main, close, "\u2223", "\\rvert");
+defineSymbol(math, main, close, "\u2225", "\\rVert");
+defineSymbol(math, main, rel, "=", "=");
+defineSymbol(math, main, rel, ":", ":");
+defineSymbol(math, main, rel, "\u2248", "\\approx", true);
+defineSymbol(math, main, rel, "\u2245", "\\cong", true);
+defineSymbol(math, main, rel, "\u2265", "\\ge");
+defineSymbol(math, main, rel, "\u2265", "\\geq", true);
+defineSymbol(math, main, rel, "\u2190", "\\gets");
+defineSymbol(math, main, rel, ">", "\\gt", true);
+defineSymbol(math, main, rel, "\u2208", "\\in", true);
+defineSymbol(math, main, rel, "\ue020", "\\@not");
+defineSymbol(math, main, rel, "\u2282", "\\subset", true);
+defineSymbol(math, main, rel, "\u2283", "\\supset", true);
+defineSymbol(math, main, rel, "\u2286", "\\subseteq", true);
+defineSymbol(math, main, rel, "\u2287", "\\supseteq", true);
+defineSymbol(math, ams, rel, "\u2288", "\\nsubseteq", true);
+defineSymbol(math, ams, rel, "\u2289", "\\nsupseteq", true);
+defineSymbol(math, main, rel, "\u22a8", "\\models");
+defineSymbol(math, main, rel, "\u2190", "\\leftarrow", true);
+defineSymbol(math, main, rel, "\u2264", "\\le");
+defineSymbol(math, main, rel, "\u2264", "\\leq", true);
+defineSymbol(math, main, rel, "<", "\\lt", true);
+defineSymbol(math, main, rel, "\u2192", "\\rightarrow", true);
+defineSymbol(math, main, rel, "\u2192", "\\to");
+defineSymbol(math, ams, rel, "\u2271", "\\ngeq", true);
+defineSymbol(math, ams, rel, "\u2270", "\\nleq", true);
+defineSymbol(math, main, spacing, "\u00a0", "\\ ");
+defineSymbol(math, main, spacing, "\u00a0", "~");
+defineSymbol(math, main, spacing, "\u00a0", "\\space"); // Ref: LaTeX Source 2e: \DeclareRobustCommand{\nobreakspace}{%
+
+defineSymbol(math, main, spacing, "\u00a0", "\\nobreakspace");
+defineSymbol(text$1, main, spacing, "\u00a0", "\\ ");
+defineSymbol(text$1, main, spacing, "\u00a0", " ");
+defineSymbol(text$1, main, spacing, "\u00a0", "~");
+defineSymbol(text$1, main, spacing, "\u00a0", "\\space");
+defineSymbol(text$1, main, spacing, "\u00a0", "\\nobreakspace");
+defineSymbol(math, main, spacing, null, "\\nobreak");
+defineSymbol(math, main, spacing, null, "\\allowbreak");
+defineSymbol(math, main, punct, ",", ",");
+defineSymbol(math, main, punct, ";", ";");
+defineSymbol(math, ams, bin, "\u22bc", "\\barwedge", true);
+defineSymbol(math, ams, bin, "\u22bb", "\\veebar", true);
+defineSymbol(math, main, bin, "\u2299", "\\odot", true);
+defineSymbol(math, main, bin, "\u2295", "\\oplus", true);
+defineSymbol(math, main, bin, "\u2297", "\\otimes", true);
+defineSymbol(math, main, textord, "\u2202", "\\partial", true);
+defineSymbol(math, main, bin, "\u2298", "\\oslash", true);
+defineSymbol(math, ams, bin, "\u229a", "\\circledcirc", true);
+defineSymbol(math, ams, bin, "\u22a1", "\\boxdot", true);
+defineSymbol(math, main, bin, "\u25b3", "\\bigtriangleup");
+defineSymbol(math, main, bin, "\u25bd", "\\bigtriangledown");
+defineSymbol(math, main, bin, "\u2020", "\\dagger");
+defineSymbol(math, main, bin, "\u22c4", "\\diamond");
+defineSymbol(math, main, bin, "\u22c6", "\\star");
+defineSymbol(math, main, bin, "\u25c3", "\\triangleleft");
+defineSymbol(math, main, bin, "\u25b9", "\\triangleright");
+defineSymbol(math, main, open, "{", "\\{");
+defineSymbol(text$1, main, textord, "{", "\\{");
+defineSymbol(text$1, main, textord, "{", "\\textbraceleft");
+defineSymbol(math, main, close, "}", "\\}");
+defineSymbol(text$1, main, textord, "}", "\\}");
+defineSymbol(text$1, main, textord, "}", "\\textbraceright");
+defineSymbol(math, main, open, "{", "\\lbrace");
+defineSymbol(math, main, close, "}", "\\rbrace");
+defineSymbol(math, main, open, "[", "\\lbrack", true);
+defineSymbol(text$1, main, textord, "[", "\\lbrack", true);
+defineSymbol(math, main, close, "]", "\\rbrack", true);
+defineSymbol(text$1, main, textord, "]", "\\rbrack", true);
+defineSymbol(math, main, open, "(", "\\lparen", true);
+defineSymbol(math, main, close, ")", "\\rparen", true);
+defineSymbol(text$1, main, textord, "<", "\\textless", true); // in T1 fontenc
+
+defineSymbol(text$1, main, textord, ">", "\\textgreater", true); // in T1 fontenc
+
+defineSymbol(math, main, open, "\u230a", "\\lfloor", true);
+defineSymbol(math, main, close, "\u230b", "\\rfloor", true);
+defineSymbol(math, main, open, "\u2308", "\\lceil", true);
+defineSymbol(math, main, close, "\u2309", "\\rceil", true);
+defineSymbol(math, main, textord, "\\", "\\backslash");
+defineSymbol(math, main, textord, "\u2223", "|");
+defineSymbol(math, main, textord, "\u2223", "\\vert");
+defineSymbol(text$1, main, textord, "|", "\\textbar", true); // in T1 fontenc
+
+defineSymbol(math, main, textord, "\u2225", "\\|");
+defineSymbol(math, main, textord, "\u2225", "\\Vert");
+defineSymbol(text$1, main, textord, "\u2225", "\\textbardbl");
+defineSymbol(text$1, main, textord, "~", "\\textasciitilde");
+defineSymbol(text$1, main, textord, "\\", "\\textbackslash");
+defineSymbol(text$1, main, textord, "^", "\\textasciicircum");
+defineSymbol(math, main, rel, "\u2191", "\\uparrow", true);
+defineSymbol(math, main, rel, "\u21d1", "\\Uparrow", true);
+defineSymbol(math, main, rel, "\u2193", "\\downarrow", true);
+defineSymbol(math, main, rel, "\u21d3", "\\Downarrow", true);
+defineSymbol(math, main, rel, "\u2195", "\\updownarrow", true);
+defineSymbol(math, main, rel, "\u21d5", "\\Updownarrow", true);
+defineSymbol(math, main, op, "\u2210", "\\coprod");
+defineSymbol(math, main, op, "\u22c1", "\\bigvee");
+defineSymbol(math, main, op, "\u22c0", "\\bigwedge");
+defineSymbol(math, main, op, "\u2a04", "\\biguplus");
+defineSymbol(math, main, op, "\u22c2", "\\bigcap");
+defineSymbol(math, main, op, "\u22c3", "\\bigcup");
+defineSymbol(math, main, op, "\u222b", "\\int");
+defineSymbol(math, main, op, "\u222b", "\\intop");
+defineSymbol(math, main, op, "\u222c", "\\iint");
+defineSymbol(math, main, op, "\u222d", "\\iiint");
+defineSymbol(math, main, op, "\u220f", "\\prod");
+defineSymbol(math, main, op, "\u2211", "\\sum");
+defineSymbol(math, main, op, "\u2a02", "\\bigotimes");
+defineSymbol(math, main, op, "\u2a01", "\\bigoplus");
+defineSymbol(math, main, op, "\u2a00", "\\bigodot");
+defineSymbol(math, main, op, "\u222e", "\\oint");
+defineSymbol(math, main, op, "\u2a06", "\\bigsqcup");
+defineSymbol(math, main, op, "\u222b", "\\smallint");
+defineSymbol(text$1, main, inner, "\u2026", "\\textellipsis");
+defineSymbol(math, main, inner, "\u2026", "\\mathellipsis");
+defineSymbol(text$1, main, inner, "\u2026", "\\ldots", true);
+defineSymbol(math, main, inner, "\u2026", "\\ldots", true);
+defineSymbol(math, main, inner, "\u22ef", "\\@cdots", true);
+defineSymbol(math, main, inner, "\u22f1", "\\ddots", true);
+defineSymbol(math, main, textord, "\u22ee", "\\varvdots"); // \vdots is a macro
+
+defineSymbol(math, main, accent, "\u02ca", "\\acute");
+defineSymbol(math, main, accent, "\u02cb", "\\grave");
+defineSymbol(math, main, accent, "\u00a8", "\\ddot");
+defineSymbol(math, main, accent, "\u007e", "\\tilde");
+defineSymbol(math, main, accent, "\u02c9", "\\bar");
+defineSymbol(math, main, accent, "\u02d8", "\\breve");
+defineSymbol(math, main, accent, "\u02c7", "\\check");
+defineSymbol(math, main, accent, "\u005e", "\\hat");
+defineSymbol(math, main, accent, "\u20d7", "\\vec");
+defineSymbol(math, main, accent, "\u02d9", "\\dot");
+defineSymbol(math, main, accent, "\u02da", "\\mathring"); // \imath and \jmath should be invariant to \mathrm, \mathbf, etc., so use PUA
+
+defineSymbol(math, main, mathord, "\ue131", "\\@imath");
+defineSymbol(math, main, mathord, "\ue237", "\\@jmath");
+defineSymbol(math, main, textord, "\u0131", "\u0131");
+defineSymbol(math, main, textord, "\u0237", "\u0237");
+defineSymbol(text$1, main, textord, "\u0131", "\\i", true);
+defineSymbol(text$1, main, textord, "\u0237", "\\j", true);
+defineSymbol(text$1, main, textord, "\u00df", "\\ss", true);
+defineSymbol(text$1, main, textord, "\u00e6", "\\ae", true);
+defineSymbol(text$1, main, textord, "\u0153", "\\oe", true);
+defineSymbol(text$1, main, textord, "\u00f8", "\\o", true);
+defineSymbol(text$1, main, textord, "\u00c6", "\\AE", true);
+defineSymbol(text$1, main, textord, "\u0152", "\\OE", true);
+defineSymbol(text$1, main, textord, "\u00d8", "\\O", true);
+defineSymbol(text$1, main, accent, "\u02ca", "\\'"); // acute
+
+defineSymbol(text$1, main, accent, "\u02cb", "\\`"); // grave
+
+defineSymbol(text$1, main, accent, "\u02c6", "\\^"); // circumflex
+
+defineSymbol(text$1, main, accent, "\u02dc", "\\~"); // tilde
+
+defineSymbol(text$1, main, accent, "\u02c9", "\\="); // macron
+
+defineSymbol(text$1, main, accent, "\u02d8", "\\u"); // breve
+
+defineSymbol(text$1, main, accent, "\u02d9", "\\."); // dot above
+
+defineSymbol(text$1, main, accent, "\u02da", "\\r"); // ring above
+
+defineSymbol(text$1, main, accent, "\u02c7", "\\v"); // caron
+
+defineSymbol(text$1, main, accent, "\u00a8", '\\"'); // diaresis
+
+defineSymbol(text$1, main, accent, "\u02dd", "\\H"); // double acute
+
+defineSymbol(text$1, main, accent, "\u25ef", "\\textcircled"); // \bigcirc glyph
+// These ligatures are detected and created in Parser.js's `formLigatures`.
+
+const ligatures = {
+  "--": true,
+  "---": true,
+  "``": true,
+  "''": true
+};
+defineSymbol(text$1, main, textord, "\u2013", "--", true);
+defineSymbol(text$1, main, textord, "\u2013", "\\textendash");
+defineSymbol(text$1, main, textord, "\u2014", "---", true);
+defineSymbol(text$1, main, textord, "\u2014", "\\textemdash");
+defineSymbol(text$1, main, textord, "\u2018", "`", true);
+defineSymbol(text$1, main, textord, "\u2018", "\\textquoteleft");
+defineSymbol(text$1, main, textord, "\u2019", "'", true);
+defineSymbol(text$1, main, textord, "\u2019", "\\textquoteright");
+defineSymbol(text$1, main, textord, "\u201c", "``", true);
+defineSymbol(text$1, main, textord, "\u201c", "\\textquotedblleft");
+defineSymbol(text$1, main, textord, "\u201d", "''", true);
+defineSymbol(text$1, main, textord, "\u201d", "\\textquotedblright"); //  \degree from gensymb package
+
+defineSymbol(math, main, textord, "\u00b0", "\\degree", true);
+defineSymbol(text$1, main, textord, "\u00b0", "\\degree"); // \textdegree from inputenc package
+
+defineSymbol(text$1, main, textord, "\u00b0", "\\textdegree", true); // TODO: In LaTeX, \pounds can generate a different character in text and math
+// mode, but among our fonts, only Main-Regular defines this character "163".
+
+defineSymbol(math, main, textord, "\u00a3", "\\pounds");
+defineSymbol(math, main, textord, "\u00a3", "\\mathsterling", true);
+defineSymbol(text$1, main, textord, "\u00a3", "\\pounds");
+defineSymbol(text$1, main, textord, "\u00a3", "\\textsterling", true);
+defineSymbol(math, ams, textord, "\u2720", "\\maltese");
+defineSymbol(text$1, ams, textord, "\u2720", "\\maltese"); // There are lots of symbols which are the same, so we add them in afterwards.
+// All of these are textords in math mode
+
+const mathTextSymbols = "0123456789/@.\"";
+
+for (let i = 0; i < mathTextSymbols.length; i++) {
+  const ch = mathTextSymbols.charAt(i);
+  defineSymbol(math, main, textord, ch, ch);
+} // All of these are textords in text mode
+
+
+const textSymbols = "0123456789!@*()-=+\";:?/.,";
+
+for (let i = 0; i < textSymbols.length; i++) {
+  const ch = textSymbols.charAt(i);
+  defineSymbol(text$1, main, textord, ch, ch);
+} // All of these are textords in text mode, and mathords in math mode
+
+
+const letters = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
+
+for (let i = 0; i < letters.length; i++) {
+  const ch = letters.charAt(i);
+  defineSymbol(math, main, mathord, ch, ch);
+  defineSymbol(text$1, main, textord, ch, ch);
+} // Blackboard bold and script letters in Unicode range
+
+
+defineSymbol(math, ams, textord, "C", "\u2102"); // blackboard bold
+
+defineSymbol(text$1, ams, textord, "C", "\u2102");
+defineSymbol(math, ams, textord, "H", "\u210D");
+defineSymbol(text$1, ams, textord, "H", "\u210D");
+defineSymbol(math, ams, textord, "N", "\u2115");
+defineSymbol(text$1, ams, textord, "N", "\u2115");
+defineSymbol(math, ams, textord, "P", "\u2119");
+defineSymbol(text$1, ams, textord, "P", "\u2119");
+defineSymbol(math, ams, textord, "Q", "\u211A");
+defineSymbol(text$1, ams, textord, "Q", "\u211A");
+defineSymbol(math, ams, textord, "R", "\u211D");
+defineSymbol(text$1, ams, textord, "R", "\u211D");
+defineSymbol(math, ams, textord, "Z", "\u2124");
+defineSymbol(text$1, ams, textord, "Z", "\u2124");
+defineSymbol(math, main, mathord, "h", "\u210E"); // italic h, Planck constant
+
+defineSymbol(text$1, main, mathord, "h", "\u210E"); // The next loop loads wide (surrogate pair) characters.
+// We support some letters in the Unicode range U+1D400 to U+1D7FF,
+// Mathematical Alphanumeric Symbols.
+// Some editors do not deal well with wide characters. So don't write the
+// string into this file. Instead, create the string from the surrogate pair.
+
+let wideChar = "";
+
+for (let i = 0; i < letters.length; i++) {
+  const ch = letters.charAt(i); // The hex numbers in the next line are a surrogate pair.
+  // 0xD835 is the high surrogate for all letters in the range we support.
+  // 0xDC00 is the low surrogate for bold A.
+
+  wideChar = String.fromCharCode(0xD835, 0xDC00 + i); // A-Z a-z bold
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDC34 + i); // A-Z a-z italic
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDC68 + i); // A-Z a-z bold italic
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDD04 + i); // A-Z a-z Fractur
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDDA0 + i); // A-Z a-z sans-serif
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDDD4 + i); // A-Z a-z sans bold
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDE08 + i); // A-Z a-z sans italic
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDE70 + i); // A-Z a-z monospace
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+
+  if (i < 26) {
+    // KaTeX fonts have only capital letters for blackboard bold and script.
+    // See exception for k below.
+    wideChar = String.fromCharCode(0xD835, 0xDD38 + i); // A-Z double struck
+
+    defineSymbol(math, main, mathord, ch, wideChar);
+    defineSymbol(text$1, main, textord, ch, wideChar);
+    wideChar = String.fromCharCode(0xD835, 0xDC9C + i); // A-Z script
+
+    defineSymbol(math, main, mathord, ch, wideChar);
+    defineSymbol(text$1, main, textord, ch, wideChar);
+  } // TODO: Add bold script when it is supported by a KaTeX font.
+
+} // "k" is the only double struck lower case letter in the KaTeX fonts.
+
+
+wideChar = String.fromCharCode(0xD835, 0xDD5C); // k double struck
+
+defineSymbol(math, main, mathord, "k", wideChar);
+defineSymbol(text$1, main, textord, "k", wideChar); // Next, some wide character numerals
+
+for (let i = 0; i < 10; i++) {
+  const ch = i.toString();
+  wideChar = String.fromCharCode(0xD835, 0xDFCE + i); // 0-9 bold
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDFE2 + i); // 0-9 sans serif
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDFEC + i); // 0-9 bold sans
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+  wideChar = String.fromCharCode(0xD835, 0xDFF6 + i); // 0-9 monospace
+
+  defineSymbol(math, main, mathord, ch, wideChar);
+  defineSymbol(text$1, main, textord, ch, wideChar);
+} // We add these Latin-1 letters as symbols for backwards-compatibility,
+// but they are not actually in the font, nor are they supported by the
+// Unicode accent mechanism, so they fall back to Times font and look ugly.
+// TODO(edemaine): Fix this.
+
+
+const extraLatin = "\u00c7\u00d0\u00de\u00e7\u00fe";
+
+for (let i = 0; i < extraLatin.length; i++) {
+  const ch = extraLatin.charAt(i);
+  defineSymbol(math, main, mathord, ch, ch);
+  defineSymbol(text$1, main, textord, ch, ch);
+}
+
+/**
+ * This file provides support for Unicode range U+1D400 to U+1D7FF,
+ * Mathematical Alphanumeric Symbols.
+ *
+ * Function wideCharacterFont takes a wide character as input and returns
+ * the font information necessary to render it properly.
+ */
+/**
+ * Data below is from https://www.unicode.org/charts/PDF/U1D400.pdf
+ * That document sorts characters into groups by font type, say bold or italic.
+ *
+ * In the arrays below, each subarray consists three elements:
+ *      * The CSS class of that group when in math mode.
+ *      * The CSS class of that group when in text mode.
+ *      * The font name, so that KaTeX can get font metrics.
+ */
+
+const wideLatinLetterData = [["mathbf", "textbf", "Main-Bold"], // A-Z bold upright
+["mathbf", "textbf", "Main-Bold"], // a-z bold upright
+["mathnormal", "textit", "Math-Italic"], // A-Z italic
+["mathnormal", "textit", "Math-Italic"], // a-z italic
+["boldsymbol", "boldsymbol", "Main-BoldItalic"], // A-Z bold italic
+["boldsymbol", "boldsymbol", "Main-BoldItalic"], // a-z bold italic
+// Map fancy A-Z letters to script, not calligraphic.
+// This aligns with unicode-math and math fonts (except Cambria Math).
+["mathscr", "textscr", "Script-Regular"], // A-Z script
+["", "", ""], // a-z script.  No font
+["", "", ""], // A-Z bold script. No font
+["", "", ""], // a-z bold script. No font
+["mathfrak", "textfrak", "Fraktur-Regular"], // A-Z Fraktur
+["mathfrak", "textfrak", "Fraktur-Regular"], // a-z Fraktur
+["mathbb", "textbb", "AMS-Regular"], // A-Z double-struck
+["mathbb", "textbb", "AMS-Regular"], // k double-struck
+["", "", ""], // A-Z bold Fraktur No font metrics
+["", "", ""], // a-z bold Fraktur.   No font.
+["mathsf", "textsf", "SansSerif-Regular"], // A-Z sans-serif
+["mathsf", "textsf", "SansSerif-Regular"], // a-z sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // A-Z bold sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // a-z bold sans-serif
+["mathitsf", "textitsf", "SansSerif-Italic"], // A-Z italic sans-serif
+["mathitsf", "textitsf", "SansSerif-Italic"], // a-z italic sans-serif
+["", "", ""], // A-Z bold italic sans. No font
+["", "", ""], // a-z bold italic sans. No font
+["mathtt", "texttt", "Typewriter-Regular"], // A-Z monospace
+["mathtt", "texttt", "Typewriter-Regular"]];
+const wideNumeralData = [["mathbf", "textbf", "Main-Bold"], // 0-9 bold
+["", "", ""], // 0-9 double-struck. No KaTeX font.
+["mathsf", "textsf", "SansSerif-Regular"], // 0-9 sans-serif
+["mathboldsf", "textboldsf", "SansSerif-Bold"], // 0-9 bold sans-serif
+["mathtt", "texttt", "Typewriter-Regular"]];
+const wideCharacterFont = function wideCharacterFont(wideChar, mode) {
+  // IE doesn't support codePointAt(). So work with the surrogate pair.
+  const H = wideChar.charCodeAt(0); // high surrogate
+
+  const L = wideChar.charCodeAt(1); // low surrogate
+
+  const codePoint = (H - 0xD800) * 0x400 + (L - 0xDC00) + 0x10000;
+  const j = mode === "math" ? 0 : 1; // column index for CSS class.
+
+  if (0x1D400 <= codePoint && codePoint < 0x1D6A4) {
+    // wideLatinLetterData contains exactly 26 chars on each row.
+    // So we can calculate the relevant row. No traverse necessary.
+    const i = Math.floor((codePoint - 0x1D400) / 26);
+    return [wideLatinLetterData[i][2], wideLatinLetterData[i][j]];
+  } else if (0x1D7CE <= codePoint && codePoint <= 0x1D7FF) {
+    // Numerals, ten per row.
+    const i = Math.floor((codePoint - 0x1D7CE) / 10);
+    return [wideNumeralData[i][2], wideNumeralData[i][j]];
+  } else if (codePoint === 0x1D6A5 || codePoint === 0x1D6A6) {
+    // dotless i or j
+    return [wideLatinLetterData[0][2], wideLatinLetterData[0][j]];
+  } else if (0x1D6A6 < codePoint && codePoint < 0x1D7CE) {
+    // Greek letters. Not supported, yet.
+    return ["", ""];
+  } else {
+    // We don't support any wide characters outside 1D400–1D7FF.
+    throw new ParseError("Unsupported character: " + wideChar);
+  }
+};
+
+/**
+ * This file contains information about the options that the Parser carries
+ * around with it while parsing. Data is held in an `Options` object, and when
+ * recursing, a new `Options` object can be created with the `.with*` and
+ * `.reset` functions.
+ */
+const sizeStyleMap = [// Each element contains [textsize, scriptsize, scriptscriptsize].
+// The size mappings are taken from TeX with \normalsize=10pt.
+[1, 1, 1], // size1: [5, 5, 5]              \tiny
+[2, 1, 1], // size2: [6, 5, 5]
+[3, 1, 1], // size3: [7, 5, 5]              \scriptsize
+[4, 2, 1], // size4: [8, 6, 5]              \footnotesize
+[5, 2, 1], // size5: [9, 6, 5]              \small
+[6, 3, 1], // size6: [10, 7, 5]             \normalsize
+[7, 4, 2], // size7: [12, 8, 6]             \large
+[8, 6, 3], // size8: [14.4, 10, 7]          \Large
+[9, 7, 6], // size9: [17.28, 12, 10]        \LARGE
+[10, 8, 7], // size10: [20.74, 14.4, 12]     \huge
+[11, 10, 9]];
+const sizeMultipliers = [// fontMetrics.js:getGlobalMetrics also uses size indexes, so if
+// you change size indexes, change that function.
+0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.44, 1.728, 2.074, 2.488];
+
+const sizeAtStyle = function sizeAtStyle(size, style) {
+  return style.size < 2 ? size : sizeStyleMap[size - 1][style.size - 1];
+}; // In these types, "" (empty string) means "no change".
+
+
+/**
+ * This is the main options class. It contains the current style, size, color,
+ * and font.
+ *
+ * Options objects should not be modified. To create a new Options with
+ * different properties, call a `.having*` method.
+ */
+class Options {
+  // A font family applies to a group of fonts (i.e. SansSerif), while a font
+  // represents a specific font (i.e. SansSerif Bold).
+  // See: https://tex.stackexchange.com/questions/22350/difference-between-textrm-and-mathrm
+
+  /**
+   * The base size index.
+   */
+  constructor(data) {
+    this.style = void 0;
+    this.color = void 0;
+    this.size = void 0;
+    this.textSize = void 0;
+    this.phantom = void 0;
+    this.font = void 0;
+    this.fontFamily = void 0;
+    this.fontWeight = void 0;
+    this.fontShape = void 0;
+    this.sizeMultiplier = void 0;
+    this.maxSize = void 0;
+    this.minRuleThickness = void 0;
+    this._fontMetrics = void 0;
+    this.style = data.style;
+    this.color = data.color;
+    this.size = data.size || Options.BASESIZE;
+    this.textSize = data.textSize || this.size;
+    this.phantom = !!data.phantom;
+    this.font = data.font || "";
+    this.fontFamily = data.fontFamily || "";
+    this.fontWeight = data.fontWeight || '';
+    this.fontShape = data.fontShape || '';
+    this.sizeMultiplier = sizeMultipliers[this.size - 1];
+    this.maxSize = data.maxSize;
+    this.minRuleThickness = data.minRuleThickness;
+    this._fontMetrics = undefined;
+  }
+  /**
+   * Returns a new options object with the same properties as "this".  Properties
+   * from "extension" will be copied to the new options object.
+   */
+
+
+  extend(extension) {
+    const data = {
+      style: this.style,
+      size: this.size,
+      textSize: this.textSize,
+      color: this.color,
+      phantom: this.phantom,
+      font: this.font,
+      fontFamily: this.fontFamily,
+      fontWeight: this.fontWeight,
+      fontShape: this.fontShape,
+      maxSize: this.maxSize,
+      minRuleThickness: this.minRuleThickness
+    };
+
+    for (const key in extension) {
+      if (extension.hasOwnProperty(key)) {
+        data[key] = extension[key];
+      }
+    }
+
+    return new Options(data);
+  }
+  /**
+   * Return an options object with the given style. If `this.style === style`,
+   * returns `this`.
+   */
+
+
+  havingStyle(style) {
+    if (this.style === style) {
+      return this;
+    } else {
+      return this.extend({
+        style: style,
+        size: sizeAtStyle(this.textSize, style)
+      });
+    }
+  }
+  /**
+   * Return an options object with a cramped version of the current style. If
+   * the current style is cramped, returns `this`.
+   */
+
+
+  havingCrampedStyle() {
+    return this.havingStyle(this.style.cramp());
+  }
+  /**
+   * Return an options object with the given size and in at least `\textstyle`.
+   * Returns `this` if appropriate.
+   */
+
+
+  havingSize(size) {
+    if (this.size === size && this.textSize === size) {
+      return this;
+    } else {
+      return this.extend({
+        style: this.style.text(),
+        size: size,
+        textSize: size,
+        sizeMultiplier: sizeMultipliers[size - 1]
+      });
+    }
+  }
+  /**
+   * Like `this.havingSize(BASESIZE).havingStyle(style)`. If `style` is omitted,
+   * changes to at least `\textstyle`.
+   */
+
+
+  havingBaseStyle(style) {
+    style = style || this.style.text();
+    const wantSize = sizeAtStyle(Options.BASESIZE, style);
+
+    if (this.size === wantSize && this.textSize === Options.BASESIZE && this.style === style) {
+      return this;
+    } else {
+      return this.extend({
+        style: style,
+        size: wantSize
+      });
+    }
+  }
+  /**
+   * Remove the effect of sizing changes such as \Huge.
+   * Keep the effect of the current style, such as \scriptstyle.
+   */
+
+
+  havingBaseSizing() {
+    let size;
+
+    switch (this.style.id) {
+      case 4:
+      case 5:
+        size = 3; // normalsize in scriptstyle
+
+        break;
+
+      case 6:
+      case 7:
+        size = 1; // normalsize in scriptscriptstyle
+
+        break;
+
+      default:
+        size = 6;
+      // normalsize in textstyle or displaystyle
+    }
+
+    return this.extend({
+      style: this.style.text(),
+      size: size
+    });
+  }
+  /**
+   * Create a new options object with the given color.
+   */
+
+
+  withColor(color) {
+    return this.extend({
+      color: color
+    });
+  }
+  /**
+   * Create a new options object with "phantom" set to true.
+   */
+
+
+  withPhantom() {
+    return this.extend({
+      phantom: true
+    });
+  }
+  /**
+   * Creates a new options object with the given math font or old text font.
+   * @type {[type]}
+   */
+
+
+  withFont(font) {
+    return this.extend({
+      font
+    });
+  }
+  /**
+   * Create a new options objects with the given fontFamily.
+   */
+
+
+  withTextFontFamily(fontFamily) {
+    return this.extend({
+      fontFamily,
+      font: ""
+    });
+  }
+  /**
+   * Creates a new options object with the given font weight
+   */
+
+
+  withTextFontWeight(fontWeight) {
+    return this.extend({
+      fontWeight,
+      font: ""
+    });
+  }
+  /**
+   * Creates a new options object with the given font weight
+   */
+
+
+  withTextFontShape(fontShape) {
+    return this.extend({
+      fontShape,
+      font: ""
+    });
+  }
+  /**
+   * Return the CSS sizing classes required to switch from enclosing options
+   * `oldOptions` to `this`. Returns an array of classes.
+   */
+
+
+  sizingClasses(oldOptions) {
+    if (oldOptions.size !== this.size) {
+      return ["sizing", "reset-size" + oldOptions.size, "size" + this.size];
+    } else {
+      return [];
+    }
+  }
+  /**
+   * Return the CSS sizing classes required to switch to the base size. Like
+   * `this.havingSize(BASESIZE).sizingClasses(this)`.
+   */
+
+
+  baseSizingClasses() {
+    if (this.size !== Options.BASESIZE) {
+      return ["sizing", "reset-size" + this.size, "size" + Options.BASESIZE];
+    } else {
+      return [];
+    }
+  }
+  /**
+   * Return the font metrics for this size.
+   */
+
+
+  fontMetrics() {
+    if (!this._fontMetrics) {
+      this._fontMetrics = getGlobalMetrics(this.size);
+    }
+
+    return this._fontMetrics;
+  }
+  /**
+   * Gets the CSS color of the current options object
+   */
+
+
+  getColor() {
+    if (this.phantom) {
+      return "transparent";
+    } else {
+      return this.color;
+    }
+  }
+
+}
+
+Options.BASESIZE = 6;
+
+/**
+ * This file does conversion between units.  In particular, it provides
+ * calculateSize to convert other units into ems.
+ */
+// Thus, multiplying a length by this number converts the length from units
+// into pts.  Dividing the result by ptPerEm gives the number of ems
+// *assuming* a font size of ptPerEm (normal size, normal style).
+
+const ptPerUnit = {
+  // https://en.wikibooks.org/wiki/LaTeX/Lengths and
+  // https://tex.stackexchange.com/a/8263
+  "pt": 1,
+  // TeX point
+  "mm": 7227 / 2540,
+  // millimeter
+  "cm": 7227 / 254,
+  // centimeter
+  "in": 72.27,
+  // inch
+  "bp": 803 / 800,
+  // big (PostScript) points
+  "pc": 12,
+  // pica
+  "dd": 1238 / 1157,
+  // didot
+  "cc": 14856 / 1157,
+  // cicero (12 didot)
+  "nd": 685 / 642,
+  // new didot
+  "nc": 1370 / 107,
+  // new cicero (12 new didot)
+  "sp": 1 / 65536,
+  // scaled point (TeX's internal smallest unit)
+  // https://tex.stackexchange.com/a/41371
+  "px": 803 / 800 // \pdfpxdimen defaults to 1 bp in pdfTeX and LuaTeX
+
+}; // Dictionary of relative units, for fast validity testing.
+
+const relativeUnit = {
+  "ex": true,
+  "em": true,
+  "mu": true
+};
+
+/**
+ * Determine whether the specified unit (either a string defining the unit
+ * or a "size" parse node containing a unit field) is valid.
+ */
+const validUnit = function validUnit(unit) {
+  if (typeof unit !== "string") {
+    unit = unit.unit;
+  }
+
+  return unit in ptPerUnit || unit in relativeUnit || unit === "ex";
+};
+/*
+ * Convert a "size" parse node (with numeric "number" and string "unit" fields,
+ * as parsed by functions.js argType "size") into a CSS em value for the
+ * current style/scale.  `options` gives the current options.
+ */
+
+const calculateSize = function calculateSize(sizeValue, options) {
+  let scale;
+
+  if (sizeValue.unit in ptPerUnit) {
+    // Absolute units
+    scale = ptPerUnit[sizeValue.unit] // Convert unit to pt
+    / options.fontMetrics().ptPerEm // Convert pt to CSS em
+    / options.sizeMultiplier; // Unscale to make absolute units
+  } else if (sizeValue.unit === "mu") {
+    // `mu` units scale with scriptstyle/scriptscriptstyle.
+    scale = options.fontMetrics().cssEmPerMu;
+  } else {
+    // Other relative units always refer to the *textstyle* font
+    // in the current size.
+    let unitOptions;
+
+    if (options.style.isTight()) {
+      // isTight() means current style is script/scriptscript.
+      unitOptions = options.havingStyle(options.style.text());
+    } else {
+      unitOptions = options;
+    } // TODO: In TeX these units are relative to the quad of the current
+    // *text* font, e.g. cmr10. KaTeX instead uses values from the
+    // comparably-sized *Computer Modern symbol* font. At 10pt, these
+    // match. At 7pt and 5pt, they differ: cmr7=1.138894, cmsy7=1.170641;
+    // cmr5=1.361133, cmsy5=1.472241. Consider $\scriptsize a\kern1emb$.
+    // TeX \showlists shows a kern of 1.13889 * fontsize;
+    // KaTeX shows a kern of 1.171 * fontsize.
+
+
+    if (sizeValue.unit === "ex") {
+      scale = unitOptions.fontMetrics().xHeight;
+    } else if (sizeValue.unit === "em") {
+      scale = unitOptions.fontMetrics().quad;
+    } else {
+      throw new ParseError("Invalid unit: '" + sizeValue.unit + "'");
+    }
+
+    if (unitOptions !== options) {
+      scale *= unitOptions.sizeMultiplier / options.sizeMultiplier;
+    }
+  }
+
+  return Math.min(sizeValue.number * scale, options.maxSize);
+};
+
+/* eslint no-console:0 */
+
+/**
+ * Looks up the given symbol in fontMetrics, after applying any symbol
+ * replacements defined in symbol.js
+ */
+const lookupSymbol = function lookupSymbol(value, // TODO(#963): Use a union type for this.
+fontName, mode) {
+  // Replace the value with its replaced value from symbol.js
+  if (symbols[mode][value] && symbols[mode][value].replace) {
+    value = symbols[mode][value].replace;
+  }
+
+  return {
+    value: value,
+    metrics: getCharacterMetrics(value, fontName, mode)
+  };
+};
+/**
+ * Makes a symbolNode after translation via the list of symbols in symbols.js.
+ * Correctly pulls out metrics for the character, and optionally takes a list of
+ * classes to be attached to the node.
+ *
+ * TODO: make argument order closer to makeSpan
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ * TODO(#953): Make `options` mandatory and always pass it in.
+ */
+
+
+const makeSymbol = function makeSymbol(value, fontName, mode, options, classes) {
+  const lookup = lookupSymbol(value, fontName, mode);
+  const metrics = lookup.metrics;
+  value = lookup.value;
+  let symbolNode;
+
+  if (metrics) {
+    let italic = metrics.italic;
+
+    if (mode === "text" || options && options.font === "mathit") {
+      italic = 0;
+    }
+
+    symbolNode = new SymbolNode(value, metrics.height, metrics.depth, italic, metrics.skew, metrics.width, classes);
+  } else {
+    // TODO(emily): Figure out a good way to only print this in development
+    typeof console !== "undefined" && console.warn("No character metrics " + `for '${value}' in style '${fontName}' and mode '${mode}'`);
+    symbolNode = new SymbolNode(value, 0, 0, 0, 0, 0, classes);
+  }
+
+  if (options) {
+    symbolNode.maxFontSize = options.sizeMultiplier;
+
+    if (options.style.isTight()) {
+      symbolNode.classes.push("mtight");
+    }
+
+    const color = options.getColor();
+
+    if (color) {
+      symbolNode.style.color = color;
+    }
+  }
+
+  return symbolNode;
+};
+/**
+ * Makes a symbol in Main-Regular or AMS-Regular.
+ * Used for rel, bin, open, close, inner, and punct.
+ */
+
+
+const mathsym = function mathsym(value, mode, options, classes) {
+  if (classes === void 0) {
+    classes = [];
+  }
+
+  // Decide what font to render the symbol in by its entry in the symbols
+  // table.
+  // Have a special case for when the value = \ because the \ is used as a
+  // textord in unsupported command errors but cannot be parsed as a regular
+  // text ordinal and is therefore not present as a symbol in the symbols
+  // table for text, as well as a special case for boldsymbol because it
+  // can be used for bold + and -
+  if (options.font === "boldsymbol" && lookupSymbol(value, "Main-Bold", mode).metrics) {
+    return makeSymbol(value, "Main-Bold", mode, options, classes.concat(["mathbf"]));
+  } else if (value === "\\" || symbols[mode][value].font === "main") {
+    return makeSymbol(value, "Main-Regular", mode, options, classes);
+  } else {
+    return makeSymbol(value, "AMS-Regular", mode, options, classes.concat(["amsrm"]));
+  }
+};
+/**
+ * Determines which of the two font names (Main-Bold and Math-BoldItalic) and
+ * corresponding style tags (mathbf or boldsymbol) to use for font "boldsymbol",
+ * depending on the symbol.  Use this function instead of fontMap for font
+ * "boldsymbol".
+ */
+
+
+const boldsymbol = function boldsymbol(value, mode, options, classes, type) {
+  if (type !== "textord" && lookupSymbol(value, "Math-BoldItalic", mode).metrics) {
+    return {
+      fontName: "Math-BoldItalic",
+      fontClass: "boldsymbol"
+    };
+  } else {
+    // Some glyphs do not exist in Math-BoldItalic so we need to use
+    // Main-Bold instead.
+    return {
+      fontName: "Main-Bold",
+      fontClass: "mathbf"
+    };
+  }
+};
+/**
+ * Makes either a mathord or textord in the correct font and color.
+ */
+
+
+const makeOrd = function makeOrd(group, options, type) {
+  const mode = group.mode;
+  const text = group.text;
+  const classes = ["mord"]; // Math mode or Old font (i.e. \rm)
+
+  const isFont = mode === "math" || mode === "text" && options.font;
+  const fontOrFamily = isFont ? options.font : options.fontFamily;
+
+  if (text.charCodeAt(0) === 0xD835) {
+    // surrogate pairs get special treatment
+    const _wideCharacterFont = wideCharacterFont(text, mode),
+          wideFontName = _wideCharacterFont[0],
+          wideFontClass = _wideCharacterFont[1];
+
+    return makeSymbol(text, wideFontName, mode, options, classes.concat(wideFontClass));
+  } else if (fontOrFamily) {
+    let fontName;
+    let fontClasses;
+
+    if (fontOrFamily === "boldsymbol") {
+      const fontData = boldsymbol(text, mode, options, classes, type);
+      fontName = fontData.fontName;
+      fontClasses = [fontData.fontClass];
+    } else if (isFont) {
+      fontName = fontMap[fontOrFamily].fontName;
+      fontClasses = [fontOrFamily];
+    } else {
+      fontName = retrieveTextFontName(fontOrFamily, options.fontWeight, options.fontShape);
+      fontClasses = [fontOrFamily, options.fontWeight, options.fontShape];
+    }
+
+    if (lookupSymbol(text, fontName, mode).metrics) {
+      return makeSymbol(text, fontName, mode, options, classes.concat(fontClasses));
+    } else if (ligatures.hasOwnProperty(text) && fontName.substr(0, 10) === "Typewriter") {
+      // Deconstruct ligatures in monospace fonts (\texttt, \tt).
+      const parts = [];
+
+      for (let i = 0; i < text.length; i++) {
+        parts.push(makeSymbol(text[i], fontName, mode, options, classes.concat(fontClasses)));
+      }
+
+      return makeFragment(parts);
+    }
+  } // Makes a symbol in the default font for mathords and textords.
+
+
+  if (type === "mathord") {
+    return makeSymbol(text, "Math-Italic", mode, options, classes.concat(["mathnormal"]));
+  } else if (type === "textord") {
+    const font = symbols[mode][text] && symbols[mode][text].font;
+
+    if (font === "ams") {
+      const fontName = retrieveTextFontName("amsrm", options.fontWeight, options.fontShape);
+      return makeSymbol(text, fontName, mode, options, classes.concat("amsrm", options.fontWeight, options.fontShape));
+    } else if (font === "main" || !font) {
+      const fontName = retrieveTextFontName("textrm", options.fontWeight, options.fontShape);
+      return makeSymbol(text, fontName, mode, options, classes.concat(options.fontWeight, options.fontShape));
+    } else {
+      // fonts added by plugins
+      const fontName = retrieveTextFontName(font, options.fontWeight, options.fontShape); // We add font name as a css class
+
+      return makeSymbol(text, fontName, mode, options, classes.concat(fontName, options.fontWeight, options.fontShape));
+    }
+  } else {
+    throw new Error("unexpected type: " + type + " in makeOrd");
+  }
+};
+/**
+ * Returns true if subsequent symbolNodes have the same classes, skew, maxFont,
+ * and styles.
+ */
+
+
+const canCombine = (prev, next) => {
+  if (createClass(prev.classes) !== createClass(next.classes) || prev.skew !== next.skew || prev.maxFontSize !== next.maxFontSize) {
+    return false;
+  }
+
+  for (const style in prev.style) {
+    if (prev.style.hasOwnProperty(style) && prev.style[style] !== next.style[style]) {
+      return false;
+    }
+  }
+
+  for (const style in next.style) {
+    if (next.style.hasOwnProperty(style) && prev.style[style] !== next.style[style]) {
+      return false;
+    }
+  }
+
+  return true;
+};
+/**
+ * Combine consequetive domTree.symbolNodes into a single symbolNode.
+ * Note: this function mutates the argument.
+ */
+
+
+const tryCombineChars = chars => {
+  for (let i = 0; i < chars.length - 1; i++) {
+    const prev = chars[i];
+    const next = chars[i + 1];
+
+    if (prev instanceof SymbolNode && next instanceof SymbolNode && canCombine(prev, next)) {
+      prev.text += next.text;
+      prev.height = Math.max(prev.height, next.height);
+      prev.depth = Math.max(prev.depth, next.depth); // Use the last character's italic correction since we use
+      // it to add padding to the right of the span created from
+      // the combined characters.
+
+      prev.italic = next.italic;
+      chars.splice(i + 1, 1);
+      i--;
+    }
+  }
+
+  return chars;
+};
+/**
+ * Calculate the height, depth, and maxFontSize of an element based on its
+ * children.
+ */
+
+
+const sizeElementFromChildren = function sizeElementFromChildren(elem) {
+  let height = 0;
+  let depth = 0;
+  let maxFontSize = 0;
+
+  for (let i = 0; i < elem.children.length; i++) {
+    const child = elem.children[i];
+
+    if (child.height > height) {
+      height = child.height;
+    }
+
+    if (child.depth > depth) {
+      depth = child.depth;
+    }
+
+    if (child.maxFontSize > maxFontSize) {
+      maxFontSize = child.maxFontSize;
+    }
+  }
+
+  elem.height = height;
+  elem.depth = depth;
+  elem.maxFontSize = maxFontSize;
+};
+/**
+ * Makes a span with the given list of classes, list of children, and options.
+ *
+ * TODO(#953): Ensure that `options` is always provided (currently some call
+ * sites don't pass it) and make the type below mandatory.
+ * TODO: add a separate argument for math class (e.g. `mop`, `mbin`), which
+ * should if present come first in `classes`.
+ */
+
+
+const makeSpan = function makeSpan(classes, children, options, style) {
+  const span = new Span(classes, children, options, style);
+  sizeElementFromChildren(span);
+  return span;
+}; // SVG one is simpler -- doesn't require height, depth, max-font setting.
+// This is also a separate method for typesafety.
+
+
+const makeSvgSpan = (classes, children, options, style) => new Span(classes, children, options, style);
+
+const makeLineSpan = function makeLineSpan(className, options, thickness) {
+  const line = makeSpan([className], [], options);
+  line.height = Math.max(thickness || options.fontMetrics().defaultRuleThickness, options.minRuleThickness);
+  line.style.borderBottomWidth = line.height + "em";
+  line.maxFontSize = 1.0;
+  return line;
+};
+/**
+ * Makes an anchor with the given href, list of classes, list of children,
+ * and options.
+ */
+
+
+const makeAnchor = function makeAnchor(href, classes, children, options) {
+  const anchor = new Anchor(href, classes, children, options);
+  sizeElementFromChildren(anchor);
+  return anchor;
+};
+/**
+ * Makes a document fragment with the given list of children.
+ */
+
+
+const makeFragment = function makeFragment(children) {
+  const fragment = new DocumentFragment(children);
+  sizeElementFromChildren(fragment);
+  return fragment;
+};
+/**
+ * Wraps group in a span if it's a document fragment, allowing to apply classes
+ * and styles
+ */
+
+
+const wrapFragment = function wrapFragment(group, options) {
+  if (group instanceof DocumentFragment) {
+    return makeSpan([], [group], options);
+  }
+
+  return group;
+}; // These are exact object types to catch typos in the names of the optional fields.
+
+
+// Computes the updated `children` list and the overall depth.
+//
+// This helper function for makeVList makes it easier to enforce type safety by
+// allowing early exits (returns) in the logic.
+const getVListChildrenAndDepth = function getVListChildrenAndDepth(params) {
+  if (params.positionType === "individualShift") {
+    const oldChildren = params.children;
+    const children = [oldChildren[0]]; // Add in kerns to the list of params.children to get each element to be
+    // shifted to the correct specified shift
+
+    const depth = -oldChildren[0].shift - oldChildren[0].elem.depth;
+    let currPos = depth;
+
+    for (let i = 1; i < oldChildren.length; i++) {
+      const diff = -oldChildren[i].shift - currPos - oldChildren[i].elem.depth;
+      const size = diff - (oldChildren[i - 1].elem.height + oldChildren[i - 1].elem.depth);
+      currPos = currPos + diff;
+      children.push({
+        type: "kern",
+        size
+      });
+      children.push(oldChildren[i]);
+    }
+
+    return {
+      children,
+      depth
+    };
+  }
+
+  let depth;
+
+  if (params.positionType === "top") {
+    // We always start at the bottom, so calculate the bottom by adding up
+    // all the sizes
+    let bottom = params.positionData;
+
+    for (let i = 0; i < params.children.length; i++) {
+      const child = params.children[i];
+      bottom -= child.type === "kern" ? child.size : child.elem.height + child.elem.depth;
+    }
+
+    depth = bottom;
+  } else if (params.positionType === "bottom") {
+    depth = -params.positionData;
+  } else {
+    const firstChild = params.children[0];
+
+    if (firstChild.type !== "elem") {
+      throw new Error('First child must have type "elem".');
+    }
+
+    if (params.positionType === "shift") {
+      depth = -firstChild.elem.depth - params.positionData;
+    } else if (params.positionType === "firstBaseline") {
+      depth = -firstChild.elem.depth;
+    } else {
+      throw new Error(`Invalid positionType ${params.positionType}.`);
+    }
+  }
+
+  return {
+    children: params.children,
+    depth
+  };
+};
+/**
+ * Makes a vertical list by stacking elements and kerns on top of each other.
+ * Allows for many different ways of specifying the positioning method.
+ *
+ * See VListParam documentation above.
+ */
+
+
+const makeVList = function makeVList(params, options) {
+  const _getVListChildrenAndD = getVListChildrenAndDepth(params),
+        children = _getVListChildrenAndD.children,
+        depth = _getVListChildrenAndD.depth; // Create a strut that is taller than any list item. The strut is added to
+  // each item, where it will determine the item's baseline. Since it has
+  // `overflow:hidden`, the strut's top edge will sit on the item's line box's
+  // top edge and the strut's bottom edge will sit on the item's baseline,
+  // with no additional line-height spacing. This allows the item baseline to
+  // be positioned precisely without worrying about font ascent and
+  // line-height.
+
+
+  let pstrutSize = 0;
+
+  for (let i = 0; i < children.length; i++) {
+    const child = children[i];
+
+    if (child.type === "elem") {
+      const elem = child.elem;
+      pstrutSize = Math.max(pstrutSize, elem.maxFontSize, elem.height);
+    }
+  }
+
+  pstrutSize += 2;
+  const pstrut = makeSpan(["pstrut"], []);
+  pstrut.style.height = pstrutSize + "em"; // Create a new list of actual children at the correct offsets
+
+  const realChildren = [];
+  let minPos = depth;
+  let maxPos = depth;
+  let currPos = depth;
+
+  for (let i = 0; i < children.length; i++) {
+    const child = children[i];
+
+    if (child.type === "kern") {
+      currPos += child.size;
+    } else {
+      const elem = child.elem;
+      const classes = child.wrapperClasses || [];
+      const style = child.wrapperStyle || {};
+      const childWrap = makeSpan(classes, [pstrut, elem], undefined, style);
+      childWrap.style.top = -pstrutSize - currPos - elem.depth + "em";
+
+      if (child.marginLeft) {
+        childWrap.style.marginLeft = child.marginLeft;
+      }
+
+      if (child.marginRight) {
+        childWrap.style.marginRight = child.marginRight;
+      }
+
+      realChildren.push(childWrap);
+      currPos += elem.height + elem.depth;
+    }
+
+    minPos = Math.min(minPos, currPos);
+    maxPos = Math.max(maxPos, currPos);
+  } // The vlist contents go in a table-cell with `vertical-align:bottom`.
+  // This cell's bottom edge will determine the containing table's baseline
+  // without overly expanding the containing line-box.
+
+
+  const vlist = makeSpan(["vlist"], realChildren);
+  vlist.style.height = maxPos + "em"; // A second row is used if necessary to represent the vlist's depth.
+
+  let rows;
+
+  if (minPos < 0) {
+    // We will define depth in an empty span with display: table-cell.
+    // It should render with the height that we define. But Chrome, in
+    // contenteditable mode only, treats that span as if it contains some
+    // text content. And that min-height over-rides our desired height.
+    // So we put another empty span inside the depth strut span.
+    const emptySpan = makeSpan([], []);
+    const depthStrut = makeSpan(["vlist"], [emptySpan]);
+    depthStrut.style.height = -minPos + "em"; // Safari wants the first row to have inline content; otherwise it
+    // puts the bottom of the *second* row on the baseline.
+
+    const topStrut = makeSpan(["vlist-s"], [new SymbolNode("\u200b")]);
+    rows = [makeSpan(["vlist-r"], [vlist, topStrut]), makeSpan(["vlist-r"], [depthStrut])];
+  } else {
+    rows = [makeSpan(["vlist-r"], [vlist])];
+  }
+
+  const vtable = makeSpan(["vlist-t"], rows);
+
+  if (rows.length === 2) {
+    vtable.classes.push("vlist-t2");
+  }
+
+  vtable.height = maxPos;
+  vtable.depth = -minPos;
+  return vtable;
+}; // Glue is a concept from TeX which is a flexible space between elements in
+// either a vertical or horizontal list. In KaTeX, at least for now, it's
+// static space between elements in a horizontal layout.
+
+
+const makeGlue = (measurement, options) => {
+  // Make an empty span for the space
+  const rule = makeSpan(["mspace"], [], options);
+  const size = calculateSize(measurement, options);
+  rule.style.marginRight = `${size}em`;
+  return rule;
+}; // Takes font options, and returns the appropriate fontLookup name
+
+
+const retrieveTextFontName = function retrieveTextFontName(fontFamily, fontWeight, fontShape) {
+  let baseFontName = "";
+
+  switch (fontFamily) {
+    case "amsrm":
+      baseFontName = "AMS";
+      break;
+
+    case "textrm":
+      baseFontName = "Main";
+      break;
+
+    case "textsf":
+      baseFontName = "SansSerif";
+      break;
+
+    case "texttt":
+      baseFontName = "Typewriter";
+      break;
+
+    default:
+      baseFontName = fontFamily;
+    // use fonts added by a plugin
+  }
+
+  let fontStylesName;
+
+  if (fontWeight === "textbf" && fontShape === "textit") {
+    fontStylesName = "BoldItalic";
+  } else if (fontWeight === "textbf") {
+    fontStylesName = "Bold";
+  } else if (fontWeight === "textit") {
+    fontStylesName = "Italic";
+  } else {
+    fontStylesName = "Regular";
+  }
+
+  return `${baseFontName}-${fontStylesName}`;
+};
+/**
+ * Maps TeX font commands to objects containing:
+ * - variant: string used for "mathvariant" attribute in buildMathML.js
+ * - fontName: the "style" parameter to fontMetrics.getCharacterMetrics
+ */
+// A map between tex font commands an MathML mathvariant attribute values
+
+
+const fontMap = {
+  // styles
+  "mathbf": {
+    variant: "bold",
+    fontName: "Main-Bold"
+  },
+  "mathrm": {
+    variant: "normal",
+    fontName: "Main-Regular"
+  },
+  "textit": {
+    variant: "italic",
+    fontName: "Main-Italic"
+  },
+  "mathit": {
+    variant: "italic",
+    fontName: "Main-Italic"
+  },
+  "mathnormal": {
+    variant: "italic",
+    fontName: "Math-Italic"
+  },
+  // "boldsymbol" is missing because they require the use of multiple fonts:
+  // Math-BoldItalic and Main-Bold.  This is handled by a special case in
+  // makeOrd which ends up calling boldsymbol.
+  // families
+  "mathbb": {
+    variant: "double-struck",
+    fontName: "AMS-Regular"
+  },
+  "mathcal": {
+    variant: "script",
+    fontName: "Caligraphic-Regular"
+  },
+  "mathfrak": {
+    variant: "fraktur",
+    fontName: "Fraktur-Regular"
+  },
+  "mathscr": {
+    variant: "script",
+    fontName: "Script-Regular"
+  },
+  "mathsf": {
+    variant: "sans-serif",
+    fontName: "SansSerif-Regular"
+  },
+  "mathtt": {
+    variant: "monospace",
+    fontName: "Typewriter-Regular"
+  }
+};
+const svgData = {
+  //   path, width, height
+  vec: ["vec", 0.471, 0.714],
+  // values from the font glyph
+  oiintSize1: ["oiintSize1", 0.957, 0.499],
+  // oval to overlay the integrand
+  oiintSize2: ["oiintSize2", 1.472, 0.659],
+  oiiintSize1: ["oiiintSize1", 1.304, 0.499],
+  oiiintSize2: ["oiiintSize2", 1.98, 0.659],
+  leftParenInner: ["leftParenInner", 0.875, 0.3],
+  rightParenInner: ["rightParenInner", 0.875, 0.3]
+};
+
+const staticSvg = function staticSvg(value, options) {
+  // Create a span with inline SVG for the element.
+  const _svgData$value = svgData[value],
+        pathName = _svgData$value[0],
+        width = _svgData$value[1],
+        height = _svgData$value[2];
+  const path = new PathNode(pathName);
+  const svgNode = new SvgNode([path], {
+    "width": width + "em",
+    "height": height + "em",
+    // Override CSS rule `.katex svg { width: 100% }`
+    "style": "width:" + width + "em",
+    "viewBox": "0 0 " + 1000 * width + " " + 1000 * height,
+    "preserveAspectRatio": "xMinYMin"
+  });
+  const span = makeSvgSpan(["overlay"], [svgNode], options);
+  span.height = height;
+  span.style.height = height + "em";
+  span.style.width = width + "em";
+  return span;
+};
+
+var buildCommon = {
+  fontMap,
+  makeSymbol,
+  mathsym,
+  makeSpan,
+  makeSvgSpan,
+  makeLineSpan,
+  makeAnchor,
+  makeFragment,
+  wrapFragment,
+  makeVList,
+  makeOrd,
+  makeGlue,
+  staticSvg,
+  svgData,
+  tryCombineChars
+};
+
+/**
+ * Describes spaces between different classes of atoms.
+ */
+const thinspace = {
+  number: 3,
+  unit: "mu"
+};
+const mediumspace = {
+  number: 4,
+  unit: "mu"
+};
+const thickspace = {
+  number: 5,
+  unit: "mu"
+}; // Making the type below exact with all optional fields doesn't work due to
+// - https://github.com/facebook/flow/issues/4582
+// - https://github.com/facebook/flow/issues/5688
+// However, since *all* fields are optional, $Shape<> works as suggested in 5688
+// above.
+
+// Spacing relationships for display and text styles
+const spacings = {
+  mord: {
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mop: {
+    mord: thinspace,
+    mop: thinspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mbin: {
+    mord: mediumspace,
+    mop: mediumspace,
+    mopen: mediumspace,
+    minner: mediumspace
+  },
+  mrel: {
+    mord: thickspace,
+    mop: thickspace,
+    mopen: thickspace,
+    minner: thickspace
+  },
+  mopen: {},
+  mclose: {
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    minner: thinspace
+  },
+  mpunct: {
+    mord: thinspace,
+    mop: thinspace,
+    mrel: thickspace,
+    mopen: thinspace,
+    mclose: thinspace,
+    mpunct: thinspace,
+    minner: thinspace
+  },
+  minner: {
+    mord: thinspace,
+    mop: thinspace,
+    mbin: mediumspace,
+    mrel: thickspace,
+    mopen: thinspace,
+    mpunct: thinspace,
+    minner: thinspace
+  }
+}; // Spacing relationships for script and scriptscript styles
+
+const tightSpacings = {
+  mord: {
+    mop: thinspace
+  },
+  mop: {
+    mord: thinspace,
+    mop: thinspace
+  },
+  mbin: {},
+  mrel: {},
+  mopen: {},
+  mclose: {
+    mop: thinspace
+  },
+  mpunct: {},
+  minner: {
+    mop: thinspace
+  }
+};
+
+/** Context provided to function handlers for error messages. */
+// Note: reverse the order of the return type union will cause a flow error.
+// See https://github.com/facebook/flow/issues/3663.
+// More general version of `HtmlBuilder` for nodes (e.g. \sum, accent types)
+// whose presence impacts super/subscripting. In this case, ParseNode<"supsub">
+// delegates its HTML building to the HtmlBuilder corresponding to these nodes.
+
+/**
+ * Final function spec for use at parse time.
+ * This is almost identical to `FunctionPropSpec`, except it
+ * 1. includes the function handler, and
+ * 2. requires all arguments except argTypes.
+ * It is generated by `defineFunction()` below.
+ */
+
+/**
+ * All registered functions.
+ * `functions.js` just exports this same dictionary again and makes it public.
+ * `Parser.js` requires this dictionary.
+ */
+const _functions = {};
+/**
+ * All HTML builders. Should be only used in the `define*` and the `build*ML`
+ * functions.
+ */
+
+const _htmlGroupBuilders = {};
+/**
+ * All MathML builders. Should be only used in the `define*` and the `build*ML`
+ * functions.
+ */
+
+const _mathmlGroupBuilders = {};
+function defineFunction(_ref) {
+  let type = _ref.type,
+      names = _ref.names,
+      props = _ref.props,
+      handler = _ref.handler,
+      htmlBuilder = _ref.htmlBuilder,
+      mathmlBuilder = _ref.mathmlBuilder;
+  // Set default values of functions
+  const data = {
+    type,
+    numArgs: props.numArgs,
+    argTypes: props.argTypes,
+    greediness: props.greediness === undefined ? 1 : props.greediness,
+    allowedInText: !!props.allowedInText,
+    allowedInMath: props.allowedInMath === undefined ? true : props.allowedInMath,
+    numOptionalArgs: props.numOptionalArgs || 0,
+    infix: !!props.infix,
+    handler: handler
+  };
+
+  for (let i = 0; i < names.length; ++i) {
+    _functions[names[i]] = data;
+  }
+
+  if (type) {
+    if (htmlBuilder) {
+      _htmlGroupBuilders[type] = htmlBuilder;
+    }
+
+    if (mathmlBuilder) {
+      _mathmlGroupBuilders[type] = mathmlBuilder;
+    }
+  }
+}
+/**
+ * Use this to register only the HTML and MathML builders for a function (e.g.
+ * if the function's ParseNode is generated in Parser.js rather than via a
+ * stand-alone handler provided to `defineFunction`).
+ */
+
+function defineFunctionBuilders(_ref2) {
+  let type = _ref2.type,
+      htmlBuilder = _ref2.htmlBuilder,
+      mathmlBuilder = _ref2.mathmlBuilder;
+  defineFunction({
+    type,
+    names: [],
+    props: {
+      numArgs: 0
+    },
+
+    handler() {
+      throw new Error('Should never be called.');
+    },
+
+    htmlBuilder,
+    mathmlBuilder
+  });
+} // Since the corresponding buildHTML/buildMathML function expects a
+// list of elements, we normalize for different kinds of arguments
+
+const ordargument = function ordargument(arg) {
+  return arg.type === "ordgroup" ? arg.body : [arg];
+};
+
+/**
+ * This file does the main work of building a domTree structure from a parse
+ * tree. The entry point is the `buildHTML` function, which takes a parse tree.
+ * Then, the buildExpression, buildGroup, and various groupBuilders functions
+ * are called, to produce a final HTML tree.
+ */
+const makeSpan$1 = buildCommon.makeSpan; // Binary atoms (first class `mbin`) change into ordinary atoms (`mord`)
+// depending on their surroundings. See TeXbook pg. 442-446, Rules 5 and 6,
+// and the text before Rule 19.
+
+const binLeftCanceller = ["leftmost", "mbin", "mopen", "mrel", "mop", "mpunct"];
+const binRightCanceller = ["rightmost", "mrel", "mclose", "mpunct"];
+const styleMap = {
+  "display": Style$1.DISPLAY,
+  "text": Style$1.TEXT,
+  "script": Style$1.SCRIPT,
+  "scriptscript": Style$1.SCRIPTSCRIPT
+};
+const DomEnum = {
+  mord: "mord",
+  mop: "mop",
+  mbin: "mbin",
+  mrel: "mrel",
+  mopen: "mopen",
+  mclose: "mclose",
+  mpunct: "mpunct",
+  minner: "minner"
+};
+
+/**
+ * Take a list of nodes, build them in order, and return a list of the built
+ * nodes. documentFragments are flattened into their contents, so the
+ * returned list contains no fragments. `isRealGroup` is true if `expression`
+ * is a real group (no atoms will be added on either side), as opposed to
+ * a partial group (e.g. one created by \color). `surrounding` is an array
+ * consisting type of nodes that will be added to the left and right.
+ */
+const buildExpression = function buildExpression(expression, options, isRealGroup, surrounding) {
+  if (surrounding === void 0) {
+    surrounding = [null, null];
+  }
+
+  // Parse expressions into `groups`.
+  const groups = [];
+
+  for (let i = 0; i < expression.length; i++) {
+    const output = buildGroup(expression[i], options);
+
+    if (output instanceof DocumentFragment) {
+      const children = output.children;
+      groups.push(...children);
+    } else {
+      groups.push(output);
+    }
+  } // If `expression` is a partial group, let the parent handle spacings
+  // to avoid processing groups multiple times.
+
+
+  if (!isRealGroup) {
+    return groups;
+  }
+
+  let glueOptions = options;
+
+  if (expression.length === 1) {
+    const node = expression[0];
+
+    if (node.type === "sizing") {
+      glueOptions = options.havingSize(node.size);
+    } else if (node.type === "styling") {
+      glueOptions = options.havingStyle(styleMap[node.style]);
+    }
+  } // Dummy spans for determining spacings between surrounding atoms.
+  // If `expression` has no atoms on the left or right, class "leftmost"
+  // or "rightmost", respectively, is used to indicate it.
+
+
+  const dummyPrev = makeSpan$1([surrounding[0] || "leftmost"], [], options);
+  const dummyNext = makeSpan$1([surrounding[1] || "rightmost"], [], options); // TODO: These code assumes that a node's math class is the first element
+  // of its `classes` array. A later cleanup should ensure this, for
+  // instance by changing the signature of `makeSpan`.
+  // Before determining what spaces to insert, perform bin cancellation.
+  // Binary operators change to ordinary symbols in some contexts.
+
+  const isRoot = isRealGroup === "root";
+  traverseNonSpaceNodes(groups, (node, prev) => {
+    const prevType = prev.classes[0];
+    const type = node.classes[0];
+
+    if (prevType === "mbin" && utils.contains(binRightCanceller, type)) {
+      prev.classes[0] = "mord";
+    } else if (type === "mbin" && utils.contains(binLeftCanceller, prevType)) {
+      node.classes[0] = "mord";
+    }
+  }, {
+    node: dummyPrev
+  }, dummyNext, isRoot);
+  traverseNonSpaceNodes(groups, (node, prev) => {
+    const prevType = getTypeOfDomTree(prev);
+    const type = getTypeOfDomTree(node); // 'mtight' indicates that the node is script or scriptscript style.
+
+    const space = prevType && type ? node.hasClass("mtight") ? tightSpacings[prevType][type] : spacings[prevType][type] : null;
+
+    if (space) {
+      // Insert glue (spacing) after the `prev`.
+      return buildCommon.makeGlue(space, glueOptions);
+    }
+  }, {
+    node: dummyPrev
+  }, dummyNext, isRoot);
+  return groups;
+}; // Depth-first traverse non-space `nodes`, calling `callback` with the current and
+// previous node as arguments, optionally returning a node to insert after the
+// previous node. `prev` is an object with the previous node and `insertAfter`
+// function to insert after it. `next` is a node that will be added to the right.
+// Used for bin cancellation and inserting spacings.
+
+const traverseNonSpaceNodes = function traverseNonSpaceNodes(nodes, callback, prev, next, isRoot) {
+  if (next) {
+    // temporarily append the right node, if exists
+    nodes.push(next);
+  }
+
+  let i = 0;
+
+  for (; i < nodes.length; i++) {
+    const node = nodes[i];
+    const partialGroup = checkPartialGroup(node);
+
+    if (partialGroup) {
+      // Recursive DFS
+      // $FlowFixMe: make nodes a $ReadOnlyArray by returning a new array
+      traverseNonSpaceNodes(partialGroup.children, callback, prev, null, isRoot);
+      continue;
+    } // Ignore explicit spaces (e.g., \;, \,) when determining what implicit
+    // spacing should go between atoms of different classes
+
+
+    const nonspace = !node.hasClass("mspace");
+
+    if (nonspace) {
+      const result = callback(node, prev.node);
+
+      if (result) {
+        if (prev.insertAfter) {
+          prev.insertAfter(result);
+        } else {
+          // insert at front
+          nodes.unshift(result);
+          i++;
+        }
+      }
+    }
+
+    if (nonspace) {
+      prev.node = node;
+    } else if (isRoot && node.hasClass("newline")) {
+      prev.node = makeSpan$1(["leftmost"]); // treat like beginning of line
+    }
+
+    prev.insertAfter = (index => n => {
+      nodes.splice(index + 1, 0, n);
+      i++;
+    })(i);
+  }
+
+  if (next) {
+    nodes.pop();
+  }
+}; // Check if given node is a partial group, i.e., does not affect spacing around.
+
+
+const checkPartialGroup = function checkPartialGroup(node) {
+  if (node instanceof DocumentFragment || node instanceof Anchor || node instanceof Span && node.hasClass("enclosing")) {
+    return node;
+  }
+
+  return null;
+}; // Return the outermost node of a domTree.
+
+
+const getOutermostNode = function getOutermostNode(node, side) {
+  const partialGroup = checkPartialGroup(node);
+
+  if (partialGroup) {
+    const children = partialGroup.children;
+
+    if (children.length) {
+      if (side === "right") {
+        return getOutermostNode(children[children.length - 1], "right");
+      } else if (side === "left") {
+        return getOutermostNode(children[0], "left");
+      }
+    }
+  }
+
+  return node;
+}; // Return math atom class (mclass) of a domTree.
+// If `side` is given, it will get the type of the outermost node at given side.
+
+
+const getTypeOfDomTree = function getTypeOfDomTree(node, side) {
+  if (!node) {
+    return null;
+  }
+
+  if (side) {
+    node = getOutermostNode(node, side);
+  } // This makes a lot of assumptions as to where the type of atom
+  // appears.  We should do a better job of enforcing this.
+
+
+  return DomEnum[node.classes[0]] || null;
+};
+const makeNullDelimiter = function makeNullDelimiter(options, classes) {
+  const moreClasses = ["nulldelimiter"].concat(options.baseSizingClasses());
+  return makeSpan$1(classes.concat(moreClasses));
+};
+/**
+ * buildGroup is the function that takes a group and calls the correct groupType
+ * function for it. It also handles the interaction of size and style changes
+ * between parents and children.
+ */
+
+const buildGroup = function buildGroup(group, options, baseOptions) {
+  if (!group) {
+    return makeSpan$1();
+  }
+
+  if (_htmlGroupBuilders[group.type]) {
+    // Call the groupBuilders function
+    let groupNode = _htmlGroupBuilders[group.type](group, options); // If the size changed between the parent and the current group, account
+    // for that size difference.
+
+    if (baseOptions && options.size !== baseOptions.size) {
+      groupNode = makeSpan$1(options.sizingClasses(baseOptions), [groupNode], options);
+      const multiplier = options.sizeMultiplier / baseOptions.sizeMultiplier;
+      groupNode.height *= multiplier;
+      groupNode.depth *= multiplier;
+    }
+
+    return groupNode;
+  } else {
+    throw new ParseError("Got group of unknown type: '" + group.type + "'");
+  }
+};
+/**
+ * Combine an array of HTML DOM nodes (e.g., the output of `buildExpression`)
+ * into an unbreakable HTML node of class .base, with proper struts to
+ * guarantee correct vertical extent.  `buildHTML` calls this repeatedly to
+ * make up the entire expression as a sequence of unbreakable units.
+ */
+
+function buildHTMLUnbreakable(children, options) {
+  // Compute height and depth of this chunk.
+  const body = makeSpan$1(["base"], children, options); // Add strut, which ensures that the top of the HTML element falls at
+  // the height of the expression, and the bottom of the HTML element
+  // falls at the depth of the expression.
+
+  const strut = makeSpan$1(["strut"]);
+  strut.style.height = body.height + body.depth + "em";
+  strut.style.verticalAlign = -body.depth + "em";
+  body.children.unshift(strut);
+  return body;
+}
+/**
+ * Take an entire parse tree, and build it into an appropriate set of HTML
+ * nodes.
+ */
+
+
+function buildHTML(tree, options) {
+  // Strip off outer tag wrapper for processing below.
+  let tag = null;
+
+  if (tree.length === 1 && tree[0].type === "tag") {
+    tag = tree[0].tag;
+    tree = tree[0].body;
+  } // Build the expression contained in the tree
+
+
+  const expression = buildExpression(tree, options, "root");
+  const children = []; // Create one base node for each chunk between potential line breaks.
+  // The TeXBook [p.173] says "A formula will be broken only after a
+  // relation symbol like $=$ or $<$ or $\rightarrow$, or after a binary
+  // operation symbol like $+$ or $-$ or $\times$, where the relation or
+  // binary operation is on the ``outer level'' of the formula (i.e., not
+  // enclosed in {...} and not part of an \over construction)."
+
+  let parts = [];
+
+  for (let i = 0; i < expression.length; i++) {
+    parts.push(expression[i]);
+
+    if (expression[i].hasClass("mbin") || expression[i].hasClass("mrel") || expression[i].hasClass("allowbreak")) {
+      // Put any post-operator glue on same line as operator.
+      // Watch for \nobreak along the way, and stop at \newline.
+      let nobreak = false;
+
+      while (i < expression.length - 1 && expression[i + 1].hasClass("mspace") && !expression[i + 1].hasClass("newline")) {
+        i++;
+        parts.push(expression[i]);
+
+        if (expression[i].hasClass("nobreak")) {
+          nobreak = true;
+        }
+      } // Don't allow break if \nobreak among the post-operator glue.
+
+
+      if (!nobreak) {
+        children.push(buildHTMLUnbreakable(parts, options));
+        parts = [];
+      }
+    } else if (expression[i].hasClass("newline")) {
+      // Write the line except the newline
+      parts.pop();
+
+      if (parts.length > 0) {
+        children.push(buildHTMLUnbreakable(parts, options));
+        parts = [];
+      } // Put the newline at the top level
+
+
+      children.push(expression[i]);
+    }
+  }
+
+  if (parts.length > 0) {
+    children.push(buildHTMLUnbreakable(parts, options));
+  } // Now, if there was a tag, build it too and append it as a final child.
+
+
+  let tagChild;
+
+  if (tag) {
+    tagChild = buildHTMLUnbreakable(buildExpression(tag, options, true));
+    tagChild.classes = ["tag"];
+    children.push(tagChild);
+  }
+
+  const htmlNode = makeSpan$1(["katex-html"], children);
+  htmlNode.setAttribute("aria-hidden", "true"); // Adjust the strut of the tag to be the maximum height of all children
+  // (the height of the enclosing htmlNode) for proper vertical alignment.
+
+  if (tagChild) {
+    const strut = tagChild.children[0];
+    strut.style.height = htmlNode.height + htmlNode.depth + "em";
+    strut.style.verticalAlign = -htmlNode.depth + "em";
+  }
+
+  return htmlNode;
+}
+
+/**
+ * These objects store data about MathML nodes. This is the MathML equivalent
+ * of the types in domTree.js. Since MathML handles its own rendering, and
+ * since we're mainly using MathML to improve accessibility, we don't manage
+ * any of the styling state that the plain DOM nodes do.
+ *
+ * The `toNode` and `toMarkup` functions work simlarly to how they do in
+ * domTree.js, creating namespaced DOM nodes and HTML text markup respectively.
+ */
+function newDocumentFragment(children) {
+  return new DocumentFragment(children);
+}
+/**
+ * This node represents a general purpose MathML node of any type. The
+ * constructor requires the type of node to create (for example, `"mo"` or
+ * `"mspace"`, corresponding to `<mo>` and `<mspace>` tags).
+ */
+
+class MathNode {
+  constructor(type, children) {
+    this.type = void 0;
+    this.attributes = void 0;
+    this.children = void 0;
+    this.type = type;
+    this.attributes = {};
+    this.children = children || [];
+  }
+  /**
+   * Sets an attribute on a MathML node. MathML depends on attributes to convey a
+   * semantic content, so this is used heavily.
+   */
+
+
+  setAttribute(name, value) {
+    this.attributes[name] = value;
+  }
+  /**
+   * Gets an attribute on a MathML node.
+   */
+
+
+  getAttribute(name) {
+    return this.attributes[name];
+  }
+  /**
+   * Converts the math node into a MathML-namespaced DOM element.
+   */
+
+
+  toNode() {
+    const node = document.createElementNS("http://www.w3.org/1998/Math/MathML", this.type);
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        node.setAttribute(attr, this.attributes[attr]);
+      }
+    }
+
+    for (let i = 0; i < this.children.length; i++) {
+      node.appendChild(this.children[i].toNode());
+    }
+
+    return node;
+  }
+  /**
+   * Converts the math node into an HTML markup string.
+   */
+
+
+  toMarkup() {
+    let markup = "<" + this.type; // Add the attributes
+
+    for (const attr in this.attributes) {
+      if (Object.prototype.hasOwnProperty.call(this.attributes, attr)) {
+        markup += " " + attr + "=\"";
+        markup += utils.escape(this.attributes[attr]);
+        markup += "\"";
+      }
+    }
+
+    markup += ">";
+
+    for (let i = 0; i < this.children.length; i++) {
+      markup += this.children[i].toMarkup();
+    }
+
+    markup += "</" + this.type + ">";
+    return markup;
+  }
+  /**
+   * Converts the math node into a string, similar to innerText, but escaped.
+   */
+
+
+  toText() {
+    return this.children.map(child => child.toText()).join("");
+  }
+
+}
+/**
+ * This node represents a piece of text.
+ */
+
+class TextNode {
+  constructor(text) {
+    this.text = void 0;
+    this.text = text;
+  }
+  /**
+   * Converts the text node into a DOM text node.
+   */
+
+
+  toNode() {
+    return document.createTextNode(this.text);
+  }
+  /**
+   * Converts the text node into escaped HTML markup
+   * (representing the text itself).
+   */
+
+
+  toMarkup() {
+    return utils.escape(this.toText());
+  }
+  /**
+   * Converts the text node into a string
+   * (representing the text iteself).
+   */
+
+
+  toText() {
+    return this.text;
+  }
+
+}
+/**
+ * This node represents a space, but may render as <mspace.../> or as text,
+ * depending on the width.
+ */
+
+class SpaceNode {
+  /**
+   * Create a Space node with width given in CSS ems.
+   */
+  constructor(width) {
+    this.width = void 0;
+    this.character = void 0;
+    this.width = width; // See https://www.w3.org/TR/2000/WD-MathML2-20000328/chapter6.html
+    // for a table of space-like characters.  We use Unicode
+    // representations instead of &LongNames; as it's not clear how to
+    // make the latter via document.createTextNode.
+
+    if (width >= 0.05555 && width <= 0.05556) {
+      this.character = "\u200a"; // &VeryThinSpace;
+    } else if (width >= 0.1666 && width <= 0.1667) {
+      this.character = "\u2009"; // &ThinSpace;
+    } else if (width >= 0.2222 && width <= 0.2223) {
+      this.character = "\u2005"; // &MediumSpace;
+    } else if (width >= 0.2777 && width <= 0.2778) {
+      this.character = "\u2005\u200a"; // &ThickSpace;
+    } else if (width >= -0.05556 && width <= -0.05555) {
+      this.character = "\u200a\u2063"; // &NegativeVeryThinSpace;
+    } else if (width >= -0.1667 && width <= -0.1666) {
+      this.character = "\u2009\u2063"; // &NegativeThinSpace;
+    } else if (width >= -0.2223 && width <= -0.2222) {
+      this.character = "\u205f\u2063"; // &NegativeMediumSpace;
+    } else if (width >= -0.2778 && width <= -0.2777) {
+      this.character = "\u2005\u2063"; // &NegativeThickSpace;
+    } else {
+      this.character = null;
+    }
+  }
+  /**
+   * Converts the math node into a MathML-namespaced DOM element.
+   */
+
+
+  toNode() {
+    if (this.character) {
+      return document.createTextNode(this.character);
+    } else {
+      const node = document.createElementNS("http://www.w3.org/1998/Math/MathML", "mspace");
+      node.setAttribute("width", this.width + "em");
+      return node;
+    }
+  }
+  /**
+   * Converts the math node into an HTML markup string.
+   */
+
+
+  toMarkup() {
+    if (this.character) {
+      return `<mtext>${this.character}</mtext>`;
+    } else {
+      return `<mspace width="${this.width}em"/>`;
+    }
+  }
+  /**
+   * Converts the math node into a string, similar to innerText.
+   */
+
+
+  toText() {
+    if (this.character) {
+      return this.character;
+    } else {
+      return " ";
+    }
+  }
+
+}
+
+var mathMLTree = {
+  MathNode,
+  TextNode,
+  SpaceNode,
+  newDocumentFragment
+};
+
+/**
+ * This file converts a parse tree into a cooresponding MathML tree. The main
+ * entry point is the `buildMathML` function, which takes a parse tree from the
+ * parser.
+ */
+
+/**
+ * Takes a symbol and converts it into a MathML text node after performing
+ * optional replacement from symbols.js.
+ */
+const makeText = function makeText(text, mode, options) {
+  if (symbols[mode][text] && symbols[mode][text].replace && text.charCodeAt(0) !== 0xD835 && !(ligatures.hasOwnProperty(text) && options && (options.fontFamily && options.fontFamily.substr(4, 2) === "tt" || options.font && options.font.substr(4, 2) === "tt"))) {
+    text = symbols[mode][text].replace;
+  }
+
+  return new mathMLTree.TextNode(text);
+};
+/**
+ * Wrap the given array of nodes in an <mrow> node if needed, i.e.,
+ * unless the array has length 1.  Always returns a single node.
+ */
+
+const makeRow = function makeRow(body) {
+  if (body.length === 1) {
+    return body[0];
+  } else {
+    return new mathMLTree.MathNode("mrow", body);
+  }
+};
+/**
+ * Returns the math variant as a string or null if none is required.
+ */
+
+const getVariant = function getVariant(group, options) {
+  // Handle \text... font specifiers as best we can.
+  // MathML has a limited list of allowable mathvariant specifiers; see
+  // https://www.w3.org/TR/MathML3/chapter3.html#presm.commatt
+  if (options.fontFamily === "texttt") {
+    return "monospace";
+  } else if (options.fontFamily === "textsf") {
+    if (options.fontShape === "textit" && options.fontWeight === "textbf") {
+      return "sans-serif-bold-italic";
+    } else if (options.fontShape === "textit") {
+      return "sans-serif-italic";
+    } else if (options.fontWeight === "textbf") {
+      return "bold-sans-serif";
+    } else {
+      return "sans-serif";
+    }
+  } else if (options.fontShape === "textit" && options.fontWeight === "textbf") {
+    return "bold-italic";
+  } else if (options.fontShape === "textit") {
+    return "italic";
+  } else if (options.fontWeight === "textbf") {
+    return "bold";
+  }
+
+  const font = options.font;
+
+  if (!font || font === "mathnormal") {
+    return null;
+  }
+
+  const mode = group.mode;
+
+  if (font === "mathit") {
+    return "italic";
+  } else if (font === "boldsymbol") {
+    return group.type === "textord" ? "bold" : "bold-italic";
+  } else if (font === "mathbf") {
+    return "bold";
+  } else if (font === "mathbb") {
+    return "double-struck";
+  } else if (font === "mathfrak") {
+    return "fraktur";
+  } else if (font === "mathscr" || font === "mathcal") {
+    // MathML makes no distinction between script and caligrahpic
+    return "script";
+  } else if (font === "mathsf") {
+    return "sans-serif";
+  } else if (font === "mathtt") {
+    return "monospace";
+  }
+
+  let text = group.text;
+
+  if (utils.contains(["\\imath", "\\jmath"], text)) {
+    return null;
+  }
+
+  if (symbols[mode][text] && symbols[mode][text].replace) {
+    text = symbols[mode][text].replace;
+  }
+
+  const fontName = buildCommon.fontMap[font].fontName;
+
+  if (getCharacterMetrics(text, fontName, mode)) {
+    return buildCommon.fontMap[font].variant;
+  }
+
+  return null;
+};
+/**
+ * Takes a list of nodes, builds them, and returns a list of the generated
+ * MathML nodes.  Also combine consecutive <mtext> outputs into a single
+ * <mtext> tag.
+ */
+
+const buildExpression$1 = function buildExpression(expression, options, isOrdgroup) {
+  if (expression.length === 1) {
+    const group = buildGroup$1(expression[0], options);
+
+    if (isOrdgroup && group instanceof MathNode && group.type === "mo") {
+      // When TeX writers want to suppress spacing on an operator,
+      // they often put the operator by itself inside braces.
+      group.setAttribute("lspace", "0em");
+      group.setAttribute("rspace", "0em");
+    }
+
+    return [group];
+  }
+
+  const groups = [];
+  let lastGroup;
+
+  for (let i = 0; i < expression.length; i++) {
+    const group = buildGroup$1(expression[i], options);
+
+    if (group instanceof MathNode && lastGroup instanceof MathNode) {
+      // Concatenate adjacent <mtext>s
+      if (group.type === 'mtext' && lastGroup.type === 'mtext' && group.getAttribute('mathvariant') === lastGroup.getAttribute('mathvariant')) {
+        lastGroup.children.push(...group.children);
+        continue; // Concatenate adjacent <mn>s
+      } else if (group.type === 'mn' && lastGroup.type === 'mn') {
+        lastGroup.children.push(...group.children);
+        continue; // Concatenate <mn>...</mn> followed by <mi>.</mi>
+      } else if (group.type === 'mi' && group.children.length === 1 && lastGroup.type === 'mn') {
+        const child = group.children[0];
+
+        if (child instanceof TextNode && child.text === '.') {
+          lastGroup.children.push(...group.children);
+          continue;
+        }
+      } else if (lastGroup.type === 'mi' && lastGroup.children.length === 1) {
+        const lastChild = lastGroup.children[0];
+
+        if (lastChild instanceof TextNode && lastChild.text === '\u0338' && (group.type === 'mo' || group.type === 'mi' || group.type === 'mn')) {
+          const child = group.children[0];
+
+          if (child instanceof TextNode && child.text.length > 0) {
+            // Overlay with combining character long solidus
+            child.text = child.text.slice(0, 1) + "\u0338" + child.text.slice(1);
+            groups.pop();
+          }
+        }
+      }
+    }
+
+    groups.push(group);
+    lastGroup = group;
+  }
+
+  return groups;
+};
+/**
+ * Equivalent to buildExpression, but wraps the elements in an <mrow>
+ * if there's more than one.  Returns a single node instead of an array.
+ */
+
+const buildExpressionRow = function buildExpressionRow(expression, options, isOrdgroup) {
+  return makeRow(buildExpression$1(expression, options, isOrdgroup));
+};
+/**
+ * Takes a group from the parser and calls the appropriate groupBuilders function
+ * on it to produce a MathML node.
+ */
+
+const buildGroup$1 = function buildGroup(group, options) {
+  if (!group) {
+    return new mathMLTree.MathNode("mrow");
+  }
+
+  if (_mathmlGroupBuilders[group.type]) {
+    // Call the groupBuilders function
+    const result = _mathmlGroupBuilders[group.type](group, options);
+    return result;
+  } else {
+    throw new ParseError("Got group of unknown type: '" + group.type + "'");
+  }
+};
+/**
+ * Takes a full parse tree and settings and builds a MathML representation of
+ * it. In particular, we put the elements from building the parse tree into a
+ * <semantics> tag so we can also include that TeX source as an annotation.
+ *
+ * Note that we actually return a domTree element with a `<math>` inside it so
+ * we can do appropriate styling.
+ */
+
+function buildMathML(tree, texExpression, options, isDisplayMode, forMathmlOnly) {
+  const expression = buildExpression$1(tree, options); // Wrap up the expression in an mrow so it is presented in the semantics
+  // tag correctly, unless it's a single <mrow> or <mtable>.
+
+  let wrapper;
+
+  if (expression.length === 1 && expression[0] instanceof MathNode && utils.contains(["mrow", "mtable"], expression[0].type)) {
+    wrapper = expression[0];
+  } else {
+    wrapper = new mathMLTree.MathNode("mrow", expression);
+  } // Build a TeX annotation of the source
+
+
+  const annotation = new mathMLTree.MathNode("annotation", [new mathMLTree.TextNode(texExpression)]);
+  annotation.setAttribute("encoding", "application/x-tex");
+  const semantics = new mathMLTree.MathNode("semantics", [wrapper, annotation]);
+  const math = new mathMLTree.MathNode("math", [semantics]);
+  math.setAttribute("xmlns", "http://www.w3.org/1998/Math/MathML");
+
+  if (isDisplayMode) {
+    math.setAttribute("display", "block");
+  } // You can't style <math> nodes, so we wrap the node in a span.
+  // NOTE: The span class is not typed to have <math> nodes as children, and
+  // we don't want to make the children type more generic since the children
+  // of span are expected to have more fields in `buildHtml` contexts.
+
+
+  const wrapperClass = forMathmlOnly ? "katex" : "katex-mathml"; // $FlowFixMe
+
+  return buildCommon.makeSpan([wrapperClass], [math]);
+}
+
+const optionsFromSettings = function optionsFromSettings(settings) {
+  return new Options({
+    style: settings.displayMode ? Style$1.DISPLAY : Style$1.TEXT,
+    maxSize: settings.maxSize,
+    minRuleThickness: settings.minRuleThickness
+  });
+};
+
+const displayWrap = function displayWrap(node, settings) {
+  if (settings.displayMode) {
+    const classes = ["katex-display"];
+
+    if (settings.leqno) {
+      classes.push("leqno");
+    }
+
+    if (settings.fleqn) {
+      classes.push("fleqn");
+    }
+
+    node = buildCommon.makeSpan(classes, [node]);
+  }
+
+  return node;
+};
+
+const buildTree = function buildTree(tree, expression, settings) {
+  const options = optionsFromSettings(settings);
+  let katexNode;
+
+  if (settings.output === "mathml") {
+    return buildMathML(tree, expression, options, settings.displayMode, true);
+  } else if (settings.output === "html") {
+    const htmlNode = buildHTML(tree, options);
+    katexNode = buildCommon.makeSpan(["katex"], [htmlNode]);
+  } else {
+    const mathMLNode = buildMathML(tree, expression, options, settings.displayMode, false);
+    const htmlNode = buildHTML(tree, options);
+    katexNode = buildCommon.makeSpan(["katex"], [mathMLNode, htmlNode]);
+  }
+
+  return displayWrap(katexNode, settings);
+};
+const buildHTMLTree = function buildHTMLTree(tree, expression, settings) {
+  const options = optionsFromSettings(settings);
+  const htmlNode = buildHTML(tree, options);
+  const katexNode = buildCommon.makeSpan(["katex"], [htmlNode]);
+  return displayWrap(katexNode, settings);
+};
+
+/**
+ * This file provides support to buildMathML.js and buildHTML.js
+ * for stretchy wide elements rendered from SVG files
+ * and other CSS trickery.
+ */
+const stretchyCodePoint = {
+  widehat: "^",
+  widecheck: "ˇ",
+  widetilde: "~",
+  utilde: "~",
+  overleftarrow: "\u2190",
+  underleftarrow: "\u2190",
+  xleftarrow: "\u2190",
+  overrightarrow: "\u2192",
+  underrightarrow: "\u2192",
+  xrightarrow: "\u2192",
+  underbrace: "\u23df",
+  overbrace: "\u23de",
+  overgroup: "\u23e0",
+  undergroup: "\u23e1",
+  overleftrightarrow: "\u2194",
+  underleftrightarrow: "\u2194",
+  xleftrightarrow: "\u2194",
+  Overrightarrow: "\u21d2",
+  xRightarrow: "\u21d2",
+  overleftharpoon: "\u21bc",
+  xleftharpoonup: "\u21bc",
+  overrightharpoon: "\u21c0",
+  xrightharpoonup: "\u21c0",
+  xLeftarrow: "\u21d0",
+  xLeftrightarrow: "\u21d4",
+  xhookleftarrow: "\u21a9",
+  xhookrightarrow: "\u21aa",
+  xmapsto: "\u21a6",
+  xrightharpoondown: "\u21c1",
+  xleftharpoondown: "\u21bd",
+  xrightleftharpoons: "\u21cc",
+  xleftrightharpoons: "\u21cb",
+  xtwoheadleftarrow: "\u219e",
+  xtwoheadrightarrow: "\u21a0",
+  xlongequal: "=",
+  xtofrom: "\u21c4",
+  xrightleftarrows: "\u21c4",
+  xrightequilibrium: "\u21cc",
+  // Not a perfect match.
+  xleftequilibrium: "\u21cb" // None better available.
+
+};
+
+const mathMLnode = function mathMLnode(label) {
+  const node = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(stretchyCodePoint[label.substr(1)])]);
+  node.setAttribute("stretchy", "true");
+  return node;
+}; // Many of the KaTeX SVG images have been adapted from glyphs in KaTeX fonts.
+// Copyright (c) 2009-2010, Design Science, Inc. (<www.mathjax.org>)
+// Copyright (c) 2014-2017 Khan Academy (<www.khanacademy.org>)
+// Licensed under the SIL Open Font License, Version 1.1.
+// See \nhttp://scripts.sil.org/OFL
+// Very Long SVGs
+//    Many of the KaTeX stretchy wide elements use a long SVG image and an
+//    overflow: hidden tactic to achieve a stretchy image while avoiding
+//    distortion of arrowheads or brace corners.
+//    The SVG typically contains a very long (400 em) arrow.
+//    The SVG is in a container span that has overflow: hidden, so the span
+//    acts like a window that exposes only part of the  SVG.
+//    The SVG always has a longer, thinner aspect ratio than the container span.
+//    After the SVG fills 100% of the height of the container span,
+//    there is a long arrow shaft left over. That left-over shaft is not shown.
+//    Instead, it is sliced off because the span's CSS has overflow: hidden.
+//    Thus, the reader sees an arrow that matches the subject matter width
+//    without distortion.
+//    Some functions, such as \cancel, need to vary their aspect ratio. These
+//    functions do not get the overflow SVG treatment.
+// Second Brush Stroke
+//    Low resolution monitors struggle to display images in fine detail.
+//    So browsers apply anti-aliasing. A long straight arrow shaft therefore
+//    will sometimes appear as if it has a blurred edge.
+//    To mitigate this, these SVG files contain a second "brush-stroke" on the
+//    arrow shafts. That is, a second long thin rectangular SVG path has been
+//    written directly on top of each arrow shaft. This reinforcement causes
+//    some of the screen pixels to display as black instead of the anti-aliased
+//    gray pixel that a  single path would generate. So we get arrow shafts
+//    whose edges appear to be sharper.
+// In the katexImagesData object just below, the dimensions all
+// correspond to path geometry inside the relevant SVG.
+// For example, \overrightarrow uses the same arrowhead as glyph U+2192
+// from the KaTeX Main font. The scaling factor is 1000.
+// That is, inside the font, that arrowhead is 522 units tall, which
+// corresponds to 0.522 em inside the document.
+
+
+const katexImagesData = {
+  //   path(s), minWidth, height, align
+  overrightarrow: [["rightarrow"], 0.888, 522, "xMaxYMin"],
+  overleftarrow: [["leftarrow"], 0.888, 522, "xMinYMin"],
+  underrightarrow: [["rightarrow"], 0.888, 522, "xMaxYMin"],
+  underleftarrow: [["leftarrow"], 0.888, 522, "xMinYMin"],
+  xrightarrow: [["rightarrow"], 1.469, 522, "xMaxYMin"],
+  xleftarrow: [["leftarrow"], 1.469, 522, "xMinYMin"],
+  Overrightarrow: [["doublerightarrow"], 0.888, 560, "xMaxYMin"],
+  xRightarrow: [["doublerightarrow"], 1.526, 560, "xMaxYMin"],
+  xLeftarrow: [["doubleleftarrow"], 1.526, 560, "xMinYMin"],
+  overleftharpoon: [["leftharpoon"], 0.888, 522, "xMinYMin"],
+  xleftharpoonup: [["leftharpoon"], 0.888, 522, "xMinYMin"],
+  xleftharpoondown: [["leftharpoondown"], 0.888, 522, "xMinYMin"],
+  overrightharpoon: [["rightharpoon"], 0.888, 522, "xMaxYMin"],
+  xrightharpoonup: [["rightharpoon"], 0.888, 522, "xMaxYMin"],
+  xrightharpoondown: [["rightharpoondown"], 0.888, 522, "xMaxYMin"],
+  xlongequal: [["longequal"], 0.888, 334, "xMinYMin"],
+  xtwoheadleftarrow: [["twoheadleftarrow"], 0.888, 334, "xMinYMin"],
+  xtwoheadrightarrow: [["twoheadrightarrow"], 0.888, 334, "xMaxYMin"],
+  overleftrightarrow: [["leftarrow", "rightarrow"], 0.888, 522],
+  overbrace: [["leftbrace", "midbrace", "rightbrace"], 1.6, 548],
+  underbrace: [["leftbraceunder", "midbraceunder", "rightbraceunder"], 1.6, 548],
+  underleftrightarrow: [["leftarrow", "rightarrow"], 0.888, 522],
+  xleftrightarrow: [["leftarrow", "rightarrow"], 1.75, 522],
+  xLeftrightarrow: [["doubleleftarrow", "doublerightarrow"], 1.75, 560],
+  xrightleftharpoons: [["leftharpoondownplus", "rightharpoonplus"], 1.75, 716],
+  xleftrightharpoons: [["leftharpoonplus", "rightharpoondownplus"], 1.75, 716],
+  xhookleftarrow: [["leftarrow", "righthook"], 1.08, 522],
+  xhookrightarrow: [["lefthook", "rightarrow"], 1.08, 522],
+  overlinesegment: [["leftlinesegment", "rightlinesegment"], 0.888, 522],
+  underlinesegment: [["leftlinesegment", "rightlinesegment"], 0.888, 522],
+  overgroup: [["leftgroup", "rightgroup"], 0.888, 342],
+  undergroup: [["leftgroupunder", "rightgroupunder"], 0.888, 342],
+  xmapsto: [["leftmapsto", "rightarrow"], 1.5, 522],
+  xtofrom: [["leftToFrom", "rightToFrom"], 1.75, 528],
+  // The next three arrows are from the mhchem package.
+  // In mhchem.sty, min-length is 2.0em. But these arrows might appear in the
+  // document as \xrightarrow or \xrightleftharpoons. Those have
+  // min-length = 1.75em, so we set min-length on these next three to match.
+  xrightleftarrows: [["baraboveleftarrow", "rightarrowabovebar"], 1.75, 901],
+  xrightequilibrium: [["baraboveshortleftharpoon", "rightharpoonaboveshortbar"], 1.75, 716],
+  xleftequilibrium: [["shortbaraboveleftharpoon", "shortrightharpoonabovebar"], 1.75, 716]
+};
+
+const groupLength = function groupLength(arg) {
+  if (arg.type === "ordgroup") {
+    return arg.body.length;
+  } else {
+    return 1;
+  }
+};
+
+const svgSpan = function svgSpan(group, options) {
+  // Create a span with inline SVG for the element.
+  function buildSvgSpan_() {
+    let viewBoxWidth = 400000; // default
+
+    const label = group.label.substr(1);
+
+    if (utils.contains(["widehat", "widecheck", "widetilde", "utilde"], label)) {
+      // Each type in the `if` statement corresponds to one of the ParseNode
+      // types below. This narrowing is required to access `grp.base`.
+      const grp = group; // There are four SVG images available for each function.
+      // Choose a taller image when there are more characters.
+
+      const numChars = groupLength(grp.base);
+      let viewBoxHeight;
+      let pathName;
+      let height;
+
+      if (numChars > 5) {
+        if (label === "widehat" || label === "widecheck") {
+          viewBoxHeight = 420;
+          viewBoxWidth = 2364;
+          height = 0.42;
+          pathName = label + "4";
+        } else {
+          viewBoxHeight = 312;
+          viewBoxWidth = 2340;
+          height = 0.34;
+          pathName = "tilde4";
+        }
+      } else {
+        const imgIndex = [1, 1, 2, 2, 3, 3][numChars];
+
+        if (label === "widehat" || label === "widecheck") {
+          viewBoxWidth = [0, 1062, 2364, 2364, 2364][imgIndex];
+          viewBoxHeight = [0, 239, 300, 360, 420][imgIndex];
+          height = [0, 0.24, 0.3, 0.3, 0.36, 0.42][imgIndex];
+          pathName = label + imgIndex;
+        } else {
+          viewBoxWidth = [0, 600, 1033, 2339, 2340][imgIndex];
+          viewBoxHeight = [0, 260, 286, 306, 312][imgIndex];
+          height = [0, 0.26, 0.286, 0.3, 0.306, 0.34][imgIndex];
+          pathName = "tilde" + imgIndex;
+        }
+      }
+
+      const path = new PathNode(pathName);
+      const svgNode = new SvgNode([path], {
+        "width": "100%",
+        "height": height + "em",
+        "viewBox": `0 0 ${viewBoxWidth} ${viewBoxHeight}`,
+        "preserveAspectRatio": "none"
+      });
+      return {
+        span: buildCommon.makeSvgSpan([], [svgNode], options),
+        minWidth: 0,
+        height
+      };
+    } else {
+      const spans = [];
+      const data = katexImagesData[label];
+      const paths = data[0],
+            minWidth = data[1],
+            viewBoxHeight = data[2];
+      const height = viewBoxHeight / 1000;
+      const numSvgChildren = paths.length;
+      let widthClasses;
+      let aligns;
+
+      if (numSvgChildren === 1) {
+        // $FlowFixMe: All these cases must be of the 4-tuple type.
+        const align1 = data[3];
+        widthClasses = ["hide-tail"];
+        aligns = [align1];
+      } else if (numSvgChildren === 2) {
+        widthClasses = ["halfarrow-left", "halfarrow-right"];
+        aligns = ["xMinYMin", "xMaxYMin"];
+      } else if (numSvgChildren === 3) {
+        widthClasses = ["brace-left", "brace-center", "brace-right"];
+        aligns = ["xMinYMin", "xMidYMin", "xMaxYMin"];
+      } else {
+        throw new Error(`Correct katexImagesData or update code here to support
+                    ${numSvgChildren} children.`);
+      }
+
+      for (let i = 0; i < numSvgChildren; i++) {
+        const path = new PathNode(paths[i]);
+        const svgNode = new SvgNode([path], {
+          "width": "400em",
+          "height": height + "em",
+          "viewBox": `0 0 ${viewBoxWidth} ${viewBoxHeight}`,
+          "preserveAspectRatio": aligns[i] + " slice"
+        });
+        const span = buildCommon.makeSvgSpan([widthClasses[i]], [svgNode], options);
+
+        if (numSvgChildren === 1) {
+          return {
+            span,
+            minWidth,
+            height
+          };
+        } else {
+          span.style.height = height + "em";
+          spans.push(span);
+        }
+      }
+
+      return {
+        span: buildCommon.makeSpan(["stretchy"], spans, options),
+        minWidth,
+        height
+      };
+    }
+  } // buildSvgSpan_()
+
+
+  const _buildSvgSpan_ = buildSvgSpan_(),
+        span = _buildSvgSpan_.span,
+        minWidth = _buildSvgSpan_.minWidth,
+        height = _buildSvgSpan_.height; // Note that we are returning span.depth = 0.
+  // Any adjustments relative to the baseline must be done in buildHTML.
+
+
+  span.height = height;
+  span.style.height = height + "em";
+
+  if (minWidth > 0) {
+    span.style.minWidth = minWidth + "em";
+  }
+
+  return span;
+};
+
+const encloseSpan = function encloseSpan(inner, label, pad, options) {
+  // Return an image span for \cancel, \bcancel, \xcancel, or \fbox
+  let img;
+  const totalHeight = inner.height + inner.depth + 2 * pad;
+
+  if (/fbox|color/.test(label)) {
+    img = buildCommon.makeSpan(["stretchy", label], [], options);
+
+    if (label === "fbox") {
+      const color = options.color && options.getColor();
+
+      if (color) {
+        img.style.borderColor = color;
+      }
+    }
+  } else {
+    // \cancel, \bcancel, or \xcancel
+    // Since \cancel's SVG is inline and it omits the viewBox attribute,
+    // its stroke-width will not vary with span area.
+    const lines = [];
+
+    if (/^[bx]cancel$/.test(label)) {
+      lines.push(new LineNode({
+        "x1": "0",
+        "y1": "0",
+        "x2": "100%",
+        "y2": "100%",
+        "stroke-width": "0.046em"
+      }));
+    }
+
+    if (/^x?cancel$/.test(label)) {
+      lines.push(new LineNode({
+        "x1": "0",
+        "y1": "100%",
+        "x2": "100%",
+        "y2": "0",
+        "stroke-width": "0.046em"
+      }));
+    }
+
+    const svgNode = new SvgNode(lines, {
+      "width": "100%",
+      "height": totalHeight + "em"
+    });
+    img = buildCommon.makeSvgSpan([], [svgNode], options);
+  }
+
+  img.height = totalHeight;
+  img.style.height = totalHeight + "em";
+  return img;
+};
+
+var stretchy = {
+  encloseSpan,
+  mathMLnode,
+  svgSpan
+};
+
+/**
+ * Asserts that the node is of the given type and returns it with stricter
+ * typing. Throws if the node's type does not match.
+ */
+function assertNodeType(node, type) {
+  if (!node || node.type !== type) {
+    throw new Error(`Expected node of type ${type}, but got ` + (node ? `node of type ${node.type}` : String(node)));
+  }
+
+  return node;
+}
+/**
+ * Returns the node more strictly typed iff it is of the given type. Otherwise,
+ * returns null.
+ */
+
+function assertSymbolNodeType(node) {
+  const typedNode = checkSymbolNodeType(node);
+
+  if (!typedNode) {
+    throw new Error(`Expected node of symbol group type, but got ` + (node ? `node of type ${node.type}` : String(node)));
+  }
+
+  return typedNode;
+}
+/**
+ * Returns the node more strictly typed iff it is of the given type. Otherwise,
+ * returns null.
+ */
+
+function checkSymbolNodeType(node) {
+  if (node && (node.type === "atom" || NON_ATOMS.hasOwnProperty(node.type))) {
+    // $FlowFixMe
+    return node;
+  }
+
+  return null;
+}
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only "accent", but
+const htmlBuilder = (grp, options) => {
+  // Accents are handled in the TeXbook pg. 443, rule 12.
+  let base;
+  let group;
+  let supSubGroup;
+
+  if (grp && grp.type === "supsub") {
+    // If our base is a character box, and we have superscripts and
+    // subscripts, the supsub will defer to us. In particular, we want
+    // to attach the superscripts and subscripts to the inner body (so
+    // that the position of the superscripts and subscripts won't be
+    // affected by the height of the accent). We accomplish this by
+    // sticking the base of the accent into the base of the supsub, and
+    // rendering that, while keeping track of where the accent is.
+    // The real accent group is the base of the supsub group
+    group = assertNodeType(grp.base, "accent"); // The character box is the base of the accent group
+
+    base = group.base; // Stick the character box into the base of the supsub group
+
+    grp.base = base; // Rerender the supsub group with its new base, and store that
+    // result.
+
+    supSubGroup = assertSpan(buildGroup(grp, options)); // reset original base
+
+    grp.base = group;
+  } else {
+    group = assertNodeType(grp, "accent");
+    base = group.base;
+  } // Build the base group
+
+
+  const body = buildGroup(base, options.havingCrampedStyle()); // Does the accent need to shift for the skew of a character?
+
+  const mustShift = group.isShifty && utils.isCharacterBox(base); // Calculate the skew of the accent. This is based on the line "If the
+  // nucleus is not a single character, let s = 0; otherwise set s to the
+  // kern amount for the nucleus followed by the \skewchar of its font."
+  // Note that our skew metrics are just the kern between each character
+  // and the skewchar.
+
+  let skew = 0;
+
+  if (mustShift) {
+    // If the base is a character box, then we want the skew of the
+    // innermost character. To do that, we find the innermost character:
+    const baseChar = utils.getBaseElem(base); // Then, we render its group to get the symbol inside it
+
+    const baseGroup = buildGroup(baseChar, options.havingCrampedStyle()); // Finally, we pull the skew off of the symbol.
+
+    skew = assertSymbolDomNode(baseGroup).skew; // Note that we now throw away baseGroup, because the layers we
+    // removed with getBaseElem might contain things like \color which
+    // we can't get rid of.
+    // TODO(emily): Find a better way to get the skew
+  } // calculate the amount of space between the body and the accent
+
+
+  let clearance = Math.min(body.height, options.fontMetrics().xHeight); // Build the accent
+
+  let accentBody;
+
+  if (!group.isStretchy) {
+    let accent;
+    let width;
+
+    if (group.label === "\\vec") {
+      // Before version 0.9, \vec used the combining font glyph U+20D7.
+      // But browsers, especially Safari, are not consistent in how they
+      // render combining characters when not preceded by a character.
+      // So now we use an SVG.
+      // If Safari reforms, we should consider reverting to the glyph.
+      accent = buildCommon.staticSvg("vec", options);
+      width = buildCommon.svgData.vec[1];
+    } else {
+      accent = buildCommon.makeOrd({
+        mode: group.mode,
+        text: group.label
+      }, options, "textord");
+      accent = assertSymbolDomNode(accent); // Remove the italic correction of the accent, because it only serves to
+      // shift the accent over to a place we don't want.
+
+      accent.italic = 0;
+      width = accent.width;
+    }
+
+    accentBody = buildCommon.makeSpan(["accent-body"], [accent]); // "Full" accents expand the width of the resulting symbol to be
+    // at least the width of the accent, and overlap directly onto the
+    // character without any vertical offset.
+
+    const accentFull = group.label === "\\textcircled";
+
+    if (accentFull) {
+      accentBody.classes.push('accent-full');
+      clearance = body.height;
+    } // Shift the accent over by the skew.
+
+
+    let left = skew; // CSS defines `.katex .accent .accent-body:not(.accent-full) { width: 0 }`
+    // so that the accent doesn't contribute to the bounding box.
+    // We need to shift the character by its width (effectively half
+    // its width) to compensate.
+
+    if (!accentFull) {
+      left -= width / 2;
+    }
+
+    accentBody.style.left = left + "em"; // \textcircled uses the \bigcirc glyph, so it needs some
+    // vertical adjustment to match LaTeX.
+
+    if (group.label === "\\textcircled") {
+      accentBody.style.top = ".2em";
+    }
+
+    accentBody = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "kern",
+        size: -clearance
+      }, {
+        type: "elem",
+        elem: accentBody
+      }]
+    }, options);
+  } else {
+    accentBody = stretchy.svgSpan(group, options);
+    accentBody = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "elem",
+        elem: accentBody,
+        wrapperClasses: ["svg-align"],
+        wrapperStyle: skew > 0 ? {
+          width: `calc(100% - ${2 * skew}em)`,
+          marginLeft: `${2 * skew}em`
+        } : undefined
+      }]
+    }, options);
+  }
+
+  const accentWrap = buildCommon.makeSpan(["mord", "accent"], [accentBody], options);
+
+  if (supSubGroup) {
+    // Here, we replace the "base" child of the supsub with our newly
+    // generated accent.
+    supSubGroup.children[0] = accentWrap; // Since we don't rerun the height calculation after replacing the
+    // accent, we manually recalculate height.
+
+    supSubGroup.height = Math.max(accentWrap.height, supSubGroup.height); // Accents should always be ords, even when their innards are not.
+
+    supSubGroup.classes[0] = "mord";
+    return supSubGroup;
+  } else {
+    return accentWrap;
+  }
+};
+
+const mathmlBuilder = (group, options) => {
+  const accentNode = group.isStretchy ? stretchy.mathMLnode(group.label) : new mathMLTree.MathNode("mo", [makeText(group.label, group.mode)]);
+  const node = new mathMLTree.MathNode("mover", [buildGroup$1(group.base, options), accentNode]);
+  node.setAttribute("accent", "true");
+  return node;
+};
+
+const NON_STRETCHY_ACCENT_REGEX = new RegExp(["\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve", "\\check", "\\hat", "\\vec", "\\dot", "\\mathring"].map(accent => `\\${accent}`).join("|")); // Accents
+
+defineFunction({
+  type: "accent",
+  names: ["\\acute", "\\grave", "\\ddot", "\\tilde", "\\bar", "\\breve", "\\check", "\\hat", "\\vec", "\\dot", "\\mathring", "\\widecheck", "\\widehat", "\\widetilde", "\\overrightarrow", "\\overleftarrow", "\\Overrightarrow", "\\overleftrightarrow", "\\overgroup", "\\overlinesegment", "\\overleftharpoon", "\\overrightharpoon"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    const base = args[0];
+    const isStretchy = !NON_STRETCHY_ACCENT_REGEX.test(context.funcName);
+    const isShifty = !isStretchy || context.funcName === "\\widehat" || context.funcName === "\\widetilde" || context.funcName === "\\widecheck";
+    return {
+      type: "accent",
+      mode: context.parser.mode,
+      label: context.funcName,
+      isStretchy: isStretchy,
+      isShifty: isShifty,
+      base: base
+    };
+  },
+  htmlBuilder,
+  mathmlBuilder
+}); // Text-mode accents
+
+defineFunction({
+  type: "accent",
+  names: ["\\'", "\\`", "\\^", "\\~", "\\=", "\\u", "\\.", '\\"', "\\r", "\\H", "\\v", "\\textcircled"],
+  props: {
+    numArgs: 1,
+    allowedInText: true,
+    allowedInMath: false
+  },
+  handler: (context, args) => {
+    const base = args[0];
+    return {
+      type: "accent",
+      mode: context.parser.mode,
+      label: context.funcName,
+      isStretchy: false,
+      isShifty: true,
+      base: base
+    };
+  },
+  htmlBuilder,
+  mathmlBuilder
+});
+
+// Horizontal overlap functions
+defineFunction({
+  type: "accentUnder",
+  names: ["\\underleftarrow", "\\underrightarrow", "\\underleftrightarrow", "\\undergroup", "\\underlinesegment", "\\utilde"],
+  props: {
+    numArgs: 1
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const base = args[0];
+    return {
+      type: "accentUnder",
+      mode: parser.mode,
+      label: funcName,
+      base: base
+    };
+  },
+  htmlBuilder: (group, options) => {
+    // Treat under accents much like underlines.
+    const innerGroup = buildGroup(group.base, options);
+    const accentBody = stretchy.svgSpan(group, options);
+    const kern = group.label === "\\utilde" ? 0.12 : 0; // Generate the vlist, with the appropriate kerns
+
+    const vlist = buildCommon.makeVList({
+      positionType: "top",
+      positionData: innerGroup.height,
+      children: [{
+        type: "elem",
+        elem: accentBody,
+        wrapperClasses: ["svg-align"]
+      }, {
+        type: "kern",
+        size: kern
+      }, {
+        type: "elem",
+        elem: innerGroup
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "accentunder"], [vlist], options);
+  },
+  mathmlBuilder: (group, options) => {
+    const accentNode = stretchy.mathMLnode(group.label);
+    const node = new mathMLTree.MathNode("munder", [buildGroup$1(group.base, options), accentNode]);
+    node.setAttribute("accentunder", "true");
+    return node;
+  }
+});
+
+// Helper function
+const paddedNode = group => {
+  const node = new mathMLTree.MathNode("mpadded", group ? [group] : []);
+  node.setAttribute("width", "+0.6em");
+  node.setAttribute("lspace", "0.3em");
+  return node;
+}; // Stretchy arrows with an optional argument
+
+
+defineFunction({
+  type: "xArrow",
+  names: ["\\xleftarrow", "\\xrightarrow", "\\xLeftarrow", "\\xRightarrow", "\\xleftrightarrow", "\\xLeftrightarrow", "\\xhookleftarrow", "\\xhookrightarrow", "\\xmapsto", "\\xrightharpoondown", "\\xrightharpoonup", "\\xleftharpoondown", "\\xleftharpoonup", "\\xrightleftharpoons", "\\xleftrightharpoons", "\\xlongequal", "\\xtwoheadrightarrow", "\\xtwoheadleftarrow", "\\xtofrom", // The next 3 functions are here to support the mhchem extension.
+  // Direct use of these functions is discouraged and may break someday.
+  "\\xrightleftarrows", "\\xrightequilibrium", "\\xleftequilibrium"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    return {
+      type: "xArrow",
+      mode: parser.mode,
+      label: funcName,
+      body: args[0],
+      below: optArgs[0]
+    };
+  },
+
+  // Flow is unable to correctly infer the type of `group`, even though it's
+  // unamibiguously determined from the passed-in `type` above.
+  htmlBuilder(group, options) {
+    const style = options.style; // Build the argument groups in the appropriate style.
+    // Ref: amsmath.dtx:   \hbox{$\scriptstyle\mkern#3mu{#6}\mkern#4mu$}%
+    // Some groups can return document fragments.  Handle those by wrapping
+    // them in a span.
+
+    let newOptions = options.havingStyle(style.sup());
+    const upperGroup = buildCommon.wrapFragment(buildGroup(group.body, newOptions, options), options);
+    upperGroup.classes.push("x-arrow-pad");
+    let lowerGroup;
+
+    if (group.below) {
+      // Build the lower group
+      newOptions = options.havingStyle(style.sub());
+      lowerGroup = buildCommon.wrapFragment(buildGroup(group.below, newOptions, options), options);
+      lowerGroup.classes.push("x-arrow-pad");
+    }
+
+    const arrowBody = stretchy.svgSpan(group, options); // Re shift: Note that stretchy.svgSpan returned arrowBody.depth = 0.
+    // The point we want on the math axis is at 0.5 * arrowBody.height.
+
+    const arrowShift = -options.fontMetrics().axisHeight + 0.5 * arrowBody.height; // 2 mu kern. Ref: amsmath.dtx: #7\if0#2\else\mkern#2mu\fi
+
+    let upperShift = -options.fontMetrics().axisHeight - 0.5 * arrowBody.height - 0.111; // 0.111 em = 2 mu
+
+    if (upperGroup.depth > 0.25 || group.label === "\\xleftequilibrium") {
+      upperShift -= upperGroup.depth; // shift up if depth encroaches
+    } // Generate the vlist
+
+
+    let vlist;
+
+    if (lowerGroup) {
+      const lowerShift = -options.fontMetrics().axisHeight + lowerGroup.height + 0.5 * arrowBody.height + 0.111;
+      vlist = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: upperGroup,
+          shift: upperShift
+        }, {
+          type: "elem",
+          elem: arrowBody,
+          shift: arrowShift
+        }, {
+          type: "elem",
+          elem: lowerGroup,
+          shift: lowerShift
+        }]
+      }, options);
+    } else {
+      vlist = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: upperGroup,
+          shift: upperShift
+        }, {
+          type: "elem",
+          elem: arrowBody,
+          shift: arrowShift
+        }]
+      }, options);
+    } // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+
+    vlist.children[0].children[0].children[1].classes.push("svg-align");
+    return buildCommon.makeSpan(["mrel", "x-arrow"], [vlist], options);
+  },
+
+  mathmlBuilder(group, options) {
+    const arrowNode = stretchy.mathMLnode(group.label);
+    let node;
+
+    if (group.body) {
+      const upperNode = paddedNode(buildGroup$1(group.body, options));
+
+      if (group.below) {
+        const lowerNode = paddedNode(buildGroup$1(group.below, options));
+        node = new mathMLTree.MathNode("munderover", [arrowNode, lowerNode, upperNode]);
+      } else {
+        node = new mathMLTree.MathNode("mover", [arrowNode, upperNode]);
+      }
+    } else if (group.below) {
+      const lowerNode = paddedNode(buildGroup$1(group.below, options));
+      node = new mathMLTree.MathNode("munder", [arrowNode, lowerNode]);
+    } else {
+      // This should never happen.
+      // Parser.js throws an error if there is no argument.
+      node = paddedNode();
+      node = new mathMLTree.MathNode("mover", [arrowNode, node]);
+    }
+
+    return node;
+  }
+
+});
+
+// {123} and converts into symbol with code 123.  It is used by the *macro*
+// \char defined in macros.js.
+
+defineFunction({
+  type: "textord",
+  names: ["\\@char"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    const arg = assertNodeType(args[0], "ordgroup");
+    const group = arg.body;
+    let number = "";
+
+    for (let i = 0; i < group.length; i++) {
+      const node = assertNodeType(group[i], "textord");
+      number += node.text;
+    }
+
+    const code = parseInt(number);
+
+    if (isNaN(code)) {
+      throw new ParseError(`\\@char has non-numeric argument ${number}`);
+    }
+
+    return {
+      type: "textord",
+      mode: parser.mode,
+      text: String.fromCharCode(code)
+    };
+  }
+
+});
+
+const htmlBuilder$1 = (group, options) => {
+  const elements = buildExpression(group.body, options.withColor(group.color), false); // \color isn't supposed to affect the type of the elements it contains.
+  // To accomplish this, we wrap the results in a fragment, so the inner
+  // elements will be able to directly interact with their neighbors. For
+  // example, `\color{red}{2 +} 3` has the same spacing as `2 + 3`
+
+  return buildCommon.makeFragment(elements);
+};
+
+const mathmlBuilder$1 = (group, options) => {
+  const inner = buildExpression$1(group.body, options.withColor(group.color));
+  const node = new mathMLTree.MathNode("mstyle", inner);
+  node.setAttribute("mathcolor", group.color);
+  return node;
+};
+
+defineFunction({
+  type: "color",
+  names: ["\\textcolor"],
+  props: {
+    numArgs: 2,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "original"]
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    const color = assertNodeType(args[0], "color-token").color;
+    const body = args[1];
+    return {
+      type: "color",
+      mode: parser.mode,
+      color,
+      body: ordargument(body)
+    };
+  },
+
+  htmlBuilder: htmlBuilder$1,
+  mathmlBuilder: mathmlBuilder$1
+});
+defineFunction({
+  type: "color",
+  names: ["\\color"],
+  props: {
+    numArgs: 1,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color"]
+  },
+
+  handler(_ref2, args) {
+    let parser = _ref2.parser,
+        breakOnTokenText = _ref2.breakOnTokenText;
+    const color = assertNodeType(args[0], "color-token").color; // Set macro \current@color in current namespace to store the current
+    // color, mimicking the behavior of color.sty.
+    // This is currently used just to correctly color a \right
+    // that follows a \color command.
+
+    parser.gullet.macros.set("\\current@color", color); // Parse out the implicit body that should be colored.
+
+    const body = parser.parseExpression(true, breakOnTokenText);
+    return {
+      type: "color",
+      mode: parser.mode,
+      color,
+      body
+    };
+  },
+
+  htmlBuilder: htmlBuilder$1,
+  mathmlBuilder: mathmlBuilder$1
+});
+
+// Row breaks within tabular environments, and line breaks at top level
+// same signature, we implement them as one megafunction, with newRow
+// indicating whether we're in the \cr case, and newLine indicating whether
+// to break the line in the \newline case.
+
+defineFunction({
+  type: "cr",
+  names: ["\\cr", "\\newline"],
+  props: {
+    numArgs: 0,
+    numOptionalArgs: 1,
+    argTypes: ["size"],
+    allowedInText: true
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const size = optArgs[0];
+    const newRow = funcName === "\\cr";
+    let newLine = false;
+
+    if (!newRow) {
+      if (parser.settings.displayMode && parser.settings.useStrictBehavior("newLineInDisplayMode", "In LaTeX, \\\\ or \\newline " + "does nothing in display mode")) {
+        newLine = false;
+      } else {
+        newLine = true;
+      }
+    }
+
+    return {
+      type: "cr",
+      mode: parser.mode,
+      newLine,
+      newRow,
+      size: size && assertNodeType(size, "size").value
+    };
+  },
+
+  // The following builders are called only at the top level,
+  // not within tabular/array environments.
+  htmlBuilder(group, options) {
+    if (group.newRow) {
+      throw new ParseError("\\cr valid only within a tabular/array environment");
+    }
+
+    const span = buildCommon.makeSpan(["mspace"], [], options);
+
+    if (group.newLine) {
+      span.classes.push("newline");
+
+      if (group.size) {
+        span.style.marginTop = calculateSize(group.size, options) + "em";
+      }
+    }
+
+    return span;
+  },
+
+  mathmlBuilder(group, options) {
+    const node = new mathMLTree.MathNode("mspace");
+
+    if (group.newLine) {
+      node.setAttribute("linebreak", "newline");
+
+      if (group.size) {
+        node.setAttribute("height", calculateSize(group.size, options) + "em");
+      }
+    }
+
+    return node;
+  }
+
+});
+
+const globalMap = {
+  "\\global": "\\global",
+  "\\long": "\\\\globallong",
+  "\\\\globallong": "\\\\globallong",
+  "\\def": "\\gdef",
+  "\\gdef": "\\gdef",
+  "\\edef": "\\xdef",
+  "\\xdef": "\\xdef",
+  "\\let": "\\\\globallet",
+  "\\futurelet": "\\\\globalfuture"
+};
+
+const checkControlSequence = tok => {
+  const name = tok.text;
+
+  if (/^(?:[\\{}$&#^_]|EOF)$/.test(name)) {
+    throw new ParseError("Expected a control sequence", tok);
+  }
+
+  return name;
+};
+
+const getRHS = parser => {
+  let tok = parser.gullet.popToken();
+
+  if (tok.text === "=") {
+    // consume optional equals
+    tok = parser.gullet.popToken();
+
+    if (tok.text === " ") {
+      // consume one optional space
+      tok = parser.gullet.popToken();
+    }
+  }
+
+  return tok;
+};
+
+const letCommand = (parser, name, tok, global) => {
+  let macro = parser.gullet.macros.get(tok.text);
+
+  if (macro == null) {
+    // don't expand it later even if a macro with the same name is defined
+    // e.g., \let\foo=\frac \def\frac{\relax} \frac12
+    tok.noexpand = true;
+    macro = {
+      tokens: [tok],
+      numArgs: 0,
+      // reproduce the same behavior in expansion
+      unexpandable: !parser.gullet.isExpandable(tok.text)
+    };
+  }
+
+  parser.gullet.macros.set(name, macro, global);
+}; // <assignment> -> <non-macro assignment>|<macro assignment>
+// <non-macro assignment> -> <simple assignment>|\global<non-macro assignment>
+// <macro assignment> -> <definition>|<prefix><macro assignment>
+// <prefix> -> \global|\long|\outer
+
+
+defineFunction({
+  type: "internal",
+  names: ["\\global", "\\long", "\\\\globallong"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    parser.consumeSpaces();
+    const token = parser.fetch();
+
+    if (globalMap[token.text]) {
+      // KaTeX doesn't have \par, so ignore \long
+      if (funcName === "\\global" || funcName === "\\\\globallong") {
+        token.text = globalMap[token.text];
+      }
+
+      return assertNodeType(parser.parseFunction(), "internal");
+    }
+
+    throw new ParseError(`Invalid token after macro prefix`, token);
+  }
+
+}); // Basic support for macro definitions: \def, \gdef, \edef, \xdef
+// <definition> -> <def><control sequence><definition text>
+// <def> -> \def|\gdef|\edef|\xdef
+// <definition text> -> <parameter text><left brace><balanced text><right brace>
+
+defineFunction({
+  type: "internal",
+  names: ["\\def", "\\gdef", "\\edef", "\\xdef"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref2) {
+    let parser = _ref2.parser,
+        funcName = _ref2.funcName;
+    let arg = parser.gullet.consumeArgs(1)[0];
+
+    if (arg.length !== 1) {
+      throw new ParseError("\\gdef's first argument must be a macro name");
+    }
+
+    const name = arg[0].text; // Count argument specifiers, and check they are in the order #1 #2 ...
+
+    let numArgs = 0;
+    arg = parser.gullet.consumeArgs(1)[0];
+
+    while (arg.length === 1 && arg[0].text === "#") {
+      arg = parser.gullet.consumeArgs(1)[0];
+
+      if (arg.length !== 1) {
+        throw new ParseError(`Invalid argument number length "${arg.length}"`);
+      }
+
+      if (!/^[1-9]$/.test(arg[0].text)) {
+        throw new ParseError(`Invalid argument number "${arg[0].text}"`);
+      }
+
+      numArgs++;
+
+      if (parseInt(arg[0].text) !== numArgs) {
+        throw new ParseError(`Argument number "${arg[0].text}" out of order`);
+      }
+
+      arg = parser.gullet.consumeArgs(1)[0];
+    }
+
+    if (funcName === "\\edef" || funcName === "\\xdef") {
+      arg = parser.gullet.expandTokens(arg);
+      arg.reverse(); // to fit in with stack order
+    } // Final arg is the expansion of the macro
+
+
+    parser.gullet.macros.set(name, {
+      tokens: arg,
+      numArgs
+    }, funcName === globalMap[funcName]);
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+
+}); // <simple assignment> -> <let assignment>
+// <let assignment> -> \futurelet<control sequence><token><token>
+//     | \let<control sequence><equals><one optional space><token>
+// <equals> -> <optional spaces>|<optional spaces>=
+
+defineFunction({
+  type: "internal",
+  names: ["\\let", "\\\\globallet"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref3) {
+    let parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    const name = checkControlSequence(parser.gullet.popToken());
+    parser.gullet.consumeSpaces();
+    const tok = getRHS(parser);
+    letCommand(parser, name, tok, funcName === "\\\\globallet");
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+
+}); // ref: https://www.tug.org/TUGboat/tb09-3/tb22bechtolsheim.pdf
+
+defineFunction({
+  type: "internal",
+  names: ["\\futurelet", "\\\\globalfuture"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref4) {
+    let parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    const name = checkControlSequence(parser.gullet.popToken());
+    const middle = parser.gullet.popToken();
+    const tok = parser.gullet.popToken();
+    letCommand(parser, name, tok, funcName === "\\\\globalfuture");
+    parser.gullet.pushToken(tok);
+    parser.gullet.pushToken(middle);
+    return {
+      type: "internal",
+      mode: parser.mode
+    };
+  }
+
+});
+
+/**
+ * This file deals with creating delimiters of various sizes. The TeXbook
+ * discusses these routines on page 441-442, in the "Another subroutine sets box
+ * x to a specified variable delimiter" paragraph.
+ *
+ * There are three main routines here. `makeSmallDelim` makes a delimiter in the
+ * normal font, but in either text, script, or scriptscript style.
+ * `makeLargeDelim` makes a delimiter in textstyle, but in one of the Size1,
+ * Size2, Size3, or Size4 fonts. `makeStackedDelim` makes a delimiter out of
+ * smaller pieces that are stacked on top of one another.
+ *
+ * The functions take a parameter `center`, which determines if the delimiter
+ * should be centered around the axis.
+ *
+ * Then, there are three exposed functions. `sizedDelim` makes a delimiter in
+ * one of the given sizes. This is used for things like `\bigl`.
+ * `customSizedDelim` makes a delimiter with a given total height+depth. It is
+ * called in places like `\sqrt`. `leftRightDelim` makes an appropriate
+ * delimiter which surrounds an expression of a given height an depth. It is
+ * used in `\left` and `\right`.
+ */
+
+/**
+ * Get the metrics for a given symbol and font, after transformation (i.e.
+ * after following replacement from symbols.js)
+ */
+const getMetrics = function getMetrics(symbol, font, mode) {
+  const replace = symbols.math[symbol] && symbols.math[symbol].replace;
+  const metrics = getCharacterMetrics(replace || symbol, font, mode);
+
+  if (!metrics) {
+    throw new Error(`Unsupported symbol ${symbol} and font size ${font}.`);
+  }
+
+  return metrics;
+};
+/**
+ * Puts a delimiter span in a given style, and adds appropriate height, depth,
+ * and maxFontSizes.
+ */
+
+
+const styleWrap = function styleWrap(delim, toStyle, options, classes) {
+  const newOptions = options.havingBaseStyle(toStyle);
+  const span = buildCommon.makeSpan(classes.concat(newOptions.sizingClasses(options)), [delim], options);
+  const delimSizeMultiplier = newOptions.sizeMultiplier / options.sizeMultiplier;
+  span.height *= delimSizeMultiplier;
+  span.depth *= delimSizeMultiplier;
+  span.maxFontSize = newOptions.sizeMultiplier;
+  return span;
+};
+
+const centerSpan = function centerSpan(span, options, style) {
+  const newOptions = options.havingBaseStyle(style);
+  const shift = (1 - options.sizeMultiplier / newOptions.sizeMultiplier) * options.fontMetrics().axisHeight;
+  span.classes.push("delimcenter");
+  span.style.top = shift + "em";
+  span.height -= shift;
+  span.depth += shift;
+};
+/**
+ * Makes a small delimiter. This is a delimiter that comes in the Main-Regular
+ * font, but is restyled to either be in textstyle, scriptstyle, or
+ * scriptscriptstyle.
+ */
+
+
+const makeSmallDelim = function makeSmallDelim(delim, style, center, options, mode, classes) {
+  const text = buildCommon.makeSymbol(delim, "Main-Regular", mode, options);
+  const span = styleWrap(text, style, options, classes);
+
+  if (center) {
+    centerSpan(span, options, style);
+  }
+
+  return span;
+};
+/**
+ * Builds a symbol in the given font size (note size is an integer)
+ */
+
+
+const mathrmSize = function mathrmSize(value, size, mode, options) {
+  return buildCommon.makeSymbol(value, "Size" + size + "-Regular", mode, options);
+};
+/**
+ * Makes a large delimiter. This is a delimiter that comes in the Size1, Size2,
+ * Size3, or Size4 fonts. It is always rendered in textstyle.
+ */
+
+
+const makeLargeDelim = function makeLargeDelim(delim, size, center, options, mode, classes) {
+  const inner = mathrmSize(delim, size, mode, options);
+  const span = styleWrap(buildCommon.makeSpan(["delimsizing", "size" + size], [inner], options), Style$1.TEXT, options, classes);
+
+  if (center) {
+    centerSpan(span, options, Style$1.TEXT);
+  }
+
+  return span;
+};
+/**
+ * Make an inner span with the given offset and in the given font. This is used
+ * in `makeStackedDelim` to make the stacking pieces for the delimiter.
+ */
+
+
+const makeInner = function makeInner(symbol, font, mode) {
+  let sizeClass; // Apply the correct CSS class to choose the right font.
+
+  if (font === "Size1-Regular") {
+    sizeClass = "delim-size1";
+  } else
+    /* if (font === "Size4-Regular") */
+    {
+      sizeClass = "delim-size4";
+    }
+
+  const inner = buildCommon.makeSpan(["delimsizinginner", sizeClass], [buildCommon.makeSpan([], [buildCommon.makeSymbol(symbol, font, mode)])]); // Since this will be passed into `makeVList` in the end, wrap the element
+  // in the appropriate tag that VList uses.
+
+  return {
+    type: "elem",
+    elem: inner
+  };
+}; // Helper for makeStackedDelim
+
+
+const lap = {
+  type: "kern",
+  size: -0.005
+};
+/**
+ * Make a stacked delimiter out of a given delimiter, with the total height at
+ * least `heightTotal`. This routine is mentioned on page 442 of the TeXbook.
+ */
+
+const makeStackedDelim = function makeStackedDelim(delim, heightTotal, center, options, mode, classes) {
+  // There are four parts, the top, an optional middle, a repeated part, and a
+  // bottom.
+  let top;
+  let middle;
+  let repeat;
+  let bottom;
+  top = repeat = bottom = delim;
+  middle = null; // Also keep track of what font the delimiters are in
+
+  let font = "Size1-Regular"; // We set the parts and font based on the symbol. Note that we use
+  // '\u23d0' instead of '|' and '\u2016' instead of '\\|' for the
+  // repeats of the arrows
+
+  if (delim === "\\uparrow") {
+    repeat = bottom = "\u23d0";
+  } else if (delim === "\\Uparrow") {
+    repeat = bottom = "\u2016";
+  } else if (delim === "\\downarrow") {
+    top = repeat = "\u23d0";
+  } else if (delim === "\\Downarrow") {
+    top = repeat = "\u2016";
+  } else if (delim === "\\updownarrow") {
+    top = "\\uparrow";
+    repeat = "\u23d0";
+    bottom = "\\downarrow";
+  } else if (delim === "\\Updownarrow") {
+    top = "\\Uparrow";
+    repeat = "\u2016";
+    bottom = "\\Downarrow";
+  } else if (delim === "[" || delim === "\\lbrack") {
+    top = "\u23a1";
+    repeat = "\u23a2";
+    bottom = "\u23a3";
+    font = "Size4-Regular";
+  } else if (delim === "]" || delim === "\\rbrack") {
+    top = "\u23a4";
+    repeat = "\u23a5";
+    bottom = "\u23a6";
+    font = "Size4-Regular";
+  } else if (delim === "\\lfloor" || delim === "\u230a") {
+    repeat = top = "\u23a2";
+    bottom = "\u23a3";
+    font = "Size4-Regular";
+  } else if (delim === "\\lceil" || delim === "\u2308") {
+    top = "\u23a1";
+    repeat = bottom = "\u23a2";
+    font = "Size4-Regular";
+  } else if (delim === "\\rfloor" || delim === "\u230b") {
+    repeat = top = "\u23a5";
+    bottom = "\u23a6";
+    font = "Size4-Regular";
+  } else if (delim === "\\rceil" || delim === "\u2309") {
+    top = "\u23a4";
+    repeat = bottom = "\u23a5";
+    font = "Size4-Regular";
+  } else if (delim === "(" || delim === "\\lparen") {
+    top = "\u239b";
+    repeat = "\u239c";
+    bottom = "\u239d";
+    font = "Size4-Regular";
+  } else if (delim === ")" || delim === "\\rparen") {
+    top = "\u239e";
+    repeat = "\u239f";
+    bottom = "\u23a0";
+    font = "Size4-Regular";
+  } else if (delim === "\\{" || delim === "\\lbrace") {
+    top = "\u23a7";
+    middle = "\u23a8";
+    bottom = "\u23a9";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\}" || delim === "\\rbrace") {
+    top = "\u23ab";
+    middle = "\u23ac";
+    bottom = "\u23ad";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\lgroup" || delim === "\u27ee") {
+    top = "\u23a7";
+    bottom = "\u23a9";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\rgroup" || delim === "\u27ef") {
+    top = "\u23ab";
+    bottom = "\u23ad";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\lmoustache" || delim === "\u23b0") {
+    top = "\u23a7";
+    bottom = "\u23ad";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } else if (delim === "\\rmoustache" || delim === "\u23b1") {
+    top = "\u23ab";
+    bottom = "\u23a9";
+    repeat = "\u23aa";
+    font = "Size4-Regular";
+  } // Get the metrics of the four sections
+
+
+  const topMetrics = getMetrics(top, font, mode);
+  const topHeightTotal = topMetrics.height + topMetrics.depth;
+  const repeatMetrics = getMetrics(repeat, font, mode);
+  const repeatHeightTotal = repeatMetrics.height + repeatMetrics.depth;
+  const bottomMetrics = getMetrics(bottom, font, mode);
+  const bottomHeightTotal = bottomMetrics.height + bottomMetrics.depth;
+  let middleHeightTotal = 0;
+  let middleFactor = 1;
+
+  if (middle !== null) {
+    const middleMetrics = getMetrics(middle, font, mode);
+    middleHeightTotal = middleMetrics.height + middleMetrics.depth;
+    middleFactor = 2; // repeat symmetrically above and below middle
+  } // Calcuate the minimal height that the delimiter can have.
+  // It is at least the size of the top, bottom, and optional middle combined.
+
+
+  const minHeight = topHeightTotal + bottomHeightTotal + middleHeightTotal; // Compute the number of copies of the repeat symbol we will need
+
+  const repeatCount = Math.max(0, Math.ceil((heightTotal - minHeight) / (middleFactor * repeatHeightTotal))); // Compute the total height of the delimiter including all the symbols
+
+  const realHeightTotal = minHeight + repeatCount * middleFactor * repeatHeightTotal; // The center of the delimiter is placed at the center of the axis. Note
+  // that in this context, "center" means that the delimiter should be
+  // centered around the axis in the current style, while normally it is
+  // centered around the axis in textstyle.
+
+  let axisHeight = options.fontMetrics().axisHeight;
+
+  if (center) {
+    axisHeight *= options.sizeMultiplier;
+  } // Calculate the depth
+
+
+  const depth = realHeightTotal / 2 - axisHeight; // This function differs from the TeX procedure in one way.
+  // We shift each repeat element downwards by 0.005em, to prevent a gap
+  // due to browser floating point rounding error.
+  // Then, at the last element-to element joint, we add one extra repeat
+  // element to cover the gap created by the shifts.
+  // Find the shift needed to align the upper end of the extra element at a point
+  // 0.005em above the lower end of the top element.
+
+  const shiftOfExtraElement = (repeatCount + 1) * 0.005 - repeatHeightTotal; // Now, we start building the pieces that will go into the vlist
+  // Keep a list of the inner pieces
+
+  const inners = []; // Add the bottom symbol
+
+  inners.push(makeInner(bottom, font, mode));
+
+  if (middle === null) {
+    // Add that many symbols
+    for (let i = 0; i < repeatCount; i++) {
+      inners.push(lap); // overlap
+
+      inners.push(makeInner(repeat, font, mode));
+    }
+  } else {
+    // When there is a middle bit, we need the middle part and two repeated
+    // sections
+    for (let i = 0; i < repeatCount; i++) {
+      inners.push(lap);
+      inners.push(makeInner(repeat, font, mode));
+    } // Insert one extra repeat element.
+
+
+    inners.push({
+      type: "kern",
+      size: shiftOfExtraElement
+    });
+    inners.push(makeInner(repeat, font, mode));
+    inners.push(lap); // Now insert the middle of the brace.
+
+    inners.push(makeInner(middle, font, mode));
+
+    for (let i = 0; i < repeatCount; i++) {
+      inners.push(lap);
+      inners.push(makeInner(repeat, font, mode));
+    }
+  } // To cover the gap create by the overlaps, insert one more repeat element,
+  // at a position that juts 0.005 above the bottom of the top element.
+
+
+  if ((repeat === "\u239c" || repeat === "\u239f") && repeatCount === 0) {
+    // Parentheses need a short repeat element in order to avoid an overrun.
+    // We'll make a 0.3em tall element from a SVG.
+    const overlap = buildCommon.svgData.leftParenInner[2] / 2;
+    inners.push({
+      type: "kern",
+      size: -overlap
+    });
+    const pathName = repeat === "\u239c" ? "leftParenInner" : "rightParenInner";
+    const innerSpan = buildCommon.staticSvg(pathName, options);
+    inners.push({
+      type: "elem",
+      elem: innerSpan
+    });
+    inners.push({
+      type: "kern",
+      size: -overlap
+    });
+  } else {
+    inners.push({
+      type: "kern",
+      size: shiftOfExtraElement
+    });
+    inners.push(makeInner(repeat, font, mode));
+    inners.push(lap);
+  } // Add the top symbol
+
+
+  inners.push(makeInner(top, font, mode)); // Finally, build the vlist
+
+  const newOptions = options.havingBaseStyle(Style$1.TEXT);
+  const inner = buildCommon.makeVList({
+    positionType: "bottom",
+    positionData: depth,
+    children: inners
+  }, newOptions);
+  return styleWrap(buildCommon.makeSpan(["delimsizing", "mult"], [inner], newOptions), Style$1.TEXT, options, classes);
+}; // All surds have 0.08em padding above the viniculum inside the SVG.
+// That keeps browser span height rounding error from pinching the line.
+
+
+const vbPad = 80; // padding above the surd, measured inside the viewBox.
+
+const emPad = 0.08; // padding, in ems, measured in the document.
+
+const sqrtSvg = function sqrtSvg(sqrtName, height, viewBoxHeight, extraViniculum, options) {
+  const path = sqrtPath(sqrtName, extraViniculum, viewBoxHeight);
+  const pathNode = new PathNode(sqrtName, path);
+  const svg = new SvgNode([pathNode], {
+    // Note: 1000:1 ratio of viewBox to document em width.
+    "width": "400em",
+    "height": height + "em",
+    "viewBox": "0 0 400000 " + viewBoxHeight,
+    "preserveAspectRatio": "xMinYMin slice"
+  });
+  return buildCommon.makeSvgSpan(["hide-tail"], [svg], options);
+};
+/**
+ * Make a sqrt image of the given height,
+ */
+
+
+const makeSqrtImage = function makeSqrtImage(height, options) {
+  // Define a newOptions that removes the effect of size changes such as \Huge.
+  // We don't pick different a height surd for \Huge. For it, we scale up.
+  const newOptions = options.havingBaseSizing(); // Pick the desired surd glyph from a sequence of surds.
+
+  const delim = traverseSequence("\\surd", height * newOptions.sizeMultiplier, stackLargeDelimiterSequence, newOptions);
+  let sizeMultiplier = newOptions.sizeMultiplier; // default
+  // The standard sqrt SVGs each have a 0.04em thick viniculum.
+  // If Settings.minRuleThickness is larger than that, we add extraViniculum.
+
+  const extraViniculum = Math.max(0, options.minRuleThickness - options.fontMetrics().sqrtRuleThickness); // Create a span containing an SVG image of a sqrt symbol.
+
+  let span;
+  let spanHeight = 0;
+  let texHeight = 0;
+  let viewBoxHeight = 0;
+  let advanceWidth; // We create viewBoxes with 80 units of "padding" above each surd.
+  // Then browser rounding error on the parent span height will not
+  // encroach on the ink of the viniculum. But that padding is not
+  // included in the TeX-like `height` used for calculation of
+  // vertical alignment. So texHeight = span.height < span.style.height.
+
+  if (delim.type === "small") {
+    // Get an SVG that is derived from glyph U+221A in font KaTeX-Main.
+    // 1000 unit normal glyph height.
+    viewBoxHeight = 1000 + 1000 * extraViniculum + vbPad;
+
+    if (height < 1.0) {
+      sizeMultiplier = 1.0; // mimic a \textfont radical
+    } else if (height < 1.4) {
+      sizeMultiplier = 0.7; // mimic a \scriptfont radical
+    }
+
+    spanHeight = (1.0 + extraViniculum + emPad) / sizeMultiplier;
+    texHeight = (1.00 + extraViniculum) / sizeMultiplier;
+    span = sqrtSvg("sqrtMain", spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "0.853em";
+    advanceWidth = 0.833 / sizeMultiplier; // from the font.
+  } else if (delim.type === "large") {
+    // These SVGs come from fonts: KaTeX_Size1, _Size2, etc.
+    viewBoxHeight = (1000 + vbPad) * sizeToMaxHeight[delim.size];
+    texHeight = (sizeToMaxHeight[delim.size] + extraViniculum) / sizeMultiplier;
+    spanHeight = (sizeToMaxHeight[delim.size] + extraViniculum + emPad) / sizeMultiplier;
+    span = sqrtSvg("sqrtSize" + delim.size, spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "1.02em";
+    advanceWidth = 1.0 / sizeMultiplier; // 1.0 from the font.
+  } else {
+    // Tall sqrt. In TeX, this would be stacked using multiple glyphs.
+    // We'll use a single SVG to accomplish the same thing.
+    spanHeight = height + extraViniculum + emPad;
+    texHeight = height + extraViniculum;
+    viewBoxHeight = Math.floor(1000 * height + extraViniculum) + vbPad;
+    span = sqrtSvg("sqrtTall", spanHeight, viewBoxHeight, extraViniculum, options);
+    span.style.minWidth = "0.742em";
+    advanceWidth = 1.056;
+  }
+
+  span.height = texHeight;
+  span.style.height = spanHeight + "em";
+  return {
+    span,
+    advanceWidth,
+    // Calculate the actual line width.
+    // This actually should depend on the chosen font -- e.g. \boldmath
+    // should use the thicker surd symbols from e.g. KaTeX_Main-Bold, and
+    // have thicker rules.
+    ruleWidth: (options.fontMetrics().sqrtRuleThickness + extraViniculum) * sizeMultiplier
+  };
+}; // There are three kinds of delimiters, delimiters that stack when they become
+// too large
+
+
+const stackLargeDelimiters = ["(", "\\lparen", ")", "\\rparen", "[", "\\lbrack", "]", "\\rbrack", "\\{", "\\lbrace", "\\}", "\\rbrace", "\\lfloor", "\\rfloor", "\u230a", "\u230b", "\\lceil", "\\rceil", "\u2308", "\u2309", "\\surd"]; // delimiters that always stack
+
+const stackAlwaysDelimiters = ["\\uparrow", "\\downarrow", "\\updownarrow", "\\Uparrow", "\\Downarrow", "\\Updownarrow", "|", "\\|", "\\vert", "\\Vert", "\\lvert", "\\rvert", "\\lVert", "\\rVert", "\\lgroup", "\\rgroup", "\u27ee", "\u27ef", "\\lmoustache", "\\rmoustache", "\u23b0", "\u23b1"]; // and delimiters that never stack
+
+const stackNeverDelimiters = ["<", ">", "\\langle", "\\rangle", "/", "\\backslash", "\\lt", "\\gt"]; // Metrics of the different sizes. Found by looking at TeX's output of
+// $\bigl| // \Bigl| \biggl| \Biggl| \showlists$
+// Used to create stacked delimiters of appropriate sizes in makeSizedDelim.
+
+const sizeToMaxHeight = [0, 1.2, 1.8, 2.4, 3.0];
+/**
+ * Used to create a delimiter of a specific size, where `size` is 1, 2, 3, or 4.
+ */
+
+const makeSizedDelim = function makeSizedDelim(delim, size, options, mode, classes) {
+  // < and > turn into \langle and \rangle in delimiters
+  if (delim === "<" || delim === "\\lt" || delim === "\u27e8") {
+    delim = "\\langle";
+  } else if (delim === ">" || delim === "\\gt" || delim === "\u27e9") {
+    delim = "\\rangle";
+  } // Sized delimiters are never centered.
+
+
+  if (utils.contains(stackLargeDelimiters, delim) || utils.contains(stackNeverDelimiters, delim)) {
+    return makeLargeDelim(delim, size, false, options, mode, classes);
+  } else if (utils.contains(stackAlwaysDelimiters, delim)) {
+    return makeStackedDelim(delim, sizeToMaxHeight[size], false, options, mode, classes);
+  } else {
+    throw new ParseError("Illegal delimiter: '" + delim + "'");
+  }
+};
+/**
+ * There are three different sequences of delimiter sizes that the delimiters
+ * follow depending on the kind of delimiter. This is used when creating custom
+ * sized delimiters to decide whether to create a small, large, or stacked
+ * delimiter.
+ *
+ * In real TeX, these sequences aren't explicitly defined, but are instead
+ * defined inside the font metrics. Since there are only three sequences that
+ * are possible for the delimiters that TeX defines, it is easier to just encode
+ * them explicitly here.
+ */
+
+
+// Delimiters that never stack try small delimiters and large delimiters only
+const stackNeverDelimiterSequence = [{
+  type: "small",
+  style: Style$1.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: Style$1.SCRIPT
+}, {
+  type: "small",
+  style: Style$1.TEXT
+}, {
+  type: "large",
+  size: 1
+}, {
+  type: "large",
+  size: 2
+}, {
+  type: "large",
+  size: 3
+}, {
+  type: "large",
+  size: 4
+}]; // Delimiters that always stack try the small delimiters first, then stack
+
+const stackAlwaysDelimiterSequence = [{
+  type: "small",
+  style: Style$1.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: Style$1.SCRIPT
+}, {
+  type: "small",
+  style: Style$1.TEXT
+}, {
+  type: "stack"
+}]; // Delimiters that stack when large try the small and then large delimiters, and
+// stack afterwards
+
+const stackLargeDelimiterSequence = [{
+  type: "small",
+  style: Style$1.SCRIPTSCRIPT
+}, {
+  type: "small",
+  style: Style$1.SCRIPT
+}, {
+  type: "small",
+  style: Style$1.TEXT
+}, {
+  type: "large",
+  size: 1
+}, {
+  type: "large",
+  size: 2
+}, {
+  type: "large",
+  size: 3
+}, {
+  type: "large",
+  size: 4
+}, {
+  type: "stack"
+}];
+/**
+ * Get the font used in a delimiter based on what kind of delimiter it is.
+ * TODO(#963) Use more specific font family return type once that is introduced.
+ */
+
+const delimTypeToFont = function delimTypeToFont(type) {
+  if (type.type === "small") {
+    return "Main-Regular";
+  } else if (type.type === "large") {
+    return "Size" + type.size + "-Regular";
+  } else if (type.type === "stack") {
+    return "Size4-Regular";
+  } else {
+    throw new Error(`Add support for delim type '${type.type}' here.`);
+  }
+};
+/**
+ * Traverse a sequence of types of delimiters to decide what kind of delimiter
+ * should be used to create a delimiter of the given height+depth.
+ */
+
+
+const traverseSequence = function traverseSequence(delim, height, sequence, options) {
+  // Here, we choose the index we should start at in the sequences. In smaller
+  // sizes (which correspond to larger numbers in style.size) we start earlier
+  // in the sequence. Thus, scriptscript starts at index 3-3=0, script starts
+  // at index 3-2=1, text starts at 3-1=2, and display starts at min(2,3-0)=2
+  const start = Math.min(2, 3 - options.style.size);
+
+  for (let i = start; i < sequence.length; i++) {
+    if (sequence[i].type === "stack") {
+      // This is always the last delimiter, so we just break the loop now.
+      break;
+    }
+
+    const metrics = getMetrics(delim, delimTypeToFont(sequence[i]), "math");
+    let heightDepth = metrics.height + metrics.depth; // Small delimiters are scaled down versions of the same font, so we
+    // account for the style change size.
+
+    if (sequence[i].type === "small") {
+      const newOptions = options.havingBaseStyle(sequence[i].style);
+      heightDepth *= newOptions.sizeMultiplier;
+    } // Check if the delimiter at this size works for the given height.
+
+
+    if (heightDepth > height) {
+      return sequence[i];
+    }
+  } // If we reached the end of the sequence, return the last sequence element.
+
+
+  return sequence[sequence.length - 1];
+};
+/**
+ * Make a delimiter of a given height+depth, with optional centering. Here, we
+ * traverse the sequences, and create a delimiter that the sequence tells us to.
+ */
+
+
+const makeCustomSizedDelim = function makeCustomSizedDelim(delim, height, center, options, mode, classes) {
+  if (delim === "<" || delim === "\\lt" || delim === "\u27e8") {
+    delim = "\\langle";
+  } else if (delim === ">" || delim === "\\gt" || delim === "\u27e9") {
+    delim = "\\rangle";
+  } // Decide what sequence to use
+
+
+  let sequence;
+
+  if (utils.contains(stackNeverDelimiters, delim)) {
+    sequence = stackNeverDelimiterSequence;
+  } else if (utils.contains(stackLargeDelimiters, delim)) {
+    sequence = stackLargeDelimiterSequence;
+  } else {
+    sequence = stackAlwaysDelimiterSequence;
+  } // Look through the sequence
+
+
+  const delimType = traverseSequence(delim, height, sequence, options); // Get the delimiter from font glyphs.
+  // Depending on the sequence element we decided on, call the
+  // appropriate function.
+
+  if (delimType.type === "small") {
+    return makeSmallDelim(delim, delimType.style, center, options, mode, classes);
+  } else if (delimType.type === "large") {
+    return makeLargeDelim(delim, delimType.size, center, options, mode, classes);
+  } else
+    /* if (delimType.type === "stack") */
+    {
+      return makeStackedDelim(delim, height, center, options, mode, classes);
+    }
+};
+/**
+ * Make a delimiter for use with `\left` and `\right`, given a height and depth
+ * of an expression that the delimiters surround.
+ */
+
+
+const makeLeftRightDelim = function makeLeftRightDelim(delim, height, depth, options, mode, classes) {
+  // We always center \left/\right delimiters, so the axis is always shifted
+  const axisHeight = options.fontMetrics().axisHeight * options.sizeMultiplier; // Taken from TeX source, tex.web, function make_left_right
+
+  const delimiterFactor = 901;
+  const delimiterExtend = 5.0 / options.fontMetrics().ptPerEm;
+  const maxDistFromAxis = Math.max(height - axisHeight, depth + axisHeight);
+  const totalHeight = Math.max( // In real TeX, calculations are done using integral values which are
+  // 65536 per pt, or 655360 per em. So, the division here truncates in
+  // TeX but doesn't here, producing different results. If we wanted to
+  // exactly match TeX's calculation, we could do
+  //   Math.floor(655360 * maxDistFromAxis / 500) *
+  //    delimiterFactor / 655360
+  // (To see the difference, compare
+  //    x^{x^{\left(\rule{0.1em}{0.68em}\right)}}
+  // in TeX and KaTeX)
+  maxDistFromAxis / 500 * delimiterFactor, 2 * maxDistFromAxis - delimiterExtend); // Finally, we defer to `makeCustomSizedDelim` with our calculated total
+  // height
+
+  return makeCustomSizedDelim(delim, totalHeight, true, options, mode, classes);
+};
+
+var delimiter = {
+  sqrtImage: makeSqrtImage,
+  sizedDelim: makeSizedDelim,
+  customSizedDelim: makeCustomSizedDelim,
+  leftRightDelim: makeLeftRightDelim
+};
+
+// Extra data needed for the delimiter handler down below
+const delimiterSizes = {
+  "\\bigl": {
+    mclass: "mopen",
+    size: 1
+  },
+  "\\Bigl": {
+    mclass: "mopen",
+    size: 2
+  },
+  "\\biggl": {
+    mclass: "mopen",
+    size: 3
+  },
+  "\\Biggl": {
+    mclass: "mopen",
+    size: 4
+  },
+  "\\bigr": {
+    mclass: "mclose",
+    size: 1
+  },
+  "\\Bigr": {
+    mclass: "mclose",
+    size: 2
+  },
+  "\\biggr": {
+    mclass: "mclose",
+    size: 3
+  },
+  "\\Biggr": {
+    mclass: "mclose",
+    size: 4
+  },
+  "\\bigm": {
+    mclass: "mrel",
+    size: 1
+  },
+  "\\Bigm": {
+    mclass: "mrel",
+    size: 2
+  },
+  "\\biggm": {
+    mclass: "mrel",
+    size: 3
+  },
+  "\\Biggm": {
+    mclass: "mrel",
+    size: 4
+  },
+  "\\big": {
+    mclass: "mord",
+    size: 1
+  },
+  "\\Big": {
+    mclass: "mord",
+    size: 2
+  },
+  "\\bigg": {
+    mclass: "mord",
+    size: 3
+  },
+  "\\Bigg": {
+    mclass: "mord",
+    size: 4
+  }
+};
+const delimiters = ["(", "\\lparen", ")", "\\rparen", "[", "\\lbrack", "]", "\\rbrack", "\\{", "\\lbrace", "\\}", "\\rbrace", "\\lfloor", "\\rfloor", "\u230a", "\u230b", "\\lceil", "\\rceil", "\u2308", "\u2309", "<", ">", "\\langle", "\u27e8", "\\rangle", "\u27e9", "\\lt", "\\gt", "\\lvert", "\\rvert", "\\lVert", "\\rVert", "\\lgroup", "\\rgroup", "\u27ee", "\u27ef", "\\lmoustache", "\\rmoustache", "\u23b0", "\u23b1", "/", "\\backslash", "|", "\\vert", "\\|", "\\Vert", "\\uparrow", "\\Uparrow", "\\downarrow", "\\Downarrow", "\\updownarrow", "\\Updownarrow", "."];
+
+// Delimiter functions
+function checkDelimiter(delim, context) {
+  const symDelim = checkSymbolNodeType(delim);
+
+  if (symDelim && utils.contains(delimiters, symDelim.text)) {
+    return symDelim;
+  } else if (symDelim) {
+    throw new ParseError(`Invalid delimiter '${symDelim.text}' after '${context.funcName}'`, delim);
+  } else {
+    throw new ParseError(`Invalid delimiter type '${delim.type}'`, delim);
+  }
+}
+
+defineFunction({
+  type: "delimsizing",
+  names: ["\\bigl", "\\Bigl", "\\biggl", "\\Biggl", "\\bigr", "\\Bigr", "\\biggr", "\\Biggr", "\\bigm", "\\Bigm", "\\biggm", "\\Biggm", "\\big", "\\Big", "\\bigg", "\\Bigg"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    const delim = checkDelimiter(args[0], context);
+    return {
+      type: "delimsizing",
+      mode: context.parser.mode,
+      size: delimiterSizes[context.funcName].size,
+      mclass: delimiterSizes[context.funcName].mclass,
+      delim: delim.text
+    };
+  },
+  htmlBuilder: (group, options) => {
+    if (group.delim === ".") {
+      // Empty delimiters still count as elements, even though they don't
+      // show anything.
+      return buildCommon.makeSpan([group.mclass]);
+    } // Use delimiter.sizedDelim to generate the delimiter.
+
+
+    return delimiter.sizedDelim(group.delim, group.size, options, group.mode, [group.mclass]);
+  },
+  mathmlBuilder: group => {
+    const children = [];
+
+    if (group.delim !== ".") {
+      children.push(makeText(group.delim, group.mode));
+    }
+
+    const node = new mathMLTree.MathNode("mo", children);
+
+    if (group.mclass === "mopen" || group.mclass === "mclose") {
+      // Only some of the delimsizing functions act as fences, and they
+      // return "mopen" or "mclose" mclass.
+      node.setAttribute("fence", "true");
+    } else {
+      // Explicitly disable fencing if it's not a fence, to override the
+      // defaults.
+      node.setAttribute("fence", "false");
+    }
+
+    return node;
+  }
+});
+
+function assertParsed(group) {
+  if (!group.body) {
+    throw new Error("Bug: The leftright ParseNode wasn't fully parsed.");
+  }
+}
+
+defineFunction({
+  type: "leftright-right",
+  names: ["\\right"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    // \left case below triggers parsing of \right in
+    //   `const right = parser.parseFunction();`
+    // uses this return value.
+    const color = context.parser.gullet.macros.get("\\current@color");
+
+    if (color && typeof color !== "string") {
+      throw new ParseError("\\current@color set to non-string in \\right");
+    }
+
+    return {
+      type: "leftright-right",
+      mode: context.parser.mode,
+      delim: checkDelimiter(args[0], context).text,
+      color // undefined if not set via \color
+
+    };
+  }
+});
+defineFunction({
+  type: "leftright",
+  names: ["\\left"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    const delim = checkDelimiter(args[0], context);
+    const parser = context.parser; // Parse out the implicit body
+
+    ++parser.leftrightDepth; // parseExpression stops before '\\right'
+
+    const body = parser.parseExpression(false);
+    --parser.leftrightDepth; // Check the next token
+
+    parser.expect("\\right", false);
+    const right = assertNodeType(parser.parseFunction(), "leftright-right");
+    return {
+      type: "leftright",
+      mode: parser.mode,
+      body,
+      left: delim.text,
+      right: right.delim,
+      rightColor: right.color
+    };
+  },
+  htmlBuilder: (group, options) => {
+    assertParsed(group); // Build the inner expression
+
+    const inner = buildExpression(group.body, options, true, ["mopen", "mclose"]);
+    let innerHeight = 0;
+    let innerDepth = 0;
+    let hadMiddle = false; // Calculate its height and depth
+
+    for (let i = 0; i < inner.length; i++) {
+      // Property `isMiddle` not defined on `span`. See comment in
+      // "middle"'s htmlBuilder.
+      // $FlowFixMe
+      if (inner[i].isMiddle) {
+        hadMiddle = true;
+      } else {
+        innerHeight = Math.max(inner[i].height, innerHeight);
+        innerDepth = Math.max(inner[i].depth, innerDepth);
+      }
+    } // The size of delimiters is the same, regardless of what style we are
+    // in. Thus, to correctly calculate the size of delimiter we need around
+    // a group, we scale down the inner size based on the size.
+
+
+    innerHeight *= options.sizeMultiplier;
+    innerDepth *= options.sizeMultiplier;
+    let leftDelim;
+
+    if (group.left === ".") {
+      // Empty delimiters in \left and \right make null delimiter spaces.
+      leftDelim = makeNullDelimiter(options, ["mopen"]);
+    } else {
+      // Otherwise, use leftRightDelim to generate the correct sized
+      // delimiter.
+      leftDelim = delimiter.leftRightDelim(group.left, innerHeight, innerDepth, options, group.mode, ["mopen"]);
+    } // Add it to the beginning of the expression
+
+
+    inner.unshift(leftDelim); // Handle middle delimiters
+
+    if (hadMiddle) {
+      for (let i = 1; i < inner.length; i++) {
+        const middleDelim = inner[i]; // Property `isMiddle` not defined on `span`. See comment in
+        // "middle"'s htmlBuilder.
+        // $FlowFixMe
+
+        const isMiddle = middleDelim.isMiddle;
+
+        if (isMiddle) {
+          // Apply the options that were active when \middle was called
+          inner[i] = delimiter.leftRightDelim(isMiddle.delim, innerHeight, innerDepth, isMiddle.options, group.mode, []);
+        }
+      }
+    }
+
+    let rightDelim; // Same for the right delimiter, but using color specified by \color
+
+    if (group.right === ".") {
+      rightDelim = makeNullDelimiter(options, ["mclose"]);
+    } else {
+      const colorOptions = group.rightColor ? options.withColor(group.rightColor) : options;
+      rightDelim = delimiter.leftRightDelim(group.right, innerHeight, innerDepth, colorOptions, group.mode, ["mclose"]);
+    } // Add it to the end of the expression.
+
+
+    inner.push(rightDelim);
+    return buildCommon.makeSpan(["minner"], inner, options);
+  },
+  mathmlBuilder: (group, options) => {
+    assertParsed(group);
+    const inner = buildExpression$1(group.body, options);
+
+    if (group.left !== ".") {
+      const leftNode = new mathMLTree.MathNode("mo", [makeText(group.left, group.mode)]);
+      leftNode.setAttribute("fence", "true");
+      inner.unshift(leftNode);
+    }
+
+    if (group.right !== ".") {
+      const rightNode = new mathMLTree.MathNode("mo", [makeText(group.right, group.mode)]);
+      rightNode.setAttribute("fence", "true");
+
+      if (group.rightColor) {
+        rightNode.setAttribute("mathcolor", group.rightColor);
+      }
+
+      inner.push(rightNode);
+    }
+
+    return makeRow(inner);
+  }
+});
+defineFunction({
+  type: "middle",
+  names: ["\\middle"],
+  props: {
+    numArgs: 1
+  },
+  handler: (context, args) => {
+    const delim = checkDelimiter(args[0], context);
+
+    if (!context.parser.leftrightDepth) {
+      throw new ParseError("\\middle without preceding \\left", delim);
+    }
+
+    return {
+      type: "middle",
+      mode: context.parser.mode,
+      delim: delim.text
+    };
+  },
+  htmlBuilder: (group, options) => {
+    let middleDelim;
+
+    if (group.delim === ".") {
+      middleDelim = makeNullDelimiter(options, []);
+    } else {
+      middleDelim = delimiter.sizedDelim(group.delim, 1, options, group.mode, []);
+      const isMiddle = {
+        delim: group.delim,
+        options
+      }; // Property `isMiddle` not defined on `span`. It is only used in
+      // this file above.
+      // TODO: Fix this violation of the `span` type and possibly rename
+      // things since `isMiddle` sounds like a boolean, but is a struct.
+      // $FlowFixMe
+
+      middleDelim.isMiddle = isMiddle;
+    }
+
+    return middleDelim;
+  },
+  mathmlBuilder: (group, options) => {
+    // A Firefox \middle will strech a character vertically only if it
+    // is in the fence part of the operator dictionary at:
+    // https://www.w3.org/TR/MathML3/appendixc.html.
+    // So we need to avoid U+2223 and use plain "|" instead.
+    const textNode = group.delim === "\\vert" || group.delim === "|" ? makeText("|", "text") : makeText(group.delim, group.mode);
+    const middleNode = new mathMLTree.MathNode("mo", [textNode]);
+    middleNode.setAttribute("fence", "true"); // MathML gives 5/18em spacing to each <mo> element.
+    // \middle should get delimiter spacing instead.
+
+    middleNode.setAttribute("lspace", "0.05em");
+    middleNode.setAttribute("rspace", "0.05em");
+    return middleNode;
+  }
+});
+
+const htmlBuilder$2 = (group, options) => {
+  // \cancel, \bcancel, \xcancel, \sout, \fbox, \colorbox, \fcolorbox
+  // Some groups can return document fragments.  Handle those by wrapping
+  // them in a span.
+  const inner = buildCommon.wrapFragment(buildGroup(group.body, options), options);
+  const label = group.label.substr(1);
+  const scale = options.sizeMultiplier;
+  let img;
+  let imgShift = 0; // In the LaTeX cancel package, line geometry is slightly different
+  // depending on whether the subject is wider than it is tall, or vice versa.
+  // We don't know the width of a group, so as a proxy, we test if
+  // the subject is a single character. This captures most of the
+  // subjects that should get the "tall" treatment.
+
+  const isSingleChar = utils.isCharacterBox(group.body);
+
+  if (label === "sout") {
+    img = buildCommon.makeSpan(["stretchy", "sout"]);
+    img.height = options.fontMetrics().defaultRuleThickness / scale;
+    imgShift = -0.5 * options.fontMetrics().xHeight;
+  } else {
+    // Add horizontal padding
+    if (/cancel/.test(label)) {
+      if (!isSingleChar) {
+        inner.classes.push("cancel-pad");
+      }
+    } else {
+      inner.classes.push("boxpad");
+    } // Add vertical padding
+
+
+    let vertPad = 0;
+    let ruleThickness = 0; // ref: cancel package: \advance\totalheight2\p@ % "+2"
+
+    if (/box/.test(label)) {
+      ruleThickness = Math.max(options.fontMetrics().fboxrule, // default
+      options.minRuleThickness // User override.
+      );
+      vertPad = options.fontMetrics().fboxsep + (label === "colorbox" ? 0 : ruleThickness);
+    } else {
+      vertPad = isSingleChar ? 0.2 : 0;
+    }
+
+    img = stretchy.encloseSpan(inner, label, vertPad, options);
+
+    if (/fbox|boxed|fcolorbox/.test(label)) {
+      img.style.borderStyle = "solid";
+      img.style.borderWidth = `${ruleThickness}em`;
+    }
+
+    imgShift = inner.depth + vertPad;
+
+    if (group.backgroundColor) {
+      img.style.backgroundColor = group.backgroundColor;
+
+      if (group.borderColor) {
+        img.style.borderColor = group.borderColor;
+      }
+    }
+  }
+
+  let vlist;
+
+  if (group.backgroundColor) {
+    vlist = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [// Put the color background behind inner;
+      {
+        type: "elem",
+        elem: img,
+        shift: imgShift
+      }, {
+        type: "elem",
+        elem: inner,
+        shift: 0
+      }]
+    }, options);
+  } else {
+    vlist = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [// Write the \cancel stroke on top of inner.
+      {
+        type: "elem",
+        elem: inner,
+        shift: 0
+      }, {
+        type: "elem",
+        elem: img,
+        shift: imgShift,
+        wrapperClasses: /cancel/.test(label) ? ["svg-align"] : []
+      }]
+    }, options);
+  }
+
+  if (/cancel/.test(label)) {
+    // The cancel package documentation says that cancel lines add their height
+    // to the expression, but tests show that isn't how it actually works.
+    vlist.height = inner.height;
+    vlist.depth = inner.depth;
+  }
+
+  if (/cancel/.test(label) && !isSingleChar) {
+    // cancel does not create horiz space for its line extension.
+    return buildCommon.makeSpan(["mord", "cancel-lap"], [vlist], options);
+  } else {
+    return buildCommon.makeSpan(["mord"], [vlist], options);
+  }
+};
+
+const mathmlBuilder$2 = (group, options) => {
+  let fboxsep = 0;
+  const node = new mathMLTree.MathNode(group.label.indexOf("colorbox") > -1 ? "mpadded" : "menclose", [buildGroup$1(group.body, options)]);
+
+  switch (group.label) {
+    case "\\cancel":
+      node.setAttribute("notation", "updiagonalstrike");
+      break;
+
+    case "\\bcancel":
+      node.setAttribute("notation", "downdiagonalstrike");
+      break;
+
+    case "\\sout":
+      node.setAttribute("notation", "horizontalstrike");
+      break;
+
+    case "\\fbox":
+      node.setAttribute("notation", "box");
+      break;
+
+    case "\\fcolorbox":
+    case "\\colorbox":
+      // <menclose> doesn't have a good notation option. So use <mpadded>
+      // instead. Set some attributes that come included with <menclose>.
+      fboxsep = options.fontMetrics().fboxsep * options.fontMetrics().ptPerEm;
+      node.setAttribute("width", `+${2 * fboxsep}pt`);
+      node.setAttribute("height", `+${2 * fboxsep}pt`);
+      node.setAttribute("lspace", `${fboxsep}pt`); //
+
+      node.setAttribute("voffset", `${fboxsep}pt`);
+
+      if (group.label === "\\fcolorbox") {
+        const thk = Math.max(options.fontMetrics().fboxrule, // default
+        options.minRuleThickness // user override
+        );
+        node.setAttribute("style", "border: " + thk + "em solid " + String(group.borderColor));
+      }
+
+      break;
+
+    case "\\xcancel":
+      node.setAttribute("notation", "updiagonalstrike downdiagonalstrike");
+      break;
+  }
+
+  if (group.backgroundColor) {
+    node.setAttribute("mathbackground", group.backgroundColor);
+  }
+
+  return node;
+};
+
+defineFunction({
+  type: "enclose",
+  names: ["\\colorbox"],
+  props: {
+    numArgs: 2,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "text"]
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const color = assertNodeType(args[0], "color-token").color;
+    const body = args[1];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      backgroundColor: color,
+      body
+    };
+  },
+
+  htmlBuilder: htmlBuilder$2,
+  mathmlBuilder: mathmlBuilder$2
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\fcolorbox"],
+  props: {
+    numArgs: 3,
+    allowedInText: true,
+    greediness: 3,
+    argTypes: ["color", "color", "text"]
+  },
+
+  handler(_ref2, args, optArgs) {
+    let parser = _ref2.parser,
+        funcName = _ref2.funcName;
+    const borderColor = assertNodeType(args[0], "color-token").color;
+    const backgroundColor = assertNodeType(args[1], "color-token").color;
+    const body = args[2];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      backgroundColor,
+      borderColor,
+      body
+    };
+  },
+
+  htmlBuilder: htmlBuilder$2,
+  mathmlBuilder: mathmlBuilder$2
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\fbox"],
+  props: {
+    numArgs: 1,
+    argTypes: ["hbox"],
+    allowedInText: true
+  },
+
+  handler(_ref3, args) {
+    let parser = _ref3.parser;
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: "\\fbox",
+      body: args[0]
+    };
+  }
+
+});
+defineFunction({
+  type: "enclose",
+  names: ["\\cancel", "\\bcancel", "\\xcancel", "\\sout"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(_ref4, args, optArgs) {
+    let parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    const body = args[0];
+    return {
+      type: "enclose",
+      mode: parser.mode,
+      label: funcName,
+      body
+    };
+  },
+
+  htmlBuilder: htmlBuilder$2,
+  mathmlBuilder: mathmlBuilder$2
+});
+
+/**
+ * All registered environments.
+ * `environments.js` exports this same dictionary again and makes it public.
+ * `Parser.js` requires this dictionary via `environments.js`.
+ */
+const _environments = {};
+function defineEnvironment(_ref) {
+  let type = _ref.type,
+      names = _ref.names,
+      props = _ref.props,
+      handler = _ref.handler,
+      htmlBuilder = _ref.htmlBuilder,
+      mathmlBuilder = _ref.mathmlBuilder;
+  // Set default values of environments.
+  const data = {
+    type,
+    numArgs: props.numArgs || 0,
+    greediness: 1,
+    allowedInText: false,
+    numOptionalArgs: 0,
+    handler
+  };
+
+  for (let i = 0; i < names.length; ++i) {
+    // TODO: The value type of _environments should be a type union of all
+    // possible `EnvSpec<>` possibilities instead of `EnvSpec<*>`, which is
+    // an existential type.
+    // $FlowFixMe
+    _environments[names[i]] = data;
+  }
+
+  if (htmlBuilder) {
+    _htmlGroupBuilders[type] = htmlBuilder;
+  }
+
+  if (mathmlBuilder) {
+    _mathmlGroupBuilders[type] = mathmlBuilder;
+  }
+}
+
+function getHLines(parser) {
+  // Return an array. The array length = number of hlines.
+  // Each element in the array tells if the line is dashed.
+  const hlineInfo = [];
+  parser.consumeSpaces();
+  let nxt = parser.fetch().text;
+
+  while (nxt === "\\hline" || nxt === "\\hdashline") {
+    parser.consume();
+    hlineInfo.push(nxt === "\\hdashline");
+    parser.consumeSpaces();
+    nxt = parser.fetch().text;
+  }
+
+  return hlineInfo;
+}
+/**
+ * Parse the body of the environment, with rows delimited by \\ and
+ * columns delimited by &, and create a nested list in row-major order
+ * with one group per cell.  If given an optional argument style
+ * ("text", "display", etc.), then each cell is cast into that style.
+ */
+
+
+function parseArray(parser, _ref, style) {
+  let hskipBeforeAndAfter = _ref.hskipBeforeAndAfter,
+      addJot = _ref.addJot,
+      cols = _ref.cols,
+      arraystretch = _ref.arraystretch,
+      colSeparationType = _ref.colSeparationType;
+  // Parse body of array with \\ temporarily mapped to \cr
+  parser.gullet.beginGroup();
+  parser.gullet.macros.set("\\\\", "\\cr"); // Get current arraystretch if it's not set by the environment
+
+  if (!arraystretch) {
+    const stretch = parser.gullet.expandMacroAsText("\\arraystretch");
+
+    if (stretch == null) {
+      // Default \arraystretch from lttab.dtx
+      arraystretch = 1;
+    } else {
+      arraystretch = parseFloat(stretch);
+
+      if (!arraystretch || arraystretch < 0) {
+        throw new ParseError(`Invalid \\arraystretch: ${stretch}`);
+      }
+    }
+  } // Start group for first cell
+
+
+  parser.gullet.beginGroup();
+  let row = [];
+  const body = [row];
+  const rowGaps = [];
+  const hLinesBeforeRow = []; // Test for \hline at the top of the array.
+
+  hLinesBeforeRow.push(getHLines(parser));
+
+  while (true) {
+    // eslint-disable-line no-constant-condition
+    // Parse each cell in its own group (namespace)
+    let cell = parser.parseExpression(false, "\\cr");
+    parser.gullet.endGroup();
+    parser.gullet.beginGroup();
+    cell = {
+      type: "ordgroup",
+      mode: parser.mode,
+      body: cell
+    };
+
+    if (style) {
+      cell = {
+        type: "styling",
+        mode: parser.mode,
+        style,
+        body: [cell]
+      };
+    }
+
+    row.push(cell);
+    const next = parser.fetch().text;
+
+    if (next === "&") {
+      parser.consume();
+    } else if (next === "\\end") {
+      // Arrays terminate newlines with `\crcr` which consumes a `\cr` if
+      // the last line is empty.
+      // NOTE: Currently, `cell` is the last item added into `row`.
+      if (row.length === 1 && cell.type === "styling" && cell.body[0].body.length === 0) {
+        body.pop();
+      }
+
+      if (hLinesBeforeRow.length < body.length + 1) {
+        hLinesBeforeRow.push([]);
+      }
+
+      break;
+    } else if (next === "\\cr") {
+      const cr = assertNodeType(parser.parseFunction(), "cr");
+      rowGaps.push(cr.size); // check for \hline(s) following the row separator
+
+      hLinesBeforeRow.push(getHLines(parser));
+      row = [];
+      body.push(row);
+    } else {
+      throw new ParseError("Expected & or \\\\ or \\cr or \\end", parser.nextToken);
+    }
+  } // End cell group
+
+
+  parser.gullet.endGroup(); // End array group defining \\
+
+  parser.gullet.endGroup();
+  return {
+    type: "array",
+    mode: parser.mode,
+    addJot,
+    arraystretch,
+    body,
+    cols,
+    rowGaps,
+    hskipBeforeAndAfter,
+    hLinesBeforeRow,
+    colSeparationType
+  };
+} // Decides on a style for cells in an array according to whether the given
+// environment name starts with the letter 'd'.
+
+
+function dCellStyle(envName) {
+  if (envName.substr(0, 1) === "d") {
+    return "display";
+  } else {
+    return "text";
+  }
+}
+
+const htmlBuilder$3 = function htmlBuilder(group, options) {
+  let r;
+  let c;
+  const nr = group.body.length;
+  const hLinesBeforeRow = group.hLinesBeforeRow;
+  let nc = 0;
+  let body = new Array(nr);
+  const hlines = [];
+  const ruleThickness = Math.max( // From LaTeX \showthe\arrayrulewidth. Equals 0.04 em.
+  options.fontMetrics().arrayRuleWidth, options.minRuleThickness // User override.
+  ); // Horizontal spacing
+
+  const pt = 1 / options.fontMetrics().ptPerEm;
+  let arraycolsep = 5 * pt; // default value, i.e. \arraycolsep in article.cls
+
+  if (group.colSeparationType && group.colSeparationType === "small") {
+    // We're in a {smallmatrix}. Default column space is \thickspace,
+    // i.e. 5/18em = 0.2778em, per amsmath.dtx for {smallmatrix}.
+    // But that needs adjustment because LaTeX applies \scriptstyle to the
+    // entire array, including the colspace, but this function applies
+    // \scriptstyle only inside each element.
+    const localMultiplier = options.havingStyle(Style$1.SCRIPT).sizeMultiplier;
+    arraycolsep = 0.2778 * (localMultiplier / options.sizeMultiplier);
+  } // Vertical spacing
+
+
+  const baselineskip = 12 * pt; // see size10.clo
+  // Default \jot from ltmath.dtx
+  // TODO(edemaine): allow overriding \jot via \setlength (#687)
+
+  const jot = 3 * pt;
+  const arrayskip = group.arraystretch * baselineskip;
+  const arstrutHeight = 0.7 * arrayskip; // \strutbox in ltfsstrc.dtx and
+
+  const arstrutDepth = 0.3 * arrayskip; // \@arstrutbox in lttab.dtx
+
+  let totalHeight = 0; // Set a position for \hline(s) at the top of the array, if any.
+
+  function setHLinePos(hlinesInGap) {
+    for (let i = 0; i < hlinesInGap.length; ++i) {
+      if (i > 0) {
+        totalHeight += 0.25;
+      }
+
+      hlines.push({
+        pos: totalHeight,
+        isDashed: hlinesInGap[i]
+      });
+    }
+  }
+
+  setHLinePos(hLinesBeforeRow[0]);
+
+  for (r = 0; r < group.body.length; ++r) {
+    const inrow = group.body[r];
+    let height = arstrutHeight; // \@array adds an \@arstrut
+
+    let depth = arstrutDepth; // to each tow (via the template)
+
+    if (nc < inrow.length) {
+      nc = inrow.length;
+    }
+
+    const outrow = new Array(inrow.length);
+
+    for (c = 0; c < inrow.length; ++c) {
+      const elt = buildGroup(inrow[c], options);
+
+      if (depth < elt.depth) {
+        depth = elt.depth;
+      }
+
+      if (height < elt.height) {
+        height = elt.height;
+      }
+
+      outrow[c] = elt;
+    }
+
+    const rowGap = group.rowGaps[r];
+    let gap = 0;
+
+    if (rowGap) {
+      gap = calculateSize(rowGap, options);
+
+      if (gap > 0) {
+        // \@argarraycr
+        gap += arstrutDepth;
+
+        if (depth < gap) {
+          depth = gap; // \@xargarraycr
+        }
+
+        gap = 0;
+      }
+    } // In AMS multiline environments such as aligned and gathered, rows
+    // correspond to lines that have additional \jot added to the
+    // \baselineskip via \openup.
+
+
+    if (group.addJot) {
+      depth += jot;
+    }
+
+    outrow.height = height;
+    outrow.depth = depth;
+    totalHeight += height;
+    outrow.pos = totalHeight;
+    totalHeight += depth + gap; // \@yargarraycr
+
+    body[r] = outrow; // Set a position for \hline(s), if any.
+
+    setHLinePos(hLinesBeforeRow[r + 1]);
+  }
+
+  const offset = totalHeight / 2 + options.fontMetrics().axisHeight;
+  const colDescriptions = group.cols || [];
+  const cols = [];
+  let colSep;
+  let colDescrNum;
+
+  for (c = 0, colDescrNum = 0; // Continue while either there are more columns or more column
+  // descriptions, so trailing separators don't get lost.
+  c < nc || colDescrNum < colDescriptions.length; ++c, ++colDescrNum) {
+    let colDescr = colDescriptions[colDescrNum] || {};
+    let firstSeparator = true;
+
+    while (colDescr.type === "separator") {
+      // If there is more than one separator in a row, add a space
+      // between them.
+      if (!firstSeparator) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = options.fontMetrics().doubleRuleSep + "em";
+        cols.push(colSep);
+      }
+
+      if (colDescr.separator === "|" || colDescr.separator === ":") {
+        const lineType = colDescr.separator === "|" ? "solid" : "dashed";
+        const separator = buildCommon.makeSpan(["vertical-separator"], [], options);
+        separator.style.height = totalHeight + "em";
+        separator.style.borderRightWidth = `${ruleThickness}em`;
+        separator.style.borderRightStyle = lineType;
+        separator.style.margin = `0 -${ruleThickness / 2}em`;
+        separator.style.verticalAlign = -(totalHeight - offset) + "em";
+        cols.push(separator);
+      } else {
+        throw new ParseError("Invalid separator type: " + colDescr.separator);
+      }
+
+      colDescrNum++;
+      colDescr = colDescriptions[colDescrNum] || {};
+      firstSeparator = false;
+    }
+
+    if (c >= nc) {
+      continue;
+    }
+
+    let sepwidth;
+
+    if (c > 0 || group.hskipBeforeAndAfter) {
+      sepwidth = utils.deflt(colDescr.pregap, arraycolsep);
+
+      if (sepwidth !== 0) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = sepwidth + "em";
+        cols.push(colSep);
+      }
+    }
+
+    let col = [];
+
+    for (r = 0; r < nr; ++r) {
+      const row = body[r];
+      const elem = row[c];
+
+      if (!elem) {
+        continue;
+      }
+
+      const shift = row.pos - offset;
+      elem.depth = row.depth;
+      elem.height = row.height;
+      col.push({
+        type: "elem",
+        elem: elem,
+        shift: shift
+      });
+    }
+
+    col = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: col
+    }, options);
+    col = buildCommon.makeSpan(["col-align-" + (colDescr.align || "c")], [col]);
+    cols.push(col);
+
+    if (c < nc - 1 || group.hskipBeforeAndAfter) {
+      sepwidth = utils.deflt(colDescr.postgap, arraycolsep);
+
+      if (sepwidth !== 0) {
+        colSep = buildCommon.makeSpan(["arraycolsep"], []);
+        colSep.style.width = sepwidth + "em";
+        cols.push(colSep);
+      }
+    }
+  }
+
+  body = buildCommon.makeSpan(["mtable"], cols); // Add \hline(s), if any.
+
+  if (hlines.length > 0) {
+    const line = buildCommon.makeLineSpan("hline", options, ruleThickness);
+    const dashes = buildCommon.makeLineSpan("hdashline", options, ruleThickness);
+    const vListElems = [{
+      type: "elem",
+      elem: body,
+      shift: 0
+    }];
+
+    while (hlines.length > 0) {
+      const hline = hlines.pop();
+      const lineShift = hline.pos - offset;
+
+      if (hline.isDashed) {
+        vListElems.push({
+          type: "elem",
+          elem: dashes,
+          shift: lineShift
+        });
+      } else {
+        vListElems.push({
+          type: "elem",
+          elem: line,
+          shift: lineShift
+        });
+      }
+    }
+
+    body = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: vListElems
+    }, options);
+  }
+
+  return buildCommon.makeSpan(["mord"], [body], options);
+};
+
+const alignMap = {
+  c: "center ",
+  l: "left ",
+  r: "right "
+};
+
+const mathmlBuilder$3 = function mathmlBuilder(group, options) {
+  let table = new mathMLTree.MathNode("mtable", group.body.map(function (row) {
+    return new mathMLTree.MathNode("mtr", row.map(function (cell) {
+      return new mathMLTree.MathNode("mtd", [buildGroup$1(cell, options)]);
+    }));
+  })); // Set column alignment, row spacing, column spacing, and
+  // array lines by setting attributes on the table element.
+  // Set the row spacing. In MathML, we specify a gap distance.
+  // We do not use rowGap[] because MathML automatically increases
+  // cell height with the height/depth of the element content.
+  // LaTeX \arraystretch multiplies the row baseline-to-baseline distance.
+  // We simulate this by adding (arraystretch - 1)em to the gap. This
+  // does a reasonable job of adjusting arrays containing 1 em tall content.
+  // The 0.16 and 0.09 values are found emprically. They produce an array
+  // similar to LaTeX and in which content does not interfere with \hines.
+
+  const gap = group.arraystretch === 0.5 ? 0.1 // {smallmatrix}, {subarray}
+  : 0.16 + group.arraystretch - 1 + (group.addJot ? 0.09 : 0);
+  table.setAttribute("rowspacing", gap + "em"); // MathML table lines go only between cells.
+  // To place a line on an edge we'll use <menclose>, if necessary.
+
+  let menclose = "";
+  let align = "";
+
+  if (group.cols && group.cols.length > 0) {
+    // Find column alignment, column spacing, and  vertical lines.
+    const cols = group.cols;
+    let columnLines = "";
+    let prevTypeWasAlign = false;
+    let iStart = 0;
+    let iEnd = cols.length;
+
+    if (cols[0].type === "separator") {
+      menclose += "top ";
+      iStart = 1;
+    }
+
+    if (cols[cols.length - 1].type === "separator") {
+      menclose += "bottom ";
+      iEnd -= 1;
+    }
+
+    for (let i = iStart; i < iEnd; i++) {
+      if (cols[i].type === "align") {
+        align += alignMap[cols[i].align];
+
+        if (prevTypeWasAlign) {
+          columnLines += "none ";
+        }
+
+        prevTypeWasAlign = true;
+      } else if (cols[i].type === "separator") {
+        // MathML accepts only single lines between cells.
+        // So we read only the first of consecutive separators.
+        if (prevTypeWasAlign) {
+          columnLines += cols[i].separator === "|" ? "solid " : "dashed ";
+          prevTypeWasAlign = false;
+        }
+      }
+    }
+
+    table.setAttribute("columnalign", align.trim());
+
+    if (/[sd]/.test(columnLines)) {
+      table.setAttribute("columnlines", columnLines.trim());
+    }
+  } // Set column spacing.
+
+
+  if (group.colSeparationType === "align") {
+    const cols = group.cols || [];
+    let spacing = "";
+
+    for (let i = 1; i < cols.length; i++) {
+      spacing += i % 2 ? "0em " : "1em ";
+    }
+
+    table.setAttribute("columnspacing", spacing.trim());
+  } else if (group.colSeparationType === "alignat") {
+    table.setAttribute("columnspacing", "0em");
+  } else if (group.colSeparationType === "small") {
+    table.setAttribute("columnspacing", "0.2778em");
+  } else {
+    table.setAttribute("columnspacing", "1em");
+  } // Address \hline and \hdashline
+
+
+  let rowLines = "";
+  const hlines = group.hLinesBeforeRow;
+  menclose += hlines[0].length > 0 ? "left " : "";
+  menclose += hlines[hlines.length - 1].length > 0 ? "right " : "";
+
+  for (let i = 1; i < hlines.length - 1; i++) {
+    rowLines += hlines[i].length === 0 ? "none " // MathML accepts only a single line between rows. Read one element.
+    : hlines[i][0] ? "dashed " : "solid ";
+  }
+
+  if (/[sd]/.test(rowLines)) {
+    table.setAttribute("rowlines", rowLines.trim());
+  }
+
+  if (menclose !== "") {
+    table = new mathMLTree.MathNode("menclose", [table]);
+    table.setAttribute("notation", menclose.trim());
+  }
+
+  if (group.arraystretch && group.arraystretch < 1) {
+    // A small array. Wrap in scriptstyle so row gap is not too large.
+    table = new mathMLTree.MathNode("mstyle", [table]);
+    table.setAttribute("scriptlevel", "1");
+  }
+
+  return table;
+}; // Convenience function for aligned and alignedat environments.
+
+
+const alignedHandler = function alignedHandler(context, args) {
+  const cols = [];
+  const res = parseArray(context.parser, {
+    cols,
+    addJot: true
+  }, "display"); // Determining number of columns.
+  // 1. If the first argument is given, we use it as a number of columns,
+  //    and makes sure that each row doesn't exceed that number.
+  // 2. Otherwise, just count number of columns = maximum number
+  //    of cells in each row ("aligned" mode -- isAligned will be true).
+  //
+  // At the same time, prepend empty group {} at beginning of every second
+  // cell in each row (starting with second cell) so that operators become
+  // binary.  This behavior is implemented in amsmath's \start@aligned.
+
+  let numMaths;
+  let numCols = 0;
+  const emptyGroup = {
+    type: "ordgroup",
+    mode: context.mode,
+    body: []
+  };
+
+  if (args[0] && args[0].type === "ordgroup") {
+    let arg0 = "";
+
+    for (let i = 0; i < args[0].body.length; i++) {
+      const textord = assertNodeType(args[0].body[i], "textord");
+      arg0 += textord.text;
+    }
+
+    numMaths = Number(arg0);
+    numCols = numMaths * 2;
+  }
+
+  const isAligned = !numCols;
+  res.body.forEach(function (row) {
+    for (let i = 1; i < row.length; i += 2) {
+      // Modify ordgroup node within styling node
+      const styling = assertNodeType(row[i], "styling");
+      const ordgroup = assertNodeType(styling.body[0], "ordgroup");
+      ordgroup.body.unshift(emptyGroup);
+    }
+
+    if (!isAligned) {
+      // Case 1
+      const curMaths = row.length / 2;
+
+      if (numMaths < curMaths) {
+        throw new ParseError("Too many math in a row: " + `expected ${numMaths}, but got ${curMaths}`, row[0]);
+      }
+    } else if (numCols < row.length) {
+      // Case 2
+      numCols = row.length;
+    }
+  }); // Adjusting alignment.
+  // In aligned mode, we add one \qquad between columns;
+  // otherwise we add nothing.
+
+  for (let i = 0; i < numCols; ++i) {
+    let align = "r";
+    let pregap = 0;
+
+    if (i % 2 === 1) {
+      align = "l";
+    } else if (i > 0 && isAligned) {
+      // "aligned" mode.
+      pregap = 1; // add one \quad
+    }
+
+    cols[i] = {
+      type: "align",
+      align: align,
+      pregap: pregap,
+      postgap: 0
+    };
+  }
+
+  res.colSeparationType = isAligned ? "align" : "alignat";
+  return res;
+}; // Arrays are part of LaTeX, defined in lttab.dtx so its documentation
+// is part of the source2e.pdf file of LaTeX2e source documentation.
+// {darray} is an {array} environment where cells are set in \displaystyle,
+// as defined in nccmath.sty.
+
+
+defineEnvironment({
+  type: "array",
+  names: ["array", "darray"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(context, args) {
+    // Since no types are specified above, the two possibilities are
+    // - The argument is wrapped in {} or [], in which case Parser's
+    //   parseGroup() returns an "ordgroup" wrapping some symbol node.
+    // - The argument is a bare symbol node.
+    const symNode = checkSymbolNodeType(args[0]);
+    const colalign = symNode ? [args[0]] : assertNodeType(args[0], "ordgroup").body;
+    const cols = colalign.map(function (nde) {
+      const node = assertSymbolNodeType(nde);
+      const ca = node.text;
+
+      if ("lcr".indexOf(ca) !== -1) {
+        return {
+          type: "align",
+          align: ca
+        };
+      } else if (ca === "|") {
+        return {
+          type: "separator",
+          separator: "|"
+        };
+      } else if (ca === ":") {
+        return {
+          type: "separator",
+          separator: ":"
+        };
+      }
+
+      throw new ParseError("Unknown column alignment: " + ca, nde);
+    });
+    const res = {
+      cols,
+      hskipBeforeAndAfter: true // \@preamble in lttab.dtx
+
+    };
+    return parseArray(context.parser, res, dCellStyle(context.envName));
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // The matrix environments of amsmath builds on the array environment
+// of LaTeX, which is discussed above.
+
+defineEnvironment({
+  type: "array",
+  names: ["matrix", "pmatrix", "bmatrix", "Bmatrix", "vmatrix", "Vmatrix"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(context) {
+    const delimiters = {
+      "matrix": null,
+      "pmatrix": ["(", ")"],
+      "bmatrix": ["[", "]"],
+      "Bmatrix": ["\\{", "\\}"],
+      "vmatrix": ["|", "|"],
+      "Vmatrix": ["\\Vert", "\\Vert"]
+    }[context.envName]; // \hskip -\arraycolsep in amsmath
+
+    const payload = {
+      hskipBeforeAndAfter: false
+    };
+    const res = parseArray(context.parser, payload, dCellStyle(context.envName));
+    return delimiters ? {
+      type: "leftright",
+      mode: context.mode,
+      body: [res],
+      left: delimiters[0],
+      right: delimiters[1],
+      rightColor: undefined // \right uninfluenced by \color in array
+
+    } : res;
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+});
+defineEnvironment({
+  type: "array",
+  names: ["smallmatrix"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(context) {
+    const payload = {
+      arraystretch: 0.5
+    };
+    const res = parseArray(context.parser, payload, "script");
+    res.colSeparationType = "small";
+    return res;
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+});
+defineEnvironment({
+  type: "array",
+  names: ["subarray"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(context, args) {
+    // Parsing of {subarray} is similar to {array}
+    const symNode = checkSymbolNodeType(args[0]);
+    const colalign = symNode ? [args[0]] : assertNodeType(args[0], "ordgroup").body;
+    const cols = colalign.map(function (nde) {
+      const node = assertSymbolNodeType(nde);
+      const ca = node.text; // {subarray} only recognizes "l" & "c"
+
+      if ("lc".indexOf(ca) !== -1) {
+        return {
+          type: "align",
+          align: ca
+        };
+      }
+
+      throw new ParseError("Unknown column alignment: " + ca, nde);
+    });
+
+    if (cols.length > 1) {
+      throw new ParseError("{subarray} can contain only one column");
+    }
+
+    let res = {
+      cols,
+      hskipBeforeAndAfter: false,
+      arraystretch: 0.5
+    };
+    res = parseArray(context.parser, res, "script");
+
+    if (res.body.length > 0 && res.body[0].length > 1) {
+      throw new ParseError("{subarray} can contain only one column");
+    }
+
+    return res;
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // A cases environment (in amsmath.sty) is almost equivalent to
+// \def\arraystretch{1.2}%
+// \left\{\begin{array}{@{}l@{\quad}l@{}} … \end{array}\right.
+// {dcases} is a {cases} environment where cells are set in \displaystyle,
+// as defined in mathtools.sty.
+// {rcases} is another mathtools environment. It's brace is on the right side.
+
+defineEnvironment({
+  type: "array",
+  names: ["cases", "dcases", "rcases", "drcases"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(context) {
+    const payload = {
+      arraystretch: 1.2,
+      cols: [{
+        type: "align",
+        align: "l",
+        pregap: 0,
+        // TODO(kevinb) get the current style.
+        // For now we use the metrics for TEXT style which is what we were
+        // doing before.  Before attempting to get the current style we
+        // should look at TeX's behavior especially for \over and matrices.
+        postgap: 1.0
+        /* 1em quad */
+
+      }, {
+        type: "align",
+        align: "l",
+        pregap: 0,
+        postgap: 0
+      }]
+    };
+    const res = parseArray(context.parser, payload, dCellStyle(context.envName));
+    return {
+      type: "leftright",
+      mode: context.mode,
+      body: [res],
+      left: context.envName.indexOf("r") > -1 ? "." : "\\{",
+      right: context.envName.indexOf("r") > -1 ? "\\}" : ".",
+      rightColor: undefined
+    };
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // An aligned environment is like the align* environment
+// except it operates within math mode.
+// Note that we assume \nomallineskiplimit to be zero,
+// so that \strut@ is the same as \strut.
+
+defineEnvironment({
+  type: "array",
+  names: ["aligned"],
+  props: {
+    numArgs: 0
+  },
+  handler: alignedHandler,
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // A gathered environment is like an array environment with one centered
+// column, but where rows are considered lines so get \jot line spacing
+// and contents are set in \displaystyle.
+
+defineEnvironment({
+  type: "array",
+  names: ["gathered"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(context) {
+    const res = {
+      cols: [{
+        type: "align",
+        align: "c"
+      }],
+      addJot: true
+    };
+    return parseArray(context.parser, res, "display");
+  },
+
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // alignat environment is like an align environment, but one must explicitly
+// specify maximum number of columns in each row, and can adjust spacing between
+// each columns.
+
+defineEnvironment({
+  type: "array",
+  names: ["alignedat"],
+  // One for numbered and for unnumbered;
+  // but, KaTeX doesn't supports math numbering yet,
+  // they make no difference for now.
+  props: {
+    numArgs: 1
+  },
+  handler: alignedHandler,
+  htmlBuilder: htmlBuilder$3,
+  mathmlBuilder: mathmlBuilder$3
+}); // Catch \hline outside array environment
+
+defineFunction({
+  type: "text",
+  // Doesn't matter what this is.
+  names: ["\\hline", "\\hdashline"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: true
+  },
+
+  handler(context, args) {
+    throw new ParseError(`${context.funcName} valid only within array environment`);
+  }
+
+});
+
+const environments = _environments;
+
+// defineEnvironment definitions.
+// $FlowFixMe, "environment" handler returns an environment ParseNode
+
+defineFunction({
+  type: "environment",
+  names: ["\\begin", "\\end"],
+  props: {
+    numArgs: 1,
+    argTypes: ["text"]
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const nameGroup = args[0];
+
+    if (nameGroup.type !== "ordgroup") {
+      throw new ParseError("Invalid environment name", nameGroup);
+    }
+
+    let envName = "";
+
+    for (let i = 0; i < nameGroup.body.length; ++i) {
+      envName += assertNodeType(nameGroup.body[i], "textord").text;
+    }
+
+    if (funcName === "\\begin") {
+      // begin...end is similar to left...right
+      if (!environments.hasOwnProperty(envName)) {
+        throw new ParseError("No such environment: " + envName, nameGroup);
+      } // Build the environment object. Arguments and other information will
+      // be made available to the begin and end methods using properties.
+
+
+      const env = environments[envName];
+
+      const _parser$parseArgument = parser.parseArguments("\\begin{" + envName + "}", env),
+            args = _parser$parseArgument.args,
+            optArgs = _parser$parseArgument.optArgs;
+
+      const context = {
+        mode: parser.mode,
+        envName,
+        parser
+      };
+      const result = env.handler(context, args, optArgs);
+      parser.expect("\\end", false);
+      const endNameToken = parser.nextToken;
+      const end = assertNodeType(parser.parseFunction(), "environment");
+
+      if (end.name !== envName) {
+        throw new ParseError(`Mismatch: \\begin{${envName}} matched by \\end{${end.name}}`, endNameToken);
+      }
+
+      return result;
+    }
+
+    return {
+      type: "environment",
+      mode: parser.mode,
+      name: envName,
+      nameGroup
+    };
+  }
+
+});
+
+const makeSpan$2 = buildCommon.makeSpan;
+
+function htmlBuilder$4(group, options) {
+  const elements = buildExpression(group.body, options, true);
+  return makeSpan$2([group.mclass], elements, options);
+}
+
+function mathmlBuilder$4(group, options) {
+  let node;
+  const inner = buildExpression$1(group.body, options);
+
+  if (group.mclass === "minner") {
+    return mathMLTree.newDocumentFragment(inner);
+  } else if (group.mclass === "mord") {
+    if (group.isCharacterBox) {
+      node = inner[0];
+      node.type = "mi";
+    } else {
+      node = new mathMLTree.MathNode("mi", inner);
+    }
+  } else {
+    if (group.isCharacterBox) {
+      node = inner[0];
+      node.type = "mo";
+    } else {
+      node = new mathMLTree.MathNode("mo", inner);
+    } // Set spacing based on what is the most likely adjacent atom type.
+    // See TeXbook p170.
+
+
+    if (group.mclass === "mbin") {
+      node.attributes.lspace = "0.22em"; // medium space
+
+      node.attributes.rspace = "0.22em";
+    } else if (group.mclass === "mpunct") {
+      node.attributes.lspace = "0em";
+      node.attributes.rspace = "0.17em"; // thinspace
+    } else if (group.mclass === "mopen" || group.mclass === "mclose") {
+      node.attributes.lspace = "0em";
+      node.attributes.rspace = "0em";
+    } // MathML <mo> default space is 5/18 em, so <mrel> needs no action.
+    // Ref: https://developer.mozilla.org/en-US/docs/Web/MathML/Element/mo
+
+  }
+
+  return node;
+} // Math class commands except \mathop
+
+
+defineFunction({
+  type: "mclass",
+  names: ["\\mathord", "\\mathbin", "\\mathrel", "\\mathopen", "\\mathclose", "\\mathpunct", "\\mathinner"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: "m" + funcName.substr(5),
+      // TODO(kevinb): don't prefix with 'm'
+      body: ordargument(body),
+      isCharacterBox: utils.isCharacterBox(body)
+    };
+  },
+
+  htmlBuilder: htmlBuilder$4,
+  mathmlBuilder: mathmlBuilder$4
+});
+const binrelClass = arg => {
+  // \binrel@ spacing varies with (bin|rel|ord) of the atom in the argument.
+  // (by rendering separately and with {}s before and after, and measuring
+  // the change in spacing).  We'll do roughly the same by detecting the
+  // atom type directly.
+  const atom = arg.type === "ordgroup" && arg.body.length ? arg.body[0] : arg;
+
+  if (atom.type === "atom" && (atom.family === "bin" || atom.family === "rel")) {
+    return "m" + atom.family;
+  } else {
+    return "mord";
+  }
+}; // \@binrel{x}{y} renders like y but as mbin/mrel/mord if x is mbin/mrel/mord.
+// This is equivalent to \binrel@{x}\binrel@@{y} in AMSTeX.
+
+defineFunction({
+  type: "mclass",
+  names: ["\\@binrel"],
+  props: {
+    numArgs: 2
+  },
+
+  handler(_ref2, args) {
+    let parser = _ref2.parser;
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: binrelClass(args[0]),
+      body: [args[1]],
+      isCharacterBox: utils.isCharacterBox(args[1])
+    };
+  }
+
+}); // Build a relation or stacked op by placing one symbol on top of another
+
+defineFunction({
+  type: "mclass",
+  names: ["\\stackrel", "\\overset", "\\underset"],
+  props: {
+    numArgs: 2
+  },
+
+  handler(_ref3, args) {
+    let parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    const baseArg = args[1];
+    const shiftedArg = args[0];
+    let mclass;
+
+    if (funcName !== "\\stackrel") {
+      // LaTeX applies \binrel spacing to \overset and \underset.
+      mclass = binrelClass(baseArg);
+    } else {
+      mclass = "mrel"; // for \stackrel
+    }
+
+    const baseOp = {
+      type: "op",
+      mode: baseArg.mode,
+      limits: true,
+      alwaysHandleSupSub: true,
+      parentIsSupSub: false,
+      symbol: false,
+      suppressBaseShift: funcName !== "\\stackrel",
+      body: ordargument(baseArg)
+    };
+    const supsub = {
+      type: "supsub",
+      mode: shiftedArg.mode,
+      base: baseOp,
+      sup: funcName === "\\underset" ? null : shiftedArg,
+      sub: funcName === "\\underset" ? shiftedArg : null
+    };
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass,
+      body: [supsub],
+      isCharacterBox: utils.isCharacterBox(supsub)
+    };
+  },
+
+  htmlBuilder: htmlBuilder$4,
+  mathmlBuilder: mathmlBuilder$4
+});
+
+// TODO(kevinb): implement \\sl and \\sc
+
+const htmlBuilder$5 = (group, options) => {
+  const font = group.font;
+  const newOptions = options.withFont(font);
+  return buildGroup(group.body, newOptions);
+};
+
+const mathmlBuilder$5 = (group, options) => {
+  const font = group.font;
+  const newOptions = options.withFont(font);
+  return buildGroup$1(group.body, newOptions);
+};
+
+const fontAliases = {
+  "\\Bbb": "\\mathbb",
+  "\\bold": "\\mathbf",
+  "\\frak": "\\mathfrak",
+  "\\bm": "\\boldsymbol"
+};
+defineFunction({
+  type: "font",
+  names: [// styles, except \boldsymbol defined below
+  "\\mathrm", "\\mathit", "\\mathbf", "\\mathnormal", // families
+  "\\mathbb", "\\mathcal", "\\mathfrak", "\\mathscr", "\\mathsf", "\\mathtt", // aliases, except \bm defined below
+  "\\Bbb", "\\bold", "\\frak"],
+  props: {
+    numArgs: 1,
+    greediness: 2
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    let func = funcName;
+
+    if (func in fontAliases) {
+      func = fontAliases[func];
+    }
+
+    return {
+      type: "font",
+      mode: parser.mode,
+      font: func.slice(1),
+      body
+    };
+  },
+  htmlBuilder: htmlBuilder$5,
+  mathmlBuilder: mathmlBuilder$5
+});
+defineFunction({
+  type: "mclass",
+  names: ["\\boldsymbol", "\\bm"],
+  props: {
+    numArgs: 1,
+    greediness: 2
+  },
+  handler: (_ref2, args) => {
+    let parser = _ref2.parser;
+    const body = args[0];
+    const isCharacterBox = utils.isCharacterBox(body); // amsbsy.sty's \boldsymbol uses \binrel spacing to inherit the
+    // argument's bin|rel|ord status
+
+    return {
+      type: "mclass",
+      mode: parser.mode,
+      mclass: binrelClass(body),
+      body: [{
+        type: "font",
+        mode: parser.mode,
+        font: "boldsymbol",
+        body
+      }],
+      isCharacterBox: isCharacterBox
+    };
+  }
+}); // Old font changing functions
+
+defineFunction({
+  type: "font",
+  names: ["\\rm", "\\sf", "\\tt", "\\bf", "\\it", "\\cal"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: (_ref3, args) => {
+    let parser = _ref3.parser,
+        funcName = _ref3.funcName,
+        breakOnTokenText = _ref3.breakOnTokenText;
+    const mode = parser.mode;
+    const body = parser.parseExpression(true, breakOnTokenText);
+    const style = `math${funcName.slice(1)}`;
+    return {
+      type: "font",
+      mode: mode,
+      font: style,
+      body: {
+        type: "ordgroup",
+        mode: parser.mode,
+        body
+      }
+    };
+  },
+  htmlBuilder: htmlBuilder$5,
+  mathmlBuilder: mathmlBuilder$5
+});
+
+const adjustStyle = (size, originalStyle) => {
+  // Figure out what style this fraction should be in based on the
+  // function used
+  let style = originalStyle;
+
+  if (size === "display") {
+    // Get display style as a default.
+    // If incoming style is sub/sup, use style.text() to get correct size.
+    style = style.id >= Style$1.SCRIPT.id ? style.text() : Style$1.DISPLAY;
+  } else if (size === "text" && style.size === Style$1.DISPLAY.size) {
+    // We're in a \tfrac but incoming style is displaystyle, so:
+    style = Style$1.TEXT;
+  } else if (size === "script") {
+    style = Style$1.SCRIPT;
+  } else if (size === "scriptscript") {
+    style = Style$1.SCRIPTSCRIPT;
+  }
+
+  return style;
+};
+
+const htmlBuilder$6 = (group, options) => {
+  // Fractions are handled in the TeXbook on pages 444-445, rules 15(a-e).
+  const style = adjustStyle(group.size, options.style);
+  const nstyle = style.fracNum();
+  const dstyle = style.fracDen();
+  let newOptions;
+  newOptions = options.havingStyle(nstyle);
+  const numerm = buildGroup(group.numer, newOptions, options);
+
+  if (group.continued) {
+    // \cfrac inserts a \strut into the numerator.
+    // Get \strut dimensions from TeXbook page 353.
+    const hStrut = 8.5 / options.fontMetrics().ptPerEm;
+    const dStrut = 3.5 / options.fontMetrics().ptPerEm;
+    numerm.height = numerm.height < hStrut ? hStrut : numerm.height;
+    numerm.depth = numerm.depth < dStrut ? dStrut : numerm.depth;
+  }
+
+  newOptions = options.havingStyle(dstyle);
+  const denomm = buildGroup(group.denom, newOptions, options);
+  let rule;
+  let ruleWidth;
+  let ruleSpacing;
+
+  if (group.hasBarLine) {
+    if (group.barSize) {
+      ruleWidth = calculateSize(group.barSize, options);
+      rule = buildCommon.makeLineSpan("frac-line", options, ruleWidth);
+    } else {
+      rule = buildCommon.makeLineSpan("frac-line", options);
+    }
+
+    ruleWidth = rule.height;
+    ruleSpacing = rule.height;
+  } else {
+    rule = null;
+    ruleWidth = 0;
+    ruleSpacing = options.fontMetrics().defaultRuleThickness;
+  } // Rule 15b
+
+
+  let numShift;
+  let clearance;
+  let denomShift;
+
+  if (style.size === Style$1.DISPLAY.size || group.size === "display") {
+    numShift = options.fontMetrics().num1;
+
+    if (ruleWidth > 0) {
+      clearance = 3 * ruleSpacing;
+    } else {
+      clearance = 7 * ruleSpacing;
+    }
+
+    denomShift = options.fontMetrics().denom1;
+  } else {
+    if (ruleWidth > 0) {
+      numShift = options.fontMetrics().num2;
+      clearance = ruleSpacing;
+    } else {
+      numShift = options.fontMetrics().num3;
+      clearance = 3 * ruleSpacing;
+    }
+
+    denomShift = options.fontMetrics().denom2;
+  }
+
+  let frac;
+
+  if (!rule) {
+    // Rule 15c
+    const candidateClearance = numShift - numerm.depth - (denomm.height - denomShift);
+
+    if (candidateClearance < clearance) {
+      numShift += 0.5 * (clearance - candidateClearance);
+      denomShift += 0.5 * (clearance - candidateClearance);
+    }
+
+    frac = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [{
+        type: "elem",
+        elem: denomm,
+        shift: denomShift
+      }, {
+        type: "elem",
+        elem: numerm,
+        shift: -numShift
+      }]
+    }, options);
+  } else {
+    // Rule 15d
+    const axisHeight = options.fontMetrics().axisHeight;
+
+    if (numShift - numerm.depth - (axisHeight + 0.5 * ruleWidth) < clearance) {
+      numShift += clearance - (numShift - numerm.depth - (axisHeight + 0.5 * ruleWidth));
+    }
+
+    if (axisHeight - 0.5 * ruleWidth - (denomm.height - denomShift) < clearance) {
+      denomShift += clearance - (axisHeight - 0.5 * ruleWidth - (denomm.height - denomShift));
+    }
+
+    const midShift = -(axisHeight - 0.5 * ruleWidth);
+    frac = buildCommon.makeVList({
+      positionType: "individualShift",
+      children: [{
+        type: "elem",
+        elem: denomm,
+        shift: denomShift
+      }, {
+        type: "elem",
+        elem: rule,
+        shift: midShift
+      }, {
+        type: "elem",
+        elem: numerm,
+        shift: -numShift
+      }]
+    }, options);
+  } // Since we manually change the style sometimes (with \dfrac or \tfrac),
+  // account for the possible size change here.
+
+
+  newOptions = options.havingStyle(style);
+  frac.height *= newOptions.sizeMultiplier / options.sizeMultiplier;
+  frac.depth *= newOptions.sizeMultiplier / options.sizeMultiplier; // Rule 15e
+
+  let delimSize;
+
+  if (style.size === Style$1.DISPLAY.size) {
+    delimSize = options.fontMetrics().delim1;
+  } else {
+    delimSize = options.fontMetrics().delim2;
+  }
+
+  let leftDelim;
+  let rightDelim;
+
+  if (group.leftDelim == null) {
+    leftDelim = makeNullDelimiter(options, ["mopen"]);
+  } else {
+    leftDelim = delimiter.customSizedDelim(group.leftDelim, delimSize, true, options.havingStyle(style), group.mode, ["mopen"]);
+  }
+
+  if (group.continued) {
+    rightDelim = buildCommon.makeSpan([]); // zero width for \cfrac
+  } else if (group.rightDelim == null) {
+    rightDelim = makeNullDelimiter(options, ["mclose"]);
+  } else {
+    rightDelim = delimiter.customSizedDelim(group.rightDelim, delimSize, true, options.havingStyle(style), group.mode, ["mclose"]);
+  }
+
+  return buildCommon.makeSpan(["mord"].concat(newOptions.sizingClasses(options)), [leftDelim, buildCommon.makeSpan(["mfrac"], [frac]), rightDelim], options);
+};
+
+const mathmlBuilder$6 = (group, options) => {
+  let node = new mathMLTree.MathNode("mfrac", [buildGroup$1(group.numer, options), buildGroup$1(group.denom, options)]);
+
+  if (!group.hasBarLine) {
+    node.setAttribute("linethickness", "0px");
+  } else if (group.barSize) {
+    const ruleWidth = calculateSize(group.barSize, options);
+    node.setAttribute("linethickness", ruleWidth + "em");
+  }
+
+  const style = adjustStyle(group.size, options.style);
+
+  if (style.size !== options.style.size) {
+    node = new mathMLTree.MathNode("mstyle", [node]);
+    const isDisplay = style.size === Style$1.DISPLAY.size ? "true" : "false";
+    node.setAttribute("displaystyle", isDisplay);
+    node.setAttribute("scriptlevel", "0");
+  }
+
+  if (group.leftDelim != null || group.rightDelim != null) {
+    const withDelims = [];
+
+    if (group.leftDelim != null) {
+      const leftOp = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(group.leftDelim.replace("\\", ""))]);
+      leftOp.setAttribute("fence", "true");
+      withDelims.push(leftOp);
+    }
+
+    withDelims.push(node);
+
+    if (group.rightDelim != null) {
+      const rightOp = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode(group.rightDelim.replace("\\", ""))]);
+      rightOp.setAttribute("fence", "true");
+      withDelims.push(rightOp);
+    }
+
+    return makeRow(withDelims);
+  }
+
+  return node;
+};
+
+defineFunction({
+  type: "genfrac",
+  names: ["\\cfrac", "\\dfrac", "\\frac", "\\tfrac", "\\dbinom", "\\binom", "\\tbinom", "\\\\atopfrac", // can’t be entered directly
+  "\\\\bracefrac", "\\\\brackfrac"],
+  props: {
+    numArgs: 2,
+    greediness: 2
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const numer = args[0];
+    const denom = args[1];
+    let hasBarLine;
+    let leftDelim = null;
+    let rightDelim = null;
+    let size = "auto";
+
+    switch (funcName) {
+      case "\\cfrac":
+      case "\\dfrac":
+      case "\\frac":
+      case "\\tfrac":
+        hasBarLine = true;
+        break;
+
+      case "\\\\atopfrac":
+        hasBarLine = false;
+        break;
+
+      case "\\dbinom":
+      case "\\binom":
+      case "\\tbinom":
+        hasBarLine = false;
+        leftDelim = "(";
+        rightDelim = ")";
+        break;
+
+      case "\\\\bracefrac":
+        hasBarLine = false;
+        leftDelim = "\\{";
+        rightDelim = "\\}";
+        break;
+
+      case "\\\\brackfrac":
+        hasBarLine = false;
+        leftDelim = "[";
+        rightDelim = "]";
+        break;
+
+      default:
+        throw new Error("Unrecognized genfrac command");
+    }
+
+    switch (funcName) {
+      case "\\cfrac":
+      case "\\dfrac":
+      case "\\dbinom":
+        size = "display";
+        break;
+
+      case "\\tfrac":
+      case "\\tbinom":
+        size = "text";
+        break;
+    }
+
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      continued: funcName === "\\cfrac",
+      numer,
+      denom,
+      hasBarLine,
+      leftDelim,
+      rightDelim,
+      size,
+      barSize: null
+    };
+  },
+  htmlBuilder: htmlBuilder$6,
+  mathmlBuilder: mathmlBuilder$6
+}); // Infix generalized fractions -- these are not rendered directly, but replaced
+// immediately by one of the variants above.
+
+defineFunction({
+  type: "infix",
+  names: ["\\over", "\\choose", "\\atop", "\\brace", "\\brack"],
+  props: {
+    numArgs: 0,
+    infix: true
+  },
+
+  handler(_ref2) {
+    let parser = _ref2.parser,
+        funcName = _ref2.funcName,
+        token = _ref2.token;
+    let replaceWith;
+
+    switch (funcName) {
+      case "\\over":
+        replaceWith = "\\frac";
+        break;
+
+      case "\\choose":
+        replaceWith = "\\binom";
+        break;
+
+      case "\\atop":
+        replaceWith = "\\\\atopfrac";
+        break;
+
+      case "\\brace":
+        replaceWith = "\\\\bracefrac";
+        break;
+
+      case "\\brack":
+        replaceWith = "\\\\brackfrac";
+        break;
+
+      default:
+        throw new Error("Unrecognized infix genfrac command");
+    }
+
+    return {
+      type: "infix",
+      mode: parser.mode,
+      replaceWith,
+      token
+    };
+  }
+
+});
+const stylArray = ["display", "text", "script", "scriptscript"];
+
+const delimFromValue = function delimFromValue(delimString) {
+  let delim = null;
+
+  if (delimString.length > 0) {
+    delim = delimString;
+    delim = delim === "." ? null : delim;
+  }
+
+  return delim;
+};
+
+defineFunction({
+  type: "genfrac",
+  names: ["\\genfrac"],
+  props: {
+    numArgs: 6,
+    greediness: 6,
+    argTypes: ["math", "math", "size", "text", "math", "math"]
+  },
+
+  handler(_ref3, args) {
+    let parser = _ref3.parser;
+    const numer = args[4];
+    const denom = args[5]; // Look into the parse nodes to get the desired delimiters.
+
+    const leftDelim = args[0].type === "atom" && args[0].family === "open" ? delimFromValue(args[0].text) : null;
+    const rightDelim = args[1].type === "atom" && args[1].family === "close" ? delimFromValue(args[1].text) : null;
+    const barNode = assertNodeType(args[2], "size");
+    let hasBarLine;
+    let barSize = null;
+
+    if (barNode.isBlank) {
+      // \genfrac acts differently than \above.
+      // \genfrac treats an empty size group as a signal to use a
+      // standard bar size. \above would see size = 0 and omit the bar.
+      hasBarLine = true;
+    } else {
+      barSize = barNode.value;
+      hasBarLine = barSize.number > 0;
+    } // Find out if we want displaystyle, textstyle, etc.
+
+
+    let size = "auto";
+    let styl = args[3];
+
+    if (styl.type === "ordgroup") {
+      if (styl.body.length > 0) {
+        const textOrd = assertNodeType(styl.body[0], "textord");
+        size = stylArray[Number(textOrd.text)];
+      }
+    } else {
+      styl = assertNodeType(styl, "textord");
+      size = stylArray[Number(styl.text)];
+    }
+
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      numer,
+      denom,
+      continued: false,
+      hasBarLine,
+      barSize,
+      leftDelim,
+      rightDelim,
+      size
+    };
+  },
+
+  htmlBuilder: htmlBuilder$6,
+  mathmlBuilder: mathmlBuilder$6
+}); // \above is an infix fraction that also defines a fraction bar size.
+
+defineFunction({
+  type: "infix",
+  names: ["\\above"],
+  props: {
+    numArgs: 1,
+    argTypes: ["size"],
+    infix: true
+  },
+
+  handler(_ref4, args) {
+    let parser = _ref4.parser,
+        funcName = _ref4.funcName,
+        token = _ref4.token;
+    return {
+      type: "infix",
+      mode: parser.mode,
+      replaceWith: "\\\\abovefrac",
+      size: assertNodeType(args[0], "size").value,
+      token
+    };
+  }
+
+});
+defineFunction({
+  type: "genfrac",
+  names: ["\\\\abovefrac"],
+  props: {
+    numArgs: 3,
+    argTypes: ["math", "size", "math"]
+  },
+  handler: (_ref5, args) => {
+    let parser = _ref5.parser,
+        funcName = _ref5.funcName;
+    const numer = args[0];
+    const barSize = assert(assertNodeType(args[1], "infix").size);
+    const denom = args[2];
+    const hasBarLine = barSize.number > 0;
+    return {
+      type: "genfrac",
+      mode: parser.mode,
+      numer,
+      denom,
+      continued: false,
+      hasBarLine,
+      barSize,
+      leftDelim: null,
+      rightDelim: null,
+      size: "auto"
+    };
+  },
+  htmlBuilder: htmlBuilder$6,
+  mathmlBuilder: mathmlBuilder$6
+});
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only "horizBrace", but
+const htmlBuilder$7 = (grp, options) => {
+  const style = options.style; // Pull out the `ParseNode<"horizBrace">` if `grp` is a "supsub" node.
+
+  let supSubGroup;
+  let group;
+
+  if (grp.type === "supsub") {
+    // Ref: LaTeX source2e: }}}}\limits}
+    // i.e. LaTeX treats the brace similar to an op and passes it
+    // with \limits, so we need to assign supsub style.
+    supSubGroup = grp.sup ? buildGroup(grp.sup, options.havingStyle(style.sup()), options) : buildGroup(grp.sub, options.havingStyle(style.sub()), options);
+    group = assertNodeType(grp.base, "horizBrace");
+  } else {
+    group = assertNodeType(grp, "horizBrace");
+  } // Build the base group
+
+
+  const body = buildGroup(group.base, options.havingBaseStyle(Style$1.DISPLAY)); // Create the stretchy element
+
+  const braceBody = stretchy.svgSpan(group, options); // Generate the vlist, with the appropriate kerns        ┏━━━━━━━━┓
+  // This first vlist contains the content and the brace:   equation
+
+  let vlist;
+
+  if (group.isOver) {
+    vlist = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: body
+      }, {
+        type: "kern",
+        size: 0.1
+      }, {
+        type: "elem",
+        elem: braceBody
+      }]
+    }, options); // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+    vlist.children[0].children[0].children[1].classes.push("svg-align");
+  } else {
+    vlist = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: body.depth + 0.1 + braceBody.height,
+      children: [{
+        type: "elem",
+        elem: braceBody
+      }, {
+        type: "kern",
+        size: 0.1
+      }, {
+        type: "elem",
+        elem: body
+      }]
+    }, options); // $FlowFixMe: Replace this with passing "svg-align" into makeVList.
+
+    vlist.children[0].children[0].children[0].classes.push("svg-align");
+  }
+
+  if (supSubGroup) {
+    // To write the supsub, wrap the first vlist in another vlist:
+    // They can't all go in the same vlist, because the note might be
+    // wider than the equation. We want the equation to control the
+    // brace width.
+    //      note          long note           long note
+    //   ┏━━━━━━━━┓   or    ┏━━━┓     not    ┏━━━━━━━━━┓
+    //    equation           eqn                 eqn
+    const vSpan = buildCommon.makeSpan(["mord", group.isOver ? "mover" : "munder"], [vlist], options);
+
+    if (group.isOver) {
+      vlist = buildCommon.makeVList({
+        positionType: "firstBaseline",
+        children: [{
+          type: "elem",
+          elem: vSpan
+        }, {
+          type: "kern",
+          size: 0.2
+        }, {
+          type: "elem",
+          elem: supSubGroup
+        }]
+      }, options);
+    } else {
+      vlist = buildCommon.makeVList({
+        positionType: "bottom",
+        positionData: vSpan.depth + 0.2 + supSubGroup.height + supSubGroup.depth,
+        children: [{
+          type: "elem",
+          elem: supSubGroup
+        }, {
+          type: "kern",
+          size: 0.2
+        }, {
+          type: "elem",
+          elem: vSpan
+        }]
+      }, options);
+    }
+  }
+
+  return buildCommon.makeSpan(["mord", group.isOver ? "mover" : "munder"], [vlist], options);
+};
+
+const mathmlBuilder$7 = (group, options) => {
+  const accentNode = stretchy.mathMLnode(group.label);
+  return new mathMLTree.MathNode(group.isOver ? "mover" : "munder", [buildGroup$1(group.base, options), accentNode]);
+}; // Horizontal stretchy braces
+
+
+defineFunction({
+  type: "horizBrace",
+  names: ["\\overbrace", "\\underbrace"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    return {
+      type: "horizBrace",
+      mode: parser.mode,
+      label: funcName,
+      isOver: /^\\over/.test(funcName),
+      base: args[0]
+    };
+  },
+
+  htmlBuilder: htmlBuilder$7,
+  mathmlBuilder: mathmlBuilder$7
+});
+
+defineFunction({
+  type: "href",
+  names: ["\\href"],
+  props: {
+    numArgs: 2,
+    argTypes: ["url", "original"],
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser;
+    const body = args[1];
+    const href = assertNodeType(args[0], "url").url;
+
+    if (!parser.settings.isTrusted({
+      command: "\\href",
+      url: href
+    })) {
+      return parser.formatUnsupportedCmd("\\href");
+    }
+
+    return {
+      type: "href",
+      mode: parser.mode,
+      href,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const elements = buildExpression(group.body, options, false);
+    return buildCommon.makeAnchor(group.href, [], elements, options);
+  },
+  mathmlBuilder: (group, options) => {
+    let math = buildExpressionRow(group.body, options);
+
+    if (!(math instanceof MathNode)) {
+      math = new MathNode("mrow", [math]);
+    }
+
+    math.setAttribute("href", group.href);
+    return math;
+  }
+});
+defineFunction({
+  type: "href",
+  names: ["\\url"],
+  props: {
+    numArgs: 1,
+    argTypes: ["url"],
+    allowedInText: true
+  },
+  handler: (_ref2, args) => {
+    let parser = _ref2.parser;
+    const href = assertNodeType(args[0], "url").url;
+
+    if (!parser.settings.isTrusted({
+      command: "\\url",
+      url: href
+    })) {
+      return parser.formatUnsupportedCmd("\\url");
+    }
+
+    const chars = [];
+
+    for (let i = 0; i < href.length; i++) {
+      let c = href[i];
+
+      if (c === "~") {
+        c = "\\textasciitilde";
+      }
+
+      chars.push({
+        type: "textord",
+        mode: "text",
+        text: c
+      });
+    }
+
+    const body = {
+      type: "text",
+      mode: parser.mode,
+      font: "\\texttt",
+      body: chars
+    };
+    return {
+      type: "href",
+      mode: parser.mode,
+      href,
+      body: ordargument(body)
+    };
+  }
+});
+
+defineFunction({
+  type: "html",
+  names: ["\\htmlClass", "\\htmlId", "\\htmlStyle", "\\htmlData"],
+  props: {
+    numArgs: 2,
+    argTypes: ["raw", "original"],
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName,
+        token = _ref.token;
+    const value = assertNodeType(args[0], "raw").string;
+    const body = args[1];
+
+    if (parser.settings.strict) {
+      parser.settings.reportNonstrict("htmlExtension", "HTML extension is disabled on strict mode");
+    }
+
+    let trustContext;
+    const attributes = {};
+
+    switch (funcName) {
+      case "\\htmlClass":
+        attributes.class = value;
+        trustContext = {
+          command: "\\htmlClass",
+          class: value
+        };
+        break;
+
+      case "\\htmlId":
+        attributes.id = value;
+        trustContext = {
+          command: "\\htmlId",
+          id: value
+        };
+        break;
+
+      case "\\htmlStyle":
+        attributes.style = value;
+        trustContext = {
+          command: "\\htmlStyle",
+          style: value
+        };
+        break;
+
+      case "\\htmlData":
+        {
+          const data = value.split(",");
+
+          for (let i = 0; i < data.length; i++) {
+            const keyVal = data[i].split("=");
+
+            if (keyVal.length !== 2) {
+              throw new ParseError("Error parsing key-value for \\htmlData");
+            }
+
+            attributes["data-" + keyVal[0].trim()] = keyVal[1].trim();
+          }
+
+          trustContext = {
+            command: "\\htmlData",
+            attributes
+          };
+          break;
+        }
+
+      default:
+        throw new Error("Unrecognized html command");
+    }
+
+    if (!parser.settings.isTrusted(trustContext)) {
+      return parser.formatUnsupportedCmd(funcName);
+    }
+
+    return {
+      type: "html",
+      mode: parser.mode,
+      attributes,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const elements = buildExpression(group.body, options, false);
+    const classes = ["enclosing"];
+
+    if (group.attributes.class) {
+      classes.push(...group.attributes.class.trim().split(/\s+/));
+    }
+
+    const span = buildCommon.makeSpan(classes, elements, options);
+
+    for (const attr in group.attributes) {
+      if (attr !== "class" && group.attributes.hasOwnProperty(attr)) {
+        span.setAttribute(attr, group.attributes[attr]);
+      }
+    }
+
+    return span;
+  },
+  mathmlBuilder: (group, options) => {
+    return buildExpressionRow(group.body, options);
+  }
+});
+
+defineFunction({
+  type: "htmlmathml",
+  names: ["\\html@mathml"],
+  props: {
+    numArgs: 2,
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser;
+    return {
+      type: "htmlmathml",
+      mode: parser.mode,
+      html: ordargument(args[0]),
+      mathml: ordargument(args[1])
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const elements = buildExpression(group.html, options, false);
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: (group, options) => {
+    return buildExpressionRow(group.mathml, options);
+  }
+});
+
+const sizeData = function sizeData(str) {
+  if (/^[-+]? *(\d+(\.\d*)?|\.\d+)$/.test(str)) {
+    // str is a number with no unit specified.
+    // default unit is bp, per graphix package.
+    return {
+      number: +str,
+      unit: "bp"
+    };
+  } else {
+    const match = /([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(str);
+
+    if (!match) {
+      throw new ParseError("Invalid size: '" + str + "' in \\includegraphics");
+    }
+
+    const data = {
+      number: +(match[1] + match[2]),
+      // sign + magnitude, cast to number
+      unit: match[3]
+    };
+
+    if (!validUnit(data)) {
+      throw new ParseError("Invalid unit: '" + data.unit + "' in \\includegraphics.");
+    }
+
+    return data;
+  }
+};
+
+defineFunction({
+  type: "includegraphics",
+  names: ["\\includegraphics"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1,
+    argTypes: ["raw", "url"],
+    allowedInText: false
+  },
+  handler: (_ref, args, optArgs) => {
+    let parser = _ref.parser;
+    let width = {
+      number: 0,
+      unit: "em"
+    };
+    let height = {
+      number: 0.9,
+      unit: "em"
+    }; // sorta character sized.
+
+    let totalheight = {
+      number: 0,
+      unit: "em"
+    };
+    let alt = "";
+
+    if (optArgs[0]) {
+      const attributeStr = assertNodeType(optArgs[0], "raw").string; // Parser.js does not parse key/value pairs. We get a string.
+
+      const attributes = attributeStr.split(",");
+
+      for (let i = 0; i < attributes.length; i++) {
+        const keyVal = attributes[i].split("=");
+
+        if (keyVal.length === 2) {
+          const str = keyVal[1].trim();
+
+          switch (keyVal[0].trim()) {
+            case "alt":
+              alt = str;
+              break;
+
+            case "width":
+              width = sizeData(str);
+              break;
+
+            case "height":
+              height = sizeData(str);
+              break;
+
+            case "totalheight":
+              totalheight = sizeData(str);
+              break;
+
+            default:
+              throw new ParseError("Invalid key: '" + keyVal[0] + "' in \\includegraphics.");
+          }
+        }
+      }
+    }
+
+    const src = assertNodeType(args[0], "url").url;
+
+    if (alt === "") {
+      // No alt given. Use the file name. Strip away the path.
+      alt = src;
+      alt = alt.replace(/^.*[\\/]/, '');
+      alt = alt.substring(0, alt.lastIndexOf('.'));
+    }
+
+    if (!parser.settings.isTrusted({
+      command: "\\includegraphics",
+      url: src
+    })) {
+      return parser.formatUnsupportedCmd("\\includegraphics");
+    }
+
+    return {
+      type: "includegraphics",
+      mode: parser.mode,
+      alt: alt,
+      width: width,
+      height: height,
+      totalheight: totalheight,
+      src: src
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const height = calculateSize(group.height, options);
+    let depth = 0;
+
+    if (group.totalheight.number > 0) {
+      depth = calculateSize(group.totalheight, options) - height;
+      depth = Number(depth.toFixed(2));
+    }
+
+    let width = 0;
+
+    if (group.width.number > 0) {
+      width = calculateSize(group.width, options);
+    }
+
+    const style = {
+      height: height + depth + "em"
+    };
+
+    if (width > 0) {
+      style.width = width + "em";
+    }
+
+    if (depth > 0) {
+      style.verticalAlign = -depth + "em";
+    }
+
+    const node = new Img(group.src, group.alt, style);
+    node.height = height;
+    node.depth = depth;
+    return node;
+  },
+  mathmlBuilder: (group, options) => {
+    const node = new mathMLTree.MathNode("mglyph", []);
+    node.setAttribute("alt", group.alt);
+    const height = calculateSize(group.height, options);
+    let depth = 0;
+
+    if (group.totalheight.number > 0) {
+      depth = calculateSize(group.totalheight, options) - height;
+      depth = depth.toFixed(2);
+      node.setAttribute("valign", "-" + depth + "em");
+    }
+
+    node.setAttribute("height", height + depth + "em");
+
+    if (group.width.number > 0) {
+      const width = calculateSize(group.width, options);
+      node.setAttribute("width", width + "em");
+    }
+
+    node.setAttribute("src", group.src);
+    return node;
+  }
+});
+
+// Horizontal spacing commands
+
+defineFunction({
+  type: "kern",
+  names: ["\\kern", "\\mkern", "\\hskip", "\\mskip"],
+  props: {
+    numArgs: 1,
+    argTypes: ["size"],
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const size = assertNodeType(args[0], "size");
+
+    if (parser.settings.strict) {
+      const mathFunction = funcName[1] === 'm'; // \mkern, \mskip
+
+      const muUnit = size.value.unit === 'mu';
+
+      if (mathFunction) {
+        if (!muUnit) {
+          parser.settings.reportNonstrict("mathVsTextUnits", `LaTeX's ${funcName} supports only mu units, ` + `not ${size.value.unit} units`);
+        }
+
+        if (parser.mode !== "math") {
+          parser.settings.reportNonstrict("mathVsTextUnits", `LaTeX's ${funcName} works only in math mode`);
+        }
+      } else {
+        // !mathFunction
+        if (muUnit) {
+          parser.settings.reportNonstrict("mathVsTextUnits", `LaTeX's ${funcName} doesn't support mu units`);
+        }
+      }
+    }
+
+    return {
+      type: "kern",
+      mode: parser.mode,
+      dimension: size.value
+    };
+  },
+
+  htmlBuilder(group, options) {
+    return buildCommon.makeGlue(group.dimension, options);
+  },
+
+  mathmlBuilder(group, options) {
+    const dimension = calculateSize(group.dimension, options);
+    return new mathMLTree.SpaceNode(dimension);
+  }
+
+});
+
+// Horizontal overlap functions
+defineFunction({
+  type: "lap",
+  names: ["\\mathllap", "\\mathrlap", "\\mathclap"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    return {
+      type: "lap",
+      mode: parser.mode,
+      alignment: funcName.slice(5),
+      body
+    };
+  },
+  htmlBuilder: (group, options) => {
+    // mathllap, mathrlap, mathclap
+    let inner;
+
+    if (group.alignment === "clap") {
+      // ref: https://www.math.lsu.edu/~aperlis/publications/mathclap/
+      inner = buildCommon.makeSpan([], [buildGroup(group.body, options)]); // wrap, since CSS will center a .clap > .inner > span
+
+      inner = buildCommon.makeSpan(["inner"], [inner], options);
+    } else {
+      inner = buildCommon.makeSpan(["inner"], [buildGroup(group.body, options)]);
+    }
+
+    const fix = buildCommon.makeSpan(["fix"], []);
+    let node = buildCommon.makeSpan([group.alignment], [inner, fix], options); // At this point, we have correctly set horizontal alignment of the
+    // two items involved in the lap.
+    // Next, use a strut to set the height of the HTML bounding box.
+    // Otherwise, a tall argument may be misplaced.
+    // This code resolved issue #1153
+
+    const strut = buildCommon.makeSpan(["strut"]);
+    strut.style.height = node.height + node.depth + "em";
+    strut.style.verticalAlign = -node.depth + "em";
+    node.children.unshift(strut); // Next, prevent vertical misplacement when next to something tall.
+    // This code resolves issue #1234
+
+    node = buildCommon.makeSpan(["thinbox"], [node], options);
+    return buildCommon.makeSpan(["mord", "vbox"], [node], options);
+  },
+  mathmlBuilder: (group, options) => {
+    // mathllap, mathrlap, mathclap
+    const node = new mathMLTree.MathNode("mpadded", [buildGroup$1(group.body, options)]);
+
+    if (group.alignment !== "rlap") {
+      const offset = group.alignment === "llap" ? "-1" : "-0.5";
+      node.setAttribute("lspace", offset + "width");
+    }
+
+    node.setAttribute("width", "0px");
+    return node;
+  }
+});
+
+defineFunction({
+  type: "styling",
+  names: ["\\(", "$"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: false
+  },
+
+  handler(_ref, args) {
+    let funcName = _ref.funcName,
+        parser = _ref.parser;
+    const outerMode = parser.mode;
+    parser.switchMode("math");
+    const close = funcName === "\\(" ? "\\)" : "$";
+    const body = parser.parseExpression(false, close);
+    parser.expect(close);
+    parser.switchMode(outerMode);
+    return {
+      type: "styling",
+      mode: parser.mode,
+      style: "text",
+      body
+    };
+  }
+
+}); // Check for extra closing math delimiters
+
+defineFunction({
+  type: "text",
+  // Doesn't matter what this is.
+  names: ["\\)", "\\]"],
+  props: {
+    numArgs: 0,
+    allowedInText: true,
+    allowedInMath: false
+  },
+
+  handler(context, args) {
+    throw new ParseError(`Mismatched ${context.funcName}`);
+  }
+
+});
+
+const chooseMathStyle = (group, options) => {
+  switch (options.style.size) {
+    case Style$1.DISPLAY.size:
+      return group.display;
+
+    case Style$1.TEXT.size:
+      return group.text;
+
+    case Style$1.SCRIPT.size:
+      return group.script;
+
+    case Style$1.SCRIPTSCRIPT.size:
+      return group.scriptscript;
+
+    default:
+      return group.text;
+  }
+};
+
+defineFunction({
+  type: "mathchoice",
+  names: ["\\mathchoice"],
+  props: {
+    numArgs: 4
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser;
+    return {
+      type: "mathchoice",
+      mode: parser.mode,
+      display: ordargument(args[0]),
+      text: ordargument(args[1]),
+      script: ordargument(args[2]),
+      scriptscript: ordargument(args[3])
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const body = chooseMathStyle(group, options);
+    const elements = buildExpression(body, options, false);
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: (group, options) => {
+    const body = chooseMathStyle(group, options);
+    return buildExpressionRow(body, options);
+  }
+});
+
+// For an operator with limits, assemble the base, sup, and sub into a span.
+const assembleSupSub = (base, supGroup, subGroup, options, style, slant, baseShift) => {
+  base = buildCommon.makeSpan([], [base]);
+  let sub;
+  let sup; // We manually have to handle the superscripts and subscripts. This,
+  // aside from the kern calculations, is copied from supsub.
+
+  if (supGroup) {
+    const elem = buildGroup(supGroup, options.havingStyle(style.sup()), options);
+    sup = {
+      elem,
+      kern: Math.max(options.fontMetrics().bigOpSpacing1, options.fontMetrics().bigOpSpacing3 - elem.depth)
+    };
+  }
+
+  if (subGroup) {
+    const elem = buildGroup(subGroup, options.havingStyle(style.sub()), options);
+    sub = {
+      elem,
+      kern: Math.max(options.fontMetrics().bigOpSpacing2, options.fontMetrics().bigOpSpacing4 - elem.height)
+    };
+  } // Build the final group as a vlist of the possible subscript, base,
+  // and possible superscript.
+
+
+  let finalGroup;
+
+  if (sup && sub) {
+    const bottom = options.fontMetrics().bigOpSpacing5 + sub.elem.height + sub.elem.depth + sub.kern + base.depth + baseShift;
+    finalGroup = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: bottom,
+      children: [{
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }, {
+        type: "elem",
+        elem: sub.elem,
+        marginLeft: -slant + "em"
+      }, {
+        type: "kern",
+        size: sub.kern
+      }, {
+        type: "elem",
+        elem: base
+      }, {
+        type: "kern",
+        size: sup.kern
+      }, {
+        type: "elem",
+        elem: sup.elem,
+        marginLeft: slant + "em"
+      }, {
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }]
+    }, options);
+  } else if (sub) {
+    const top = base.height - baseShift; // Shift the limits by the slant of the symbol. Note
+    // that we are supposed to shift the limits by 1/2 of the slant,
+    // but since we are centering the limits adding a full slant of
+    // margin will shift by 1/2 that.
+
+    finalGroup = buildCommon.makeVList({
+      positionType: "top",
+      positionData: top,
+      children: [{
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }, {
+        type: "elem",
+        elem: sub.elem,
+        marginLeft: -slant + "em"
+      }, {
+        type: "kern",
+        size: sub.kern
+      }, {
+        type: "elem",
+        elem: base
+      }]
+    }, options);
+  } else if (sup) {
+    const bottom = base.depth + baseShift;
+    finalGroup = buildCommon.makeVList({
+      positionType: "bottom",
+      positionData: bottom,
+      children: [{
+        type: "elem",
+        elem: base
+      }, {
+        type: "kern",
+        size: sup.kern
+      }, {
+        type: "elem",
+        elem: sup.elem,
+        marginLeft: slant + "em"
+      }, {
+        type: "kern",
+        size: options.fontMetrics().bigOpSpacing5
+      }]
+    }, options);
+  } else {
+    // This case probably shouldn't occur (this would mean the
+    // supsub was sending us a group with no superscript or
+    // subscript) but be safe.
+    return base;
+  }
+
+  return buildCommon.makeSpan(["mop", "op-limits"], [finalGroup], options);
+};
+
+// Limits, symbols
+// Most operators have a large successor symbol, but these don't.
+const noSuccessor = ["\\smallint"]; // NOTE: Unlike most `htmlBuilder`s, this one handles not only "op", but also
+// "supsub" since some of them (like \int) can affect super/subscripting.
+
+const htmlBuilder$8 = (grp, options) => {
+  // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+  let supGroup;
+  let subGroup;
+  let hasLimits = false;
+  let group;
+
+  if (grp.type === "supsub") {
+    // If we have limits, supsub will pass us its group to handle. Pull
+    // out the superscript and subscript and set the group to the op in
+    // its base.
+    supGroup = grp.sup;
+    subGroup = grp.sub;
+    group = assertNodeType(grp.base, "op");
+    hasLimits = true;
+  } else {
+    group = assertNodeType(grp, "op");
+  }
+
+  const style = options.style;
+  let large = false;
+
+  if (style.size === Style$1.DISPLAY.size && group.symbol && !utils.contains(noSuccessor, group.name)) {
+    // Most symbol operators get larger in displaystyle (rule 13)
+    large = true;
+  }
+
+  let base;
+
+  if (group.symbol) {
+    // If this is a symbol, create the symbol.
+    const fontName = large ? "Size2-Regular" : "Size1-Regular";
+    let stash = "";
+
+    if (group.name === "\\oiint" || group.name === "\\oiiint") {
+      // No font glyphs yet, so use a glyph w/o the oval.
+      // TODO: When font glyphs are available, delete this code.
+      stash = group.name.substr(1); // $FlowFixMe
+
+      group.name = stash === "oiint" ? "\\iint" : "\\iiint";
+    }
+
+    base = buildCommon.makeSymbol(group.name, fontName, "math", options, ["mop", "op-symbol", large ? "large-op" : "small-op"]);
+
+    if (stash.length > 0) {
+      // We're in \oiint or \oiiint. Overlay the oval.
+      // TODO: When font glyphs are available, delete this code.
+      const italic = base.italic;
+      const oval = buildCommon.staticSvg(stash + "Size" + (large ? "2" : "1"), options);
+      base = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: [{
+          type: "elem",
+          elem: base,
+          shift: 0
+        }, {
+          type: "elem",
+          elem: oval,
+          shift: large ? 0.08 : 0
+        }]
+      }, options); // $FlowFixMe
+
+      group.name = "\\" + stash;
+      base.classes.unshift("mop"); // $FlowFixMe
+
+      base.italic = italic;
+    }
+  } else if (group.body) {
+    // If this is a list, compose that list.
+    const inner = buildExpression(group.body, options, true);
+
+    if (inner.length === 1 && inner[0] instanceof SymbolNode) {
+      base = inner[0];
+      base.classes[0] = "mop"; // replace old mclass
+    } else {
+      base = buildCommon.makeSpan(["mop"], buildCommon.tryCombineChars(inner), options);
+    }
+  } else {
+    // Otherwise, this is a text operator. Build the text from the
+    // operator's name.
+    // TODO(emily): Add a space in the middle of some of these
+    // operators, like \limsup
+    const output = [];
+
+    for (let i = 1; i < group.name.length; i++) {
+      output.push(buildCommon.mathsym(group.name[i], group.mode, options));
+    }
+
+    base = buildCommon.makeSpan(["mop"], output, options);
+  } // If content of op is a single symbol, shift it vertically.
+
+
+  let baseShift = 0;
+  let slant = 0;
+
+  if ((base instanceof SymbolNode || group.name === "\\oiint" || group.name === "\\oiiint") && !group.suppressBaseShift) {
+    // We suppress the shift of the base of \overset and \underset. Otherwise,
+    // shift the symbol so its center lies on the axis (rule 13). It
+    // appears that our fonts have the centers of the symbols already
+    // almost on the axis, so these numbers are very small. Note we
+    // don't actually apply this here, but instead it is used either in
+    // the vlist creation or separately when there are no limits.
+    baseShift = (base.height - base.depth) / 2 - options.fontMetrics().axisHeight; // The slant of the symbol is just its italic correction.
+    // $FlowFixMe
+
+    slant = base.italic;
+  }
+
+  if (hasLimits) {
+    return assembleSupSub(base, supGroup, subGroup, options, style, slant, baseShift);
+  } else {
+    if (baseShift) {
+      base.style.position = "relative";
+      base.style.top = baseShift + "em";
+    }
+
+    return base;
+  }
+};
+
+const mathmlBuilder$8 = (group, options) => {
+  let node;
+
+  if (group.symbol) {
+    // This is a symbol. Just add the symbol.
+    node = new MathNode("mo", [makeText(group.name, group.mode)]);
+
+    if (utils.contains(noSuccessor, group.name)) {
+      node.setAttribute("largeop", "false");
+    }
+  } else if (group.body) {
+    // This is an operator with children. Add them.
+    node = new MathNode("mo", buildExpression$1(group.body, options));
+  } else {
+    // This is a text operator. Add all of the characters from the
+    // operator's name.
+    node = new MathNode("mi", [new TextNode(group.name.slice(1))]); // Append an <mo>&ApplyFunction;</mo>.
+    // ref: https://www.w3.org/TR/REC-MathML/chap3_2.html#sec3.2.4
+
+    const operator = new MathNode("mo", [makeText("\u2061", "text")]);
+
+    if (group.parentIsSupSub) {
+      node = new MathNode("mo", [node, operator]);
+    } else {
+      node = newDocumentFragment([node, operator]);
+    }
+  }
+
+  return node;
+};
+
+const singleCharBigOps = {
+  "\u220F": "\\prod",
+  "\u2210": "\\coprod",
+  "\u2211": "\\sum",
+  "\u22c0": "\\bigwedge",
+  "\u22c1": "\\bigvee",
+  "\u22c2": "\\bigcap",
+  "\u22c3": "\\bigcup",
+  "\u2a00": "\\bigodot",
+  "\u2a01": "\\bigoplus",
+  "\u2a02": "\\bigotimes",
+  "\u2a04": "\\biguplus",
+  "\u2a06": "\\bigsqcup"
+};
+defineFunction({
+  type: "op",
+  names: ["\\coprod", "\\bigvee", "\\bigwedge", "\\biguplus", "\\bigcap", "\\bigcup", "\\intop", "\\prod", "\\sum", "\\bigotimes", "\\bigoplus", "\\bigodot", "\\bigsqcup", "\\smallint", "\u220F", "\u2210", "\u2211", "\u22c0", "\u22c1", "\u22c2", "\u22c3", "\u2a00", "\u2a01", "\u2a02", "\u2a04", "\u2a06"],
+  props: {
+    numArgs: 0
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    let fName = funcName;
+
+    if (fName.length === 1) {
+      fName = singleCharBigOps[fName];
+    }
+
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: true,
+      parentIsSupSub: false,
+      symbol: true,
+      name: fName
+    };
+  },
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+}); // Note: calling defineFunction with a type that's already been defined only
+// works because the same htmlBuilder and mathmlBuilder are being used.
+
+defineFunction({
+  type: "op",
+  names: ["\\mathop"],
+  props: {
+    numArgs: 1
+  },
+  handler: (_ref2, args) => {
+    let parser = _ref2.parser;
+    const body = args[0];
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: false,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+}); // There are 2 flags for operators; whether they produce limits in
+// displaystyle, and whether they are symbols and should grow in
+// displaystyle. These four groups cover the four possible choices.
+
+const singleCharIntegrals = {
+  "\u222b": "\\int",
+  "\u222c": "\\iint",
+  "\u222d": "\\iiint",
+  "\u222e": "\\oint",
+  "\u222f": "\\oiint",
+  "\u2230": "\\oiiint"
+}; // No limits, not symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\arcsin", "\\arccos", "\\arctan", "\\arctg", "\\arcctg", "\\arg", "\\ch", "\\cos", "\\cosec", "\\cosh", "\\cot", "\\cotg", "\\coth", "\\csc", "\\ctg", "\\cth", "\\deg", "\\dim", "\\exp", "\\hom", "\\ker", "\\lg", "\\ln", "\\log", "\\sec", "\\sin", "\\sinh", "\\sh", "\\tan", "\\tanh", "\\tg", "\\th"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(_ref3) {
+    let parser = _ref3.parser,
+        funcName = _ref3.funcName;
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: false,
+      name: funcName
+    };
+  },
+
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+}); // Limits, not symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\det", "\\gcd", "\\inf", "\\lim", "\\max", "\\min", "\\Pr", "\\sup"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(_ref4) {
+    let parser = _ref4.parser,
+        funcName = _ref4.funcName;
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: true,
+      parentIsSupSub: false,
+      symbol: false,
+      name: funcName
+    };
+  },
+
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+}); // No limits, symbols
+
+defineFunction({
+  type: "op",
+  names: ["\\int", "\\iint", "\\iiint", "\\oint", "\\oiint", "\\oiiint", "\u222b", "\u222c", "\u222d", "\u222e", "\u222f", "\u2230"],
+  props: {
+    numArgs: 0
+  },
+
+  handler(_ref5) {
+    let parser = _ref5.parser,
+        funcName = _ref5.funcName;
+    let fName = funcName;
+
+    if (fName.length === 1) {
+      fName = singleCharIntegrals[fName];
+    }
+
+    return {
+      type: "op",
+      mode: parser.mode,
+      limits: false,
+      parentIsSupSub: false,
+      symbol: true,
+      name: fName
+    };
+  },
+
+  htmlBuilder: htmlBuilder$8,
+  mathmlBuilder: mathmlBuilder$8
+});
+
+// NOTE: Unlike most `htmlBuilder`s, this one handles not only
+// "operatorname", but also  "supsub" since \operatorname* can
+const htmlBuilder$9 = (grp, options) => {
+  // Operators are handled in the TeXbook pg. 443-444, rule 13(a).
+  let supGroup;
+  let subGroup;
+  let hasLimits = false;
+  let group;
+
+  if (grp.type === "supsub") {
+    // If we have limits, supsub will pass us its group to handle. Pull
+    // out the superscript and subscript and set the group to the op in
+    // its base.
+    supGroup = grp.sup;
+    subGroup = grp.sub;
+    group = assertNodeType(grp.base, "operatorname");
+    hasLimits = true;
+  } else {
+    group = assertNodeType(grp, "operatorname");
+  }
+
+  let base;
+
+  if (group.body.length > 0) {
+    const body = group.body.map(child => {
+      // $FlowFixMe: Check if the node has a string `text` property.
+      const childText = child.text;
+
+      if (typeof childText === "string") {
+        return {
+          type: "textord",
+          mode: child.mode,
+          text: childText
+        };
+      } else {
+        return child;
+      }
+    }); // Consolidate function names into symbol characters.
+
+    const expression = buildExpression(body, options.withFont("mathrm"), true);
+
+    for (let i = 0; i < expression.length; i++) {
+      const child = expression[i];
+
+      if (child instanceof SymbolNode) {
+        // Per amsopn package,
+        // change minus to hyphen and \ast to asterisk
+        child.text = child.text.replace(/\u2212/, "-").replace(/\u2217/, "*");
+      }
+    }
+
+    base = buildCommon.makeSpan(["mop"], expression, options);
+  } else {
+    base = buildCommon.makeSpan(["mop"], [], options);
+  }
+
+  if (hasLimits) {
+    return assembleSupSub(base, supGroup, subGroup, options, options.style, 0, 0);
+  } else {
+    return base;
+  }
+};
+
+const mathmlBuilder$9 = (group, options) => {
+  // The steps taken here are similar to the html version.
+  let expression = buildExpression$1(group.body, options.withFont("mathrm")); // Is expression a string or has it something like a fraction?
+
+  let isAllString = true; // default
+
+  for (let i = 0; i < expression.length; i++) {
+    const node = expression[i];
+
+    if (node instanceof mathMLTree.SpaceNode) ; else if (node instanceof mathMLTree.MathNode) {
+      switch (node.type) {
+        case "mi":
+        case "mn":
+        case "ms":
+        case "mspace":
+        case "mtext":
+          break;
+        // Do nothing yet.
+
+        case "mo":
+          {
+            const child = node.children[0];
+
+            if (node.children.length === 1 && child instanceof mathMLTree.TextNode) {
+              child.text = child.text.replace(/\u2212/, "-").replace(/\u2217/, "*");
+            } else {
+              isAllString = false;
+            }
+
+            break;
+          }
+
+        default:
+          isAllString = false;
+      }
+    } else {
+      isAllString = false;
+    }
+  }
+
+  if (isAllString) {
+    // Write a single TextNode instead of multiple nested tags.
+    const word = expression.map(node => node.toText()).join("");
+    expression = [new mathMLTree.TextNode(word)];
+  }
+
+  const identifier = new mathMLTree.MathNode("mi", expression);
+  identifier.setAttribute("mathvariant", "normal"); // \u2061 is the same as &ApplyFunction;
+  // ref: https://www.w3schools.com/charsets/ref_html_entities_a.asp
+
+  const operator = new mathMLTree.MathNode("mo", [makeText("\u2061", "text")]);
+
+  if (group.parentIsSupSub) {
+    return new mathMLTree.MathNode("mo", [identifier, operator]);
+  } else {
+    return mathMLTree.newDocumentFragment([identifier, operator]);
+  }
+}; // \operatorname
+// amsopn.dtx: \mathop{#1\kern\z@\operator@font#3}\newmcodes@
+
+
+defineFunction({
+  type: "operatorname",
+  names: ["\\operatorname", "\\operatorname*"],
+  props: {
+    numArgs: 1
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    return {
+      type: "operatorname",
+      mode: parser.mode,
+      body: ordargument(body),
+      alwaysHandleSupSub: funcName === "\\operatorname*",
+      limits: false,
+      parentIsSupSub: false
+    };
+  },
+  htmlBuilder: htmlBuilder$9,
+  mathmlBuilder: mathmlBuilder$9
+});
+
+defineFunctionBuilders({
+  type: "ordgroup",
+
+  htmlBuilder(group, options) {
+    if (group.semisimple) {
+      return buildCommon.makeFragment(buildExpression(group.body, options, false));
+    }
+
+    return buildCommon.makeSpan(["mord"], buildExpression(group.body, options, true), options);
+  },
+
+  mathmlBuilder(group, options) {
+    return buildExpressionRow(group.body, options, true);
+  }
+
+});
+
+defineFunction({
+  type: "overline",
+  names: ["\\overline"],
+  props: {
+    numArgs: 1
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    const body = args[0];
+    return {
+      type: "overline",
+      mode: parser.mode,
+      body
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Overlines are handled in the TeXbook pg 443, Rule 9.
+    // Build the inner group in the cramped style.
+    const innerGroup = buildGroup(group.body, options.havingCrampedStyle()); // Create the line above the body
+
+    const line = buildCommon.makeLineSpan("overline-line", options); // Generate the vlist, with the appropriate kerns
+
+    const defaultRuleThickness = options.fontMetrics().defaultRuleThickness;
+    const vlist = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: innerGroup
+      }, {
+        type: "kern",
+        size: 3 * defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: line
+      }, {
+        type: "kern",
+        size: defaultRuleThickness
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "overline"], [vlist], options);
+  },
+
+  mathmlBuilder(group, options) {
+    const operator = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode("\u203e")]);
+    operator.setAttribute("stretchy", "true");
+    const node = new mathMLTree.MathNode("mover", [buildGroup$1(group.body, options), operator]);
+    node.setAttribute("accent", "true");
+    return node;
+  }
+
+});
+
+defineFunction({
+  type: "phantom",
+  names: ["\\phantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let parser = _ref.parser;
+    const body = args[0];
+    return {
+      type: "phantom",
+      mode: parser.mode,
+      body: ordargument(body)
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const elements = buildExpression(group.body, options.withPhantom(), false); // \phantom isn't supposed to affect the elements it contains.
+    // See "color" for more details.
+
+    return buildCommon.makeFragment(elements);
+  },
+  mathmlBuilder: (group, options) => {
+    const inner = buildExpression$1(group.body, options);
+    return new mathMLTree.MathNode("mphantom", inner);
+  }
+});
+defineFunction({
+  type: "hphantom",
+  names: ["\\hphantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref2, args) => {
+    let parser = _ref2.parser;
+    const body = args[0];
+    return {
+      type: "hphantom",
+      mode: parser.mode,
+      body
+    };
+  },
+  htmlBuilder: (group, options) => {
+    let node = buildCommon.makeSpan([], [buildGroup(group.body, options.withPhantom())]);
+    node.height = 0;
+    node.depth = 0;
+
+    if (node.children) {
+      for (let i = 0; i < node.children.length; i++) {
+        node.children[i].height = 0;
+        node.children[i].depth = 0;
+      }
+    } // See smash for comment re: use of makeVList
+
+
+    node = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: node
+      }]
+    }, options); // For spacing, TeX treats \smash as a math group (same spacing as ord).
+
+    return buildCommon.makeSpan(["mord"], [node], options);
+  },
+  mathmlBuilder: (group, options) => {
+    const inner = buildExpression$1(ordargument(group.body), options);
+    const phantom = new mathMLTree.MathNode("mphantom", inner);
+    const node = new mathMLTree.MathNode("mpadded", [phantom]);
+    node.setAttribute("height", "0px");
+    node.setAttribute("depth", "0px");
+    return node;
+  }
+});
+defineFunction({
+  type: "vphantom",
+  names: ["\\vphantom"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref3, args) => {
+    let parser = _ref3.parser;
+    const body = args[0];
+    return {
+      type: "vphantom",
+      mode: parser.mode,
+      body
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const inner = buildCommon.makeSpan(["inner"], [buildGroup(group.body, options.withPhantom())]);
+    const fix = buildCommon.makeSpan(["fix"], []);
+    return buildCommon.makeSpan(["mord", "rlap"], [inner, fix], options);
+  },
+  mathmlBuilder: (group, options) => {
+    const inner = buildExpression$1(ordargument(group.body), options);
+    const phantom = new mathMLTree.MathNode("mphantom", inner);
+    const node = new mathMLTree.MathNode("mpadded", [phantom]);
+    node.setAttribute("width", "0px");
+    return node;
+  }
+});
+
+defineFunction({
+  type: "raisebox",
+  names: ["\\raisebox"],
+  props: {
+    numArgs: 2,
+    argTypes: ["size", "hbox"],
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    const amount = assertNodeType(args[0], "size").value;
+    const body = args[1];
+    return {
+      type: "raisebox",
+      mode: parser.mode,
+      dy: amount,
+      body
+    };
+  },
+
+  htmlBuilder(group, options) {
+    const body = buildGroup(group.body, options);
+    const dy = calculateSize(group.dy, options);
+    return buildCommon.makeVList({
+      positionType: "shift",
+      positionData: -dy,
+      children: [{
+        type: "elem",
+        elem: body
+      }]
+    }, options);
+  },
+
+  mathmlBuilder(group, options) {
+    const node = new mathMLTree.MathNode("mpadded", [buildGroup$1(group.body, options)]);
+    const dy = group.dy.number + group.dy.unit;
+    node.setAttribute("voffset", dy);
+    return node;
+  }
+
+});
+
+defineFunction({
+  type: "rule",
+  names: ["\\rule"],
+  props: {
+    numArgs: 2,
+    numOptionalArgs: 1,
+    argTypes: ["size", "size", "size"]
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser;
+    const shift = optArgs[0];
+    const width = assertNodeType(args[0], "size");
+    const height = assertNodeType(args[1], "size");
+    return {
+      type: "rule",
+      mode: parser.mode,
+      shift: shift && assertNodeType(shift, "size").value,
+      width: width.value,
+      height: height.value
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Make an empty span for the rule
+    const rule = buildCommon.makeSpan(["mord", "rule"], [], options); // Calculate the shift, width, and height of the rule, and account for units
+
+    const width = calculateSize(group.width, options);
+    const height = calculateSize(group.height, options);
+    const shift = group.shift ? calculateSize(group.shift, options) : 0; // Style the rule to the right size
+
+    rule.style.borderRightWidth = width + "em";
+    rule.style.borderTopWidth = height + "em";
+    rule.style.bottom = shift + "em"; // Record the height and width
+
+    rule.width = width;
+    rule.height = height + shift;
+    rule.depth = -shift; // Font size is the number large enough that the browser will
+    // reserve at least `absHeight` space above the baseline.
+    // The 1.125 factor was empirically determined
+
+    rule.maxFontSize = height * 1.125 * options.sizeMultiplier;
+    return rule;
+  },
+
+  mathmlBuilder(group, options) {
+    const width = calculateSize(group.width, options);
+    const height = calculateSize(group.height, options);
+    const shift = group.shift ? calculateSize(group.shift, options) : 0;
+    const color = options.color && options.getColor() || "black";
+    const rule = new mathMLTree.MathNode("mspace");
+    rule.setAttribute("mathbackground", color);
+    rule.setAttribute("width", width + "em");
+    rule.setAttribute("height", height + "em");
+    const wrapper = new mathMLTree.MathNode("mpadded", [rule]);
+
+    if (shift >= 0) {
+      wrapper.setAttribute("height", "+" + shift + "em");
+    } else {
+      wrapper.setAttribute("height", shift + "em");
+      wrapper.setAttribute("depth", "+" + -shift + "em");
+    }
+
+    wrapper.setAttribute("voffset", shift + "em");
+    return wrapper;
+  }
+
+});
+
+function sizingGroup(value, options, baseOptions) {
+  const inner = buildExpression(value, options, false);
+  const multiplier = options.sizeMultiplier / baseOptions.sizeMultiplier; // Add size-resetting classes to the inner list and set maxFontSize
+  // manually. Handle nested size changes.
+
+  for (let i = 0; i < inner.length; i++) {
+    const pos = inner[i].classes.indexOf("sizing");
+
+    if (pos < 0) {
+      Array.prototype.push.apply(inner[i].classes, options.sizingClasses(baseOptions));
+    } else if (inner[i].classes[pos + 1] === "reset-size" + options.size) {
+      // This is a nested size change: e.g., inner[i] is the "b" in
+      // `\Huge a \small b`. Override the old size (the `reset-` class)
+      // but not the new size.
+      inner[i].classes[pos + 1] = "reset-size" + baseOptions.size;
+    }
+
+    inner[i].height *= multiplier;
+    inner[i].depth *= multiplier;
+  }
+
+  return buildCommon.makeFragment(inner);
+}
+const sizeFuncs = ["\\tiny", "\\sixptsize", "\\scriptsize", "\\footnotesize", "\\small", "\\normalsize", "\\large", "\\Large", "\\LARGE", "\\huge", "\\Huge"];
+const htmlBuilder$a = (group, options) => {
+  // Handle sizing operators like \Huge. Real TeX doesn't actually allow
+  // these functions inside of math expressions, so we do some special
+  // handling.
+  const newOptions = options.havingSize(group.size);
+  return sizingGroup(group.body, newOptions, options);
+};
+defineFunction({
+  type: "sizing",
+  names: sizeFuncs,
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+  handler: (_ref, args) => {
+    let breakOnTokenText = _ref.breakOnTokenText,
+        funcName = _ref.funcName,
+        parser = _ref.parser;
+    const body = parser.parseExpression(false, breakOnTokenText);
+    return {
+      type: "sizing",
+      mode: parser.mode,
+      // Figure out what size to use based on the list of functions above
+      size: sizeFuncs.indexOf(funcName) + 1,
+      body
+    };
+  },
+  htmlBuilder: htmlBuilder$a,
+  mathmlBuilder: (group, options) => {
+    const newOptions = options.havingSize(group.size);
+    const inner = buildExpression$1(group.body, newOptions);
+    const node = new mathMLTree.MathNode("mstyle", inner); // TODO(emily): This doesn't produce the correct size for nested size
+    // changes, because we don't keep state of what style we're currently
+    // in, so we can't reset the size to normal before changing it.  Now
+    // that we're passing an options parameter we should be able to fix
+    // this.
+
+    node.setAttribute("mathsize", newOptions.sizeMultiplier + "em");
+    return node;
+  }
+});
+
+// smash, with optional [tb], as in AMS
+defineFunction({
+  type: "smash",
+  names: ["\\smash"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1,
+    allowedInText: true
+  },
+  handler: (_ref, args, optArgs) => {
+    let parser = _ref.parser;
+    let smashHeight = false;
+    let smashDepth = false;
+    const tbArg = optArgs[0] && assertNodeType(optArgs[0], "ordgroup");
+
+    if (tbArg) {
+      // Optional [tb] argument is engaged.
+      // ref: amsmath: \renewcommand{\smash}[1][tb]{%
+      //               def\mb@t{\ht}\def\mb@b{\dp}\def\mb@tb{\ht\z@\z@\dp}%
+      let letter = "";
+
+      for (let i = 0; i < tbArg.body.length; ++i) {
+        const node = tbArg.body[i]; // $FlowFixMe: Not every node type has a `text` property.
+
+        letter = node.text;
+
+        if (letter === "t") {
+          smashHeight = true;
+        } else if (letter === "b") {
+          smashDepth = true;
+        } else {
+          smashHeight = false;
+          smashDepth = false;
+          break;
+        }
+      }
+    } else {
+      smashHeight = true;
+      smashDepth = true;
+    }
+
+    const body = args[0];
+    return {
+      type: "smash",
+      mode: parser.mode,
+      body,
+      smashHeight,
+      smashDepth
+    };
+  },
+  htmlBuilder: (group, options) => {
+    const node = buildCommon.makeSpan([], [buildGroup(group.body, options)]);
+
+    if (!group.smashHeight && !group.smashDepth) {
+      return node;
+    }
+
+    if (group.smashHeight) {
+      node.height = 0; // In order to influence makeVList, we have to reset the children.
+
+      if (node.children) {
+        for (let i = 0; i < node.children.length; i++) {
+          node.children[i].height = 0;
+        }
+      }
+    }
+
+    if (group.smashDepth) {
+      node.depth = 0;
+
+      if (node.children) {
+        for (let i = 0; i < node.children.length; i++) {
+          node.children[i].depth = 0;
+        }
+      }
+    } // At this point, we've reset the TeX-like height and depth values.
+    // But the span still has an HTML line height.
+    // makeVList applies "display: table-cell", which prevents the browser
+    // from acting on that line height. So we'll call makeVList now.
+
+
+    const smashedNode = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: node
+      }]
+    }, options); // For spacing, TeX treats \hphantom as a math group (same spacing as ord).
+
+    return buildCommon.makeSpan(["mord"], [smashedNode], options);
+  },
+  mathmlBuilder: (group, options) => {
+    const node = new mathMLTree.MathNode("mpadded", [buildGroup$1(group.body, options)]);
+
+    if (group.smashHeight) {
+      node.setAttribute("height", "0px");
+    }
+
+    if (group.smashDepth) {
+      node.setAttribute("depth", "0px");
+    }
+
+    return node;
+  }
+});
+
+defineFunction({
+  type: "sqrt",
+  names: ["\\sqrt"],
+  props: {
+    numArgs: 1,
+    numOptionalArgs: 1
+  },
+
+  handler(_ref, args, optArgs) {
+    let parser = _ref.parser;
+    const index = optArgs[0];
+    const body = args[0];
+    return {
+      type: "sqrt",
+      mode: parser.mode,
+      body,
+      index
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Square roots are handled in the TeXbook pg. 443, Rule 11.
+    // First, we do the same steps as in overline to build the inner group
+    // and line
+    let inner = buildGroup(group.body, options.havingCrampedStyle());
+
+    if (inner.height === 0) {
+      // Render a small surd.
+      inner.height = options.fontMetrics().xHeight;
+    } // Some groups can return document fragments.  Handle those by wrapping
+    // them in a span.
+
+
+    inner = buildCommon.wrapFragment(inner, options); // Calculate the minimum size for the \surd delimiter
+
+    const metrics = options.fontMetrics();
+    const theta = metrics.defaultRuleThickness;
+    let phi = theta;
+
+    if (options.style.id < Style$1.TEXT.id) {
+      phi = options.fontMetrics().xHeight;
+    } // Calculate the clearance between the body and line
+
+
+    let lineClearance = theta + phi / 4;
+    const minDelimiterHeight = inner.height + inner.depth + lineClearance + theta; // Create a sqrt SVG of the required minimum size
+
+    const _delimiter$sqrtImage = delimiter.sqrtImage(minDelimiterHeight, options),
+          img = _delimiter$sqrtImage.span,
+          ruleWidth = _delimiter$sqrtImage.ruleWidth,
+          advanceWidth = _delimiter$sqrtImage.advanceWidth;
+
+    const delimDepth = img.height - ruleWidth; // Adjust the clearance based on the delimiter size
+
+    if (delimDepth > inner.height + inner.depth + lineClearance) {
+      lineClearance = (lineClearance + delimDepth - inner.height - inner.depth) / 2;
+    } // Shift the sqrt image
+
+
+    const imgShift = img.height - inner.height - lineClearance - ruleWidth;
+    inner.style.paddingLeft = advanceWidth + "em"; // Overlay the image and the argument.
+
+    const body = buildCommon.makeVList({
+      positionType: "firstBaseline",
+      children: [{
+        type: "elem",
+        elem: inner,
+        wrapperClasses: ["svg-align"]
+      }, {
+        type: "kern",
+        size: -(inner.height + imgShift)
+      }, {
+        type: "elem",
+        elem: img
+      }, {
+        type: "kern",
+        size: ruleWidth
+      }]
+    }, options);
+
+    if (!group.index) {
+      return buildCommon.makeSpan(["mord", "sqrt"], [body], options);
+    } else {
+      // Handle the optional root index
+      // The index is always in scriptscript style
+      const newOptions = options.havingStyle(Style$1.SCRIPTSCRIPT);
+      const rootm = buildGroup(group.index, newOptions, options); // The amount the index is shifted by. This is taken from the TeX
+      // source, in the definition of `\r@@t`.
+
+      const toShift = 0.6 * (body.height - body.depth); // Build a VList with the superscript shifted up correctly
+
+      const rootVList = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: -toShift,
+        children: [{
+          type: "elem",
+          elem: rootm
+        }]
+      }, options); // Add a class surrounding it so we can add on the appropriate
+      // kerning
+
+      const rootVListWrap = buildCommon.makeSpan(["root"], [rootVList]);
+      return buildCommon.makeSpan(["mord", "sqrt"], [rootVListWrap, body], options);
+    }
+  },
+
+  mathmlBuilder(group, options) {
+    const body = group.body,
+          index = group.index;
+    return index ? new mathMLTree.MathNode("mroot", [buildGroup$1(body, options), buildGroup$1(index, options)]) : new mathMLTree.MathNode("msqrt", [buildGroup$1(body, options)]);
+  }
+
+});
+
+const styleMap$1 = {
+  "display": Style$1.DISPLAY,
+  "text": Style$1.TEXT,
+  "script": Style$1.SCRIPT,
+  "scriptscript": Style$1.SCRIPTSCRIPT
+};
+defineFunction({
+  type: "styling",
+  names: ["\\displaystyle", "\\textstyle", "\\scriptstyle", "\\scriptscriptstyle"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let breakOnTokenText = _ref.breakOnTokenText,
+        funcName = _ref.funcName,
+        parser = _ref.parser;
+    // parse out the implicit body
+    const body = parser.parseExpression(true, breakOnTokenText); // TODO: Refactor to avoid duplicating styleMap in multiple places (e.g.
+    // here and in buildHTML and de-dupe the enumeration of all the styles).
+    // $FlowFixMe: The names above exactly match the styles.
+
+    const style = funcName.slice(1, funcName.length - 5);
+    return {
+      type: "styling",
+      mode: parser.mode,
+      // Figure out what style to use by pulling out the style from
+      // the function name
+      style,
+      body
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Style changes are handled in the TeXbook on pg. 442, Rule 3.
+    const newStyle = styleMap$1[group.style];
+    const newOptions = options.havingStyle(newStyle).withFont('');
+    return sizingGroup(group.body, newOptions, options);
+  },
+
+  mathmlBuilder(group, options) {
+    // Figure out what style we're changing to.
+    const newStyle = styleMap$1[group.style];
+    const newOptions = options.havingStyle(newStyle);
+    const inner = buildExpression$1(group.body, newOptions);
+    const node = new mathMLTree.MathNode("mstyle", inner);
+    const styleAttributes = {
+      "display": ["0", "true"],
+      "text": ["0", "false"],
+      "script": ["1", "false"],
+      "scriptscript": ["2", "false"]
+    };
+    const attr = styleAttributes[group.style];
+    node.setAttribute("scriptlevel", attr[0]);
+    node.setAttribute("displaystyle", attr[1]);
+    return node;
+  }
+
+});
+
+/**
+ * Sometimes, groups perform special rules when they have superscripts or
+ * subscripts attached to them. This function lets the `supsub` group know that
+ * Sometimes, groups perform special rules when they have superscripts or
+ * its inner element should handle the superscripts and subscripts instead of
+ * handling them itself.
+ */
+const htmlBuilderDelegate = function htmlBuilderDelegate(group, options) {
+  const base = group.base;
+
+  if (!base) {
+    return null;
+  } else if (base.type === "op") {
+    // Operators handle supsubs differently when they have limits
+    // (e.g. `\displaystyle\sum_2^3`)
+    const delegate = base.limits && (options.style.size === Style$1.DISPLAY.size || base.alwaysHandleSupSub);
+    return delegate ? htmlBuilder$8 : null;
+  } else if (base.type === "operatorname") {
+    const delegate = base.alwaysHandleSupSub && (options.style.size === Style$1.DISPLAY.size || base.limits);
+    return delegate ? htmlBuilder$9 : null;
+  } else if (base.type === "accent") {
+    return utils.isCharacterBox(base.base) ? htmlBuilder : null;
+  } else if (base.type === "horizBrace") {
+    const isSup = !group.sub;
+    return isSup === base.isOver ? htmlBuilder$7 : null;
+  } else {
+    return null;
+  }
+}; // Super scripts and subscripts, whose precise placement can depend on other
+// functions that precede them.
+
+
+defineFunctionBuilders({
+  type: "supsub",
+
+  htmlBuilder(group, options) {
+    // Superscript and subscripts are handled in the TeXbook on page
+    // 445-446, rules 18(a-f).
+    // Here is where we defer to the inner group if it should handle
+    // superscripts and subscripts itself.
+    const builderDelegate = htmlBuilderDelegate(group, options);
+
+    if (builderDelegate) {
+      return builderDelegate(group, options);
+    }
+
+    const valueBase = group.base,
+          valueSup = group.sup,
+          valueSub = group.sub;
+    const base = buildGroup(valueBase, options);
+    let supm;
+    let subm;
+    const metrics = options.fontMetrics(); // Rule 18a
+
+    let supShift = 0;
+    let subShift = 0;
+    const isCharacterBox = valueBase && utils.isCharacterBox(valueBase);
+
+    if (valueSup) {
+      const newOptions = options.havingStyle(options.style.sup());
+      supm = buildGroup(valueSup, newOptions, options);
+
+      if (!isCharacterBox) {
+        supShift = base.height - newOptions.fontMetrics().supDrop * newOptions.sizeMultiplier / options.sizeMultiplier;
+      }
+    }
+
+    if (valueSub) {
+      const newOptions = options.havingStyle(options.style.sub());
+      subm = buildGroup(valueSub, newOptions, options);
+
+      if (!isCharacterBox) {
+        subShift = base.depth + newOptions.fontMetrics().subDrop * newOptions.sizeMultiplier / options.sizeMultiplier;
+      }
+    } // Rule 18c
+
+
+    let minSupShift;
+
+    if (options.style === Style$1.DISPLAY) {
+      minSupShift = metrics.sup1;
+    } else if (options.style.cramped) {
+      minSupShift = metrics.sup3;
+    } else {
+      minSupShift = metrics.sup2;
+    } // scriptspace is a font-size-independent size, so scale it
+    // appropriately for use as the marginRight.
+
+
+    const multiplier = options.sizeMultiplier;
+    const marginRight = 0.5 / metrics.ptPerEm / multiplier + "em";
+    let marginLeft = null;
+
+    if (subm) {
+      // Subscripts shouldn't be shifted by the base's italic correction.
+      // Account for that by shifting the subscript back the appropriate
+      // amount. Note we only do this when the base is a single symbol.
+      const isOiint = group.base && group.base.type === "op" && group.base.name && (group.base.name === "\\oiint" || group.base.name === "\\oiiint");
+
+      if (base instanceof SymbolNode || isOiint) {
+        // $FlowFixMe
+        marginLeft = -base.italic + "em";
+      }
+    }
+
+    let supsub;
+
+    if (supm && subm) {
+      supShift = Math.max(supShift, minSupShift, supm.depth + 0.25 * metrics.xHeight);
+      subShift = Math.max(subShift, metrics.sub2);
+      const ruleWidth = metrics.defaultRuleThickness; // Rule 18e
+
+      const maxWidth = 4 * ruleWidth;
+
+      if (supShift - supm.depth - (subm.height - subShift) < maxWidth) {
+        subShift = maxWidth - (supShift - supm.depth) + subm.height;
+        const psi = 0.8 * metrics.xHeight - (supShift - supm.depth);
+
+        if (psi > 0) {
+          supShift += psi;
+          subShift -= psi;
+        }
+      }
+
+      const vlistElem = [{
+        type: "elem",
+        elem: subm,
+        shift: subShift,
+        marginRight,
+        marginLeft
+      }, {
+        type: "elem",
+        elem: supm,
+        shift: -supShift,
+        marginRight
+      }];
+      supsub = buildCommon.makeVList({
+        positionType: "individualShift",
+        children: vlistElem
+      }, options);
+    } else if (subm) {
+      // Rule 18b
+      subShift = Math.max(subShift, metrics.sub1, subm.height - 0.8 * metrics.xHeight);
+      const vlistElem = [{
+        type: "elem",
+        elem: subm,
+        marginLeft,
+        marginRight
+      }];
+      supsub = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: subShift,
+        children: vlistElem
+      }, options);
+    } else if (supm) {
+      // Rule 18c, d
+      supShift = Math.max(supShift, minSupShift, supm.depth + 0.25 * metrics.xHeight);
+      supsub = buildCommon.makeVList({
+        positionType: "shift",
+        positionData: -supShift,
+        children: [{
+          type: "elem",
+          elem: supm,
+          marginRight
+        }]
+      }, options);
+    } else {
+      throw new Error("supsub must have either sup or sub.");
+    } // Wrap the supsub vlist in a span.msupsub to reset text-align.
+
+
+    const mclass = getTypeOfDomTree(base, "right") || "mord";
+    return buildCommon.makeSpan([mclass], [base, buildCommon.makeSpan(["msupsub"], [supsub])], options);
+  },
+
+  mathmlBuilder(group, options) {
+    // Is the inner group a relevant horizonal brace?
+    let isBrace = false;
+    let isOver;
+    let isSup;
+
+    if (group.base && group.base.type === "horizBrace") {
+      isSup = !!group.sup;
+
+      if (isSup === group.base.isOver) {
+        isBrace = true;
+        isOver = group.base.isOver;
+      }
+    }
+
+    if (group.base && (group.base.type === "op" || group.base.type === "operatorname")) {
+      group.base.parentIsSupSub = true;
+    }
+
+    const children = [buildGroup$1(group.base, options)];
+
+    if (group.sub) {
+      children.push(buildGroup$1(group.sub, options));
+    }
+
+    if (group.sup) {
+      children.push(buildGroup$1(group.sup, options));
+    }
+
+    let nodeType;
+
+    if (isBrace) {
+      nodeType = isOver ? "mover" : "munder";
+    } else if (!group.sub) {
+      const base = group.base;
+
+      if (base && base.type === "op" && base.limits && (options.style === Style$1.DISPLAY || base.alwaysHandleSupSub)) {
+        nodeType = "mover";
+      } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub && (base.limits || options.style === Style$1.DISPLAY)) {
+        nodeType = "mover";
+      } else {
+        nodeType = "msup";
+      }
+    } else if (!group.sup) {
+      const base = group.base;
+
+      if (base && base.type === "op" && base.limits && (options.style === Style$1.DISPLAY || base.alwaysHandleSupSub)) {
+        nodeType = "munder";
+      } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub && (base.limits || options.style === Style$1.DISPLAY)) {
+        nodeType = "munder";
+      } else {
+        nodeType = "msub";
+      }
+    } else {
+      const base = group.base;
+
+      if (base && base.type === "op" && base.limits && options.style === Style$1.DISPLAY) {
+        nodeType = "munderover";
+      } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub && (options.style === Style$1.DISPLAY || base.limits)) {
+        nodeType = "munderover";
+      } else {
+        nodeType = "msubsup";
+      }
+    }
+
+    const node = new mathMLTree.MathNode(nodeType, children);
+    return node;
+  }
+
+});
+
+defineFunctionBuilders({
+  type: "atom",
+
+  htmlBuilder(group, options) {
+    return buildCommon.mathsym(group.text, group.mode, options, ["m" + group.family]);
+  },
+
+  mathmlBuilder(group, options) {
+    const node = new mathMLTree.MathNode("mo", [makeText(group.text, group.mode)]);
+
+    if (group.family === "bin") {
+      const variant = getVariant(group, options);
+
+      if (variant === "bold-italic") {
+        node.setAttribute("mathvariant", variant);
+      }
+    } else if (group.family === "punct") {
+      node.setAttribute("separator", "true");
+    } else if (group.family === "open" || group.family === "close") {
+      // Delims built here should not stretch vertically.
+      // See delimsizing.js for stretchy delims.
+      node.setAttribute("stretchy", "false");
+    }
+
+    return node;
+  }
+
+});
+
+// "mathord" and "textord" ParseNodes created in Parser.js from symbol Groups in
+const defaultVariant = {
+  "mi": "italic",
+  "mn": "normal",
+  "mtext": "normal"
+};
+defineFunctionBuilders({
+  type: "mathord",
+
+  htmlBuilder(group, options) {
+    return buildCommon.makeOrd(group, options, "mathord");
+  },
+
+  mathmlBuilder(group, options) {
+    const node = new mathMLTree.MathNode("mi", [makeText(group.text, group.mode, options)]);
+    const variant = getVariant(group, options) || "italic";
+
+    if (variant !== defaultVariant[node.type]) {
+      node.setAttribute("mathvariant", variant);
+    }
+
+    return node;
+  }
+
+});
+defineFunctionBuilders({
+  type: "textord",
+
+  htmlBuilder(group, options) {
+    return buildCommon.makeOrd(group, options, "textord");
+  },
+
+  mathmlBuilder(group, options) {
+    const text = makeText(group.text, group.mode, options);
+    const variant = getVariant(group, options) || "normal";
+    let node;
+
+    if (group.mode === 'text') {
+      node = new mathMLTree.MathNode("mtext", [text]);
+    } else if (/[0-9]/.test(group.text)) {
+      // TODO(kevinb) merge adjacent <mn> nodes
+      // do it as a post processing step
+      node = new mathMLTree.MathNode("mn", [text]);
+    } else if (group.text === "\\prime") {
+      node = new mathMLTree.MathNode("mo", [text]);
+    } else {
+      node = new mathMLTree.MathNode("mi", [text]);
+    }
+
+    if (variant !== defaultVariant[node.type]) {
+      node.setAttribute("mathvariant", variant);
+    }
+
+    return node;
+  }
+
+});
+
+const cssSpace = {
+  "\\nobreak": "nobreak",
+  "\\allowbreak": "allowbreak"
+}; // A lookup table to determine whether a spacing function/symbol should be
+// treated like a regular space character.  If a symbol or command is a key
+// in this table, then it should be a regular space character.  Furthermore,
+// the associated value may have a `className` specifying an extra CSS class
+// to add to the created `span`.
+
+const regularSpace = {
+  " ": {},
+  "\\ ": {},
+  "~": {
+    className: "nobreak"
+  },
+  "\\space": {},
+  "\\nobreakspace": {
+    className: "nobreak"
+  }
+}; // ParseNode<"spacing"> created in Parser.js from the "spacing" symbol Groups in
+// src/symbols.js.
+
+defineFunctionBuilders({
+  type: "spacing",
+
+  htmlBuilder(group, options) {
+    if (regularSpace.hasOwnProperty(group.text)) {
+      const className = regularSpace[group.text].className || ""; // Spaces are generated by adding an actual space. Each of these
+      // things has an entry in the symbols table, so these will be turned
+      // into appropriate outputs.
+
+      if (group.mode === "text") {
+        const ord = buildCommon.makeOrd(group, options, "textord");
+        ord.classes.push(className);
+        return ord;
+      } else {
+        return buildCommon.makeSpan(["mspace", className], [buildCommon.mathsym(group.text, group.mode, options)], options);
+      }
+    } else if (cssSpace.hasOwnProperty(group.text)) {
+      // Spaces based on just a CSS class.
+      return buildCommon.makeSpan(["mspace", cssSpace[group.text]], [], options);
+    } else {
+      throw new ParseError(`Unknown type of space "${group.text}"`);
+    }
+  },
+
+  mathmlBuilder(group, options) {
+    let node;
+
+    if (regularSpace.hasOwnProperty(group.text)) {
+      node = new mathMLTree.MathNode("mtext", [new mathMLTree.TextNode("\u00a0")]);
+    } else if (cssSpace.hasOwnProperty(group.text)) {
+      // CSS-based MathML spaces (\nobreak, \allowbreak) are ignored
+      return new mathMLTree.MathNode("mspace");
+    } else {
+      throw new ParseError(`Unknown type of space "${group.text}"`);
+    }
+
+    return node;
+  }
+
+});
+
+const pad = () => {
+  const padNode = new mathMLTree.MathNode("mtd", []);
+  padNode.setAttribute("width", "50%");
+  return padNode;
+};
+
+defineFunctionBuilders({
+  type: "tag",
+
+  mathmlBuilder(group, options) {
+    const table = new mathMLTree.MathNode("mtable", [new mathMLTree.MathNode("mtr", [pad(), new mathMLTree.MathNode("mtd", [buildExpressionRow(group.body, options)]), pad(), new mathMLTree.MathNode("mtd", [buildExpressionRow(group.tag, options)])])]);
+    table.setAttribute("width", "100%");
+    return table; // TODO: Left-aligned tags.
+    // Currently, the group and options passed here do not contain
+    // enough info to set tag alignment. `leqno` is in Settings but it is
+    // not passed to Options. On the HTML side, leqno is
+    // set by a CSS class applied in buildTree.js. That would have worked
+    // in MathML if browsers supported <mlabeledtr>. Since they don't, we
+    // need to rewrite the way this function is called.
+  }
+
+});
+
+const textFontFamilies = {
+  "\\text": undefined,
+  "\\textrm": "textrm",
+  "\\textsf": "textsf",
+  "\\texttt": "texttt",
+  "\\textnormal": "textrm"
+};
+const textFontWeights = {
+  "\\textbf": "textbf",
+  "\\textmd": "textmd"
+};
+const textFontShapes = {
+  "\\textit": "textit",
+  "\\textup": "textup"
+};
+
+const optionsWithFont = (group, options) => {
+  const font = group.font; // Checks if the argument is a font family or a font style.
+
+  if (!font) {
+    return options;
+  } else if (textFontFamilies[font]) {
+    return options.withTextFontFamily(textFontFamilies[font]);
+  } else if (textFontWeights[font]) {
+    return options.withTextFontWeight(textFontWeights[font]);
+  } else {
+    return options.withTextFontShape(textFontShapes[font]);
+  }
+};
+
+defineFunction({
+  type: "text",
+  names: [// Font families
+  "\\text", "\\textrm", "\\textsf", "\\texttt", "\\textnormal", // Font weights
+  "\\textbf", "\\textmd", // Font Shapes
+  "\\textit", "\\textup"],
+  props: {
+    numArgs: 1,
+    argTypes: ["text"],
+    greediness: 2,
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser,
+        funcName = _ref.funcName;
+    const body = args[0];
+    return {
+      type: "text",
+      mode: parser.mode,
+      body: ordargument(body),
+      font: funcName
+    };
+  },
+
+  htmlBuilder(group, options) {
+    const newOptions = optionsWithFont(group, options);
+    const inner = buildExpression(group.body, newOptions, true);
+    return buildCommon.makeSpan(["mord", "text"], buildCommon.tryCombineChars(inner), newOptions);
+  },
+
+  mathmlBuilder(group, options) {
+    const newOptions = optionsWithFont(group, options);
+    return buildExpressionRow(group.body, newOptions);
+  }
+
+});
+
+defineFunction({
+  type: "underline",
+  names: ["\\underline"],
+  props: {
+    numArgs: 1,
+    allowedInText: true
+  },
+
+  handler(_ref, args) {
+    let parser = _ref.parser;
+    return {
+      type: "underline",
+      mode: parser.mode,
+      body: args[0]
+    };
+  },
+
+  htmlBuilder(group, options) {
+    // Underlines are handled in the TeXbook pg 443, Rule 10.
+    // Build the inner group.
+    const innerGroup = buildGroup(group.body, options); // Create the line to go below the body
+
+    const line = buildCommon.makeLineSpan("underline-line", options); // Generate the vlist, with the appropriate kerns
+
+    const defaultRuleThickness = options.fontMetrics().defaultRuleThickness;
+    const vlist = buildCommon.makeVList({
+      positionType: "top",
+      positionData: innerGroup.height,
+      children: [{
+        type: "kern",
+        size: defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: line
+      }, {
+        type: "kern",
+        size: 3 * defaultRuleThickness
+      }, {
+        type: "elem",
+        elem: innerGroup
+      }]
+    }, options);
+    return buildCommon.makeSpan(["mord", "underline"], [vlist], options);
+  },
+
+  mathmlBuilder(group, options) {
+    const operator = new mathMLTree.MathNode("mo", [new mathMLTree.TextNode("\u203e")]);
+    operator.setAttribute("stretchy", "true");
+    const node = new mathMLTree.MathNode("munder", [buildGroup$1(group.body, options), operator]);
+    node.setAttribute("accentunder", "true");
+    return node;
+  }
+
+});
+
+defineFunction({
+  type: "verb",
+  names: ["\\verb"],
+  props: {
+    numArgs: 0,
+    allowedInText: true
+  },
+
+  handler(context, args, optArgs) {
+    // \verb and \verb* are dealt with directly in Parser.js.
+    // If we end up here, it's because of a failure to match the two delimiters
+    // in the regex in Lexer.js.  LaTeX raises the following error when \verb is
+    // terminated by end of line (or file).
+    throw new ParseError("\\verb ended by end of line instead of matching delimiter");
+  },
+
+  htmlBuilder(group, options) {
+    const text = makeVerb(group);
+    const body = []; // \verb enters text mode and therefore is sized like \textstyle
+
+    const newOptions = options.havingStyle(options.style.text());
+
+    for (let i = 0; i < text.length; i++) {
+      let c = text[i];
+
+      if (c === '~') {
+        c = '\\textasciitilde';
+      }
+
+      body.push(buildCommon.makeSymbol(c, "Typewriter-Regular", group.mode, newOptions, ["mord", "texttt"]));
+    }
+
+    return buildCommon.makeSpan(["mord", "text"].concat(newOptions.sizingClasses(options)), buildCommon.tryCombineChars(body), newOptions);
+  },
+
+  mathmlBuilder(group, options) {
+    const text = new mathMLTree.TextNode(makeVerb(group));
+    const node = new mathMLTree.MathNode("mtext", [text]);
+    node.setAttribute("mathvariant", "monospace");
+    return node;
+  }
+
+});
+/**
+ * Converts verb group into body string.
+ *
+ * \verb* replaces each space with an open box \u2423
+ * \verb replaces each space with a no-break space \xA0
+ */
+
+const makeVerb = group => group.body.replace(/ /g, group.star ? '\u2423' : '\xA0');
+
+/** Include this to ensure that all functions are defined. */
+const functions = _functions;
+
+/**
+ * The Lexer class handles tokenizing the input in various ways. Since our
+ * parser expects us to be able to backtrack, the lexer allows lexing from any
+ * given starting point.
+ *
+ * Its main exposed function is the `lex` function, which takes a position to
+ * lex from and a type of token to lex. It defers to the appropriate `_innerLex`
+ * function.
+ *
+ * The various `_innerLex` functions perform the actual lexing of different
+ * kinds.
+ */
+
+/* The following tokenRegex
+ * - matches typical whitespace (but not NBSP etc.) using its first group
+ * - does not match any control character \x00-\x1f except whitespace
+ * - does not match a bare backslash
+ * - matches any ASCII character except those just mentioned
+ * - does not match the BMP private use area \uE000-\uF8FF
+ * - does not match bare surrogate code units
+ * - matches any BMP character except for those just described
+ * - matches any valid Unicode surrogate pair
+ * - matches a backslash followed by one or more letters
+ * - matches a backslash followed by any BMP character, including newline
+ * Just because the Lexer matches something doesn't mean it's valid input:
+ * If there is no matching function or symbol definition, the Parser will
+ * still reject the input.
+ */
+const spaceRegexString = "[ \r\n\t]";
+const controlWordRegexString = "\\\\[a-zA-Z@]+";
+const controlSymbolRegexString = "\\\\[^\uD800-\uDFFF]";
+const controlWordWhitespaceRegexString = `${controlWordRegexString}${spaceRegexString}*`;
+const controlWordWhitespaceRegex = new RegExp(`^(${controlWordRegexString})${spaceRegexString}*$`);
+const combiningDiacriticalMarkString = "[\u0300-\u036f]";
+const combiningDiacriticalMarksEndRegex = new RegExp(`${combiningDiacriticalMarkString}+$`);
+const tokenRegexString = `(${spaceRegexString}+)|` + // whitespace
+"([!-\\[\\]-\u2027\u202A-\uD7FF\uF900-\uFFFF]" + // single codepoint
+`${combiningDiacriticalMarkString}*` + // ...plus accents
+"|[\uD800-\uDBFF][\uDC00-\uDFFF]" + // surrogate pair
+`${combiningDiacriticalMarkString}*` + // ...plus accents
+"|\\\\verb\\*([^]).*?\\3" + // \verb*
+"|\\\\verb([^*a-zA-Z]).*?\\4" + // \verb unstarred
+"|\\\\operatorname\\*" + // \operatorname*
+`|${controlWordWhitespaceRegexString}` + // \macroName + spaces
+`|${controlSymbolRegexString})`; // \\, \', etc.
+
+/** Main Lexer class */
+
+class Lexer {
+  // category codes, only supports comment characters (14) for now
+  constructor(input, settings) {
+    this.input = void 0;
+    this.settings = void 0;
+    this.tokenRegex = void 0;
+    this.catcodes = void 0;
+    // Separate accents from characters
+    this.input = input;
+    this.settings = settings;
+    this.tokenRegex = new RegExp(tokenRegexString, 'g');
+    this.catcodes = {
+      "%": 14 // comment character
+
+    };
+  }
+
+  setCatcode(char, code) {
+    this.catcodes[char] = code;
+  }
+  /**
+   * This function lexes a single token.
+   */
+
+
+  lex() {
+    const input = this.input;
+    const pos = this.tokenRegex.lastIndex;
+
+    if (pos === input.length) {
+      return new Token("EOF", new SourceLocation(this, pos, pos));
+    }
+
+    const match = this.tokenRegex.exec(input);
+
+    if (match === null || match.index !== pos) {
+      throw new ParseError(`Unexpected character: '${input[pos]}'`, new Token(input[pos], new SourceLocation(this, pos, pos + 1)));
+    }
+
+    let text = match[2] || " ";
+
+    if (this.catcodes[text] === 14) {
+      // comment character
+      const nlIndex = input.indexOf('\n', this.tokenRegex.lastIndex);
+
+      if (nlIndex === -1) {
+        this.tokenRegex.lastIndex = input.length; // EOF
+
+        this.settings.reportNonstrict("commentAtEnd", "% comment has no terminating newline; LaTeX would " + "fail because of commenting the end of math mode (e.g. $)");
+      } else {
+        this.tokenRegex.lastIndex = nlIndex + 1;
+      }
+
+      return this.lex();
+    } // Trim any trailing whitespace from control word match
+
+
+    const controlMatch = text.match(controlWordWhitespaceRegex);
+
+    if (controlMatch) {
+      text = controlMatch[1];
+    }
+
+    return new Token(text, new SourceLocation(this, pos, this.tokenRegex.lastIndex));
+  }
+
+}
+
+/**
+ * A `Namespace` refers to a space of nameable things like macros or lengths,
+ * which can be `set` either globally or local to a nested group, using an
+ * undo stack similar to how TeX implements this functionality.
+ * Performance-wise, `get` and local `set` take constant time, while global
+ * `set` takes time proportional to the depth of group nesting.
+ */
+class Namespace {
+  /**
+   * Both arguments are optional.  The first argument is an object of
+   * built-in mappings which never change.  The second argument is an object
+   * of initial (global-level) mappings, which will constantly change
+   * according to any global/top-level `set`s done.
+   */
+  constructor(builtins, globalMacros) {
+    if (builtins === void 0) {
+      builtins = {};
+    }
+
+    if (globalMacros === void 0) {
+      globalMacros = {};
+    }
+
+    this.current = void 0;
+    this.builtins = void 0;
+    this.undefStack = void 0;
+    this.current = globalMacros;
+    this.builtins = builtins;
+    this.undefStack = [];
+  }
+  /**
+   * Start a new nested group, affecting future local `set`s.
+   */
+
+
+  beginGroup() {
+    this.undefStack.push({});
+  }
+  /**
+   * End current nested group, restoring values before the group began.
+   */
+
+
+  endGroup() {
+    if (this.undefStack.length === 0) {
+      throw new ParseError("Unbalanced namespace destruction: attempt " + "to pop global namespace; please report this as a bug");
+    }
+
+    const undefs = this.undefStack.pop();
+
+    for (const undef in undefs) {
+      if (undefs.hasOwnProperty(undef)) {
+        if (undefs[undef] === undefined) {
+          delete this.current[undef];
+        } else {
+          this.current[undef] = undefs[undef];
+        }
+      }
+    }
+  }
+  /**
+   * Detect whether `name` has a definition.  Equivalent to
+   * `get(name) != null`.
+   */
+
+
+  has(name) {
+    return this.current.hasOwnProperty(name) || this.builtins.hasOwnProperty(name);
+  }
+  /**
+   * Get the current value of a name, or `undefined` if there is no value.
+   *
+   * Note: Do not use `if (namespace.get(...))` to detect whether a macro
+   * is defined, as the definition may be the empty string which evaluates
+   * to `false` in JavaScript.  Use `if (namespace.get(...) != null)` or
+   * `if (namespace.has(...))`.
+   */
+
+
+  get(name) {
+    if (this.current.hasOwnProperty(name)) {
+      return this.current[name];
+    } else {
+      return this.builtins[name];
+    }
+  }
+  /**
+   * Set the current value of a name, and optionally set it globally too.
+   * Local set() sets the current value and (when appropriate) adds an undo
+   * operation to the undo stack.  Global set() may change the undo
+   * operation at every level, so takes time linear in their number.
+   */
+
+
+  set(name, value, global) {
+    if (global === void 0) {
+      global = false;
+    }
+
+    if (global) {
+      // Global set is equivalent to setting in all groups.  Simulate this
+      // by destroying any undos currently scheduled for this name,
+      // and adding an undo with the *new* value (in case it later gets
+      // locally reset within this environment).
+      for (let i = 0; i < this.undefStack.length; i++) {
+        delete this.undefStack[i][name];
+      }
+
+      if (this.undefStack.length > 0) {
+        this.undefStack[this.undefStack.length - 1][name] = value;
+      }
+    } else {
+      // Undo this set at end of this group (possibly to `undefined`),
+      // unless an undo is already in place, in which case that older
+      // value is the correct one.
+      const top = this.undefStack[this.undefStack.length - 1];
+
+      if (top && !top.hasOwnProperty(name)) {
+        top[name] = this.current[name];
+      }
+    }
+
+    this.current[name] = value;
+  }
+
+}
+
+/**
+ * Predefined macros for KaTeX.
+ * This can be used to define some commands in terms of others.
+ */
+const builtinMacros = {};
+
+function defineMacro(name, body) {
+  builtinMacros[name] = body;
+} //////////////////////////////////////////////////////////////////////
+// macro tools
+
+defineMacro("\\noexpand", function (context) {
+  // The expansion is the token itself; but that token is interpreted
+  // as if its meaning were ‘\relax’ if it is a control sequence that
+  // would ordinarily be expanded by TeX’s expansion rules.
+  const t = context.popToken();
+
+  if (context.isExpandable(t.text)) {
+    t.noexpand = true;
+    t.treatAsRelax = true;
+  }
+
+  return {
+    tokens: [t],
+    numArgs: 0
+  };
+});
+defineMacro("\\expandafter", function (context) {
+  // TeX first reads the token that comes immediately after \expandafter,
+  // without expanding it; let’s call this token t. Then TeX reads the
+  // token that comes after t (and possibly more tokens, if that token
+  // has an argument), replacing it by its expansion. Finally TeX puts
+  // t back in front of that expansion.
+  const t = context.popToken();
+  context.expandOnce(true); // expand only an expandable token
+
+  return {
+    tokens: [t],
+    numArgs: 0
+  };
+}); // LaTeX's \@firstoftwo{#1}{#2} expands to #1, skipping #2
+// TeX source: \long\def\@firstoftwo#1#2{#1}
+
+defineMacro("\\@firstoftwo", function (context) {
+  const args = context.consumeArgs(2);
+  return {
+    tokens: args[0],
+    numArgs: 0
+  };
+}); // LaTeX's \@secondoftwo{#1}{#2} expands to #2, skipping #1
+// TeX source: \long\def\@secondoftwo#1#2{#2}
+
+defineMacro("\\@secondoftwo", function (context) {
+  const args = context.consumeArgs(2);
+  return {
+    tokens: args[1],
+    numArgs: 0
+  };
+}); // LaTeX's \@ifnextchar{#1}{#2}{#3} looks ahead to the next (unexpanded)
+// symbol that isn't a space, consuming any spaces but not consuming the
+// first nonspace character.  If that nonspace character matches #1, then
+// the macro expands to #2; otherwise, it expands to #3.
+
+defineMacro("\\@ifnextchar", function (context) {
+  const args = context.consumeArgs(3); // symbol, if, else
+
+  context.consumeSpaces();
+  const nextToken = context.future();
+
+  if (args[0].length === 1 && args[0][0].text === nextToken.text) {
+    return {
+      tokens: args[1],
+      numArgs: 0
+    };
+  } else {
+    return {
+      tokens: args[2],
+      numArgs: 0
+    };
+  }
+}); // LaTeX's \@ifstar{#1}{#2} looks ahead to the next (unexpanded) symbol.
+// If it is `*`, then it consumes the symbol, and the macro expands to #1;
+// otherwise, the macro expands to #2 (without consuming the symbol).
+// TeX source: \def\@ifstar#1{\@ifnextchar *{\@firstoftwo{#1}}}
+
+defineMacro("\\@ifstar", "\\@ifnextchar *{\\@firstoftwo{#1}}"); // LaTeX's \TextOrMath{#1}{#2} expands to #1 in text mode, #2 in math mode
+
+defineMacro("\\TextOrMath", function (context) {
+  const args = context.consumeArgs(2);
+
+  if (context.mode === 'text') {
+    return {
+      tokens: args[0],
+      numArgs: 0
+    };
+  } else {
+    return {
+      tokens: args[1],
+      numArgs: 0
+    };
+  }
+}); // Lookup table for parsing numbers in base 8 through 16
+
+const digitToNumber = {
+  "0": 0,
+  "1": 1,
+  "2": 2,
+  "3": 3,
+  "4": 4,
+  "5": 5,
+  "6": 6,
+  "7": 7,
+  "8": 8,
+  "9": 9,
+  "a": 10,
+  "A": 10,
+  "b": 11,
+  "B": 11,
+  "c": 12,
+  "C": 12,
+  "d": 13,
+  "D": 13,
+  "e": 14,
+  "E": 14,
+  "f": 15,
+  "F": 15
+}; // TeX \char makes a literal character (catcode 12) using the following forms:
+// (see The TeXBook, p. 43)
+//   \char123  -- decimal
+//   \char'123 -- octal
+//   \char"123 -- hex
+//   \char`x   -- character that can be written (i.e. isn't active)
+//   \char`\x  -- character that cannot be written (e.g. %)
+// These all refer to characters from the font, so we turn them into special
+// calls to a function \@char dealt with in the Parser.
+
+defineMacro("\\char", function (context) {
+  let token = context.popToken();
+  let base;
+  let number = '';
+
+  if (token.text === "'") {
+    base = 8;
+    token = context.popToken();
+  } else if (token.text === '"') {
+    base = 16;
+    token = context.popToken();
+  } else if (token.text === "`") {
+    token = context.popToken();
+
+    if (token.text[0] === "\\") {
+      number = token.text.charCodeAt(1);
+    } else if (token.text === "EOF") {
+      throw new ParseError("\\char` missing argument");
+    } else {
+      number = token.text.charCodeAt(0);
+    }
+  } else {
+    base = 10;
+  }
+
+  if (base) {
+    // Parse a number in the given base, starting with first `token`.
+    number = digitToNumber[token.text];
+
+    if (number == null || number >= base) {
+      throw new ParseError(`Invalid base-${base} digit ${token.text}`);
+    }
+
+    let digit;
+
+    while ((digit = digitToNumber[context.future().text]) != null && digit < base) {
+      number *= base;
+      number += digit;
+      context.popToken();
+    }
+  }
+
+  return `\\@char{${number}}`;
+}); // \newcommand{\macro}[args]{definition}
+// \renewcommand{\macro}[args]{definition}
+// TODO: Optional arguments: \newcommand{\macro}[args][default]{definition}
+
+const newcommand = (context, existsOK, nonexistsOK) => {
+  let arg = context.consumeArgs(1)[0];
+
+  if (arg.length !== 1) {
+    throw new ParseError("\\newcommand's first argument must be a macro name");
+  }
+
+  const name = arg[0].text;
+  const exists = context.isDefined(name);
+
+  if (exists && !existsOK) {
+    throw new ParseError(`\\newcommand{${name}} attempting to redefine ` + `${name}; use \\renewcommand`);
+  }
+
+  if (!exists && !nonexistsOK) {
+    throw new ParseError(`\\renewcommand{${name}} when command ${name} ` + `does not yet exist; use \\newcommand`);
+  }
+
+  let numArgs = 0;
+  arg = context.consumeArgs(1)[0];
+
+  if (arg.length === 1 && arg[0].text === "[") {
+    let argText = '';
+    let token = context.expandNextToken();
+
+    while (token.text !== "]" && token.text !== "EOF") {
+      // TODO: Should properly expand arg, e.g., ignore {}s
+      argText += token.text;
+      token = context.expandNextToken();
+    }
+
+    if (!argText.match(/^\s*[0-9]+\s*$/)) {
+      throw new ParseError(`Invalid number of arguments: ${argText}`);
+    }
+
+    numArgs = parseInt(argText);
+    arg = context.consumeArgs(1)[0];
+  } // Final arg is the expansion of the macro
+
+
+  context.macros.set(name, {
+    tokens: arg,
+    numArgs
+  });
+  return '';
+};
+
+defineMacro("\\newcommand", context => newcommand(context, false, true));
+defineMacro("\\renewcommand", context => newcommand(context, true, false));
+defineMacro("\\providecommand", context => newcommand(context, true, true)); // terminal (console) tools
+
+defineMacro("\\message", context => {
+  const arg = context.consumeArgs(1)[0]; // eslint-disable-next-line no-console
+
+  console.log(arg.reverse().map(token => token.text).join(""));
+  return '';
+});
+defineMacro("\\errmessage", context => {
+  const arg = context.consumeArgs(1)[0]; // eslint-disable-next-line no-console
+
+  console.error(arg.reverse().map(token => token.text).join(""));
+  return '';
+});
+defineMacro("\\show", context => {
+  const tok = context.popToken();
+  const name = tok.text; // eslint-disable-next-line no-console
+
+  console.log(tok, context.macros.get(name), functions[name], symbols.math[name], symbols.text[name]);
+  return '';
+}); //////////////////////////////////////////////////////////////////////
+// Grouping
+// \let\bgroup={ \let\egroup=}
+
+defineMacro("\\bgroup", "{");
+defineMacro("\\egroup", "}"); // Symbols from latex.ltx:
+// \def\lq{`}
+// \def\rq{'}
+// \def \aa {\r a}
+// \def \AA {\r A}
+
+defineMacro("\\lq", "`");
+defineMacro("\\rq", "'");
+defineMacro("\\aa", "\\r a");
+defineMacro("\\AA", "\\r A"); // Copyright (C) and registered (R) symbols. Use raw symbol in MathML.
+// \DeclareTextCommandDefault{\textcopyright}{\textcircled{c}}
+// \DeclareTextCommandDefault{\textregistered}{\textcircled{%
+//      \check@mathfonts\fontsize\sf@size\z@\math@fontsfalse\selectfont R}}
+// \DeclareRobustCommand{\copyright}{%
+//    \ifmmode{\nfss@text{\textcopyright}}\else\textcopyright\fi}
+
+defineMacro("\\textcopyright", "\\html@mathml{\\textcircled{c}}{\\char`©}");
+defineMacro("\\copyright", "\\TextOrMath{\\textcopyright}{\\text{\\textcopyright}}");
+defineMacro("\\textregistered", "\\html@mathml{\\textcircled{\\scriptsize R}}{\\char`®}"); // Characters omitted from Unicode range 1D400–1D7FF
+
+defineMacro("\u212C", "\\mathscr{B}"); // script
+
+defineMacro("\u2130", "\\mathscr{E}");
+defineMacro("\u2131", "\\mathscr{F}");
+defineMacro("\u210B", "\\mathscr{H}");
+defineMacro("\u2110", "\\mathscr{I}");
+defineMacro("\u2112", "\\mathscr{L}");
+defineMacro("\u2133", "\\mathscr{M}");
+defineMacro("\u211B", "\\mathscr{R}");
+defineMacro("\u212D", "\\mathfrak{C}"); // Fraktur
+
+defineMacro("\u210C", "\\mathfrak{H}");
+defineMacro("\u2128", "\\mathfrak{Z}"); // Define \Bbbk with a macro that works in both HTML and MathML.
+
+defineMacro("\\Bbbk", "\\Bbb{k}"); // Unicode middle dot
+// The KaTeX fonts do not contain U+00B7. Instead, \cdotp displays
+// the dot at U+22C5 and gives it punct spacing.
+
+defineMacro("\u00b7", "\\cdotp"); // \llap and \rlap render their contents in text mode
+
+defineMacro("\\llap", "\\mathllap{\\textrm{#1}}");
+defineMacro("\\rlap", "\\mathrlap{\\textrm{#1}}");
+defineMacro("\\clap", "\\mathclap{\\textrm{#1}}"); // \not is defined by base/fontmath.ltx via
+// \DeclareMathSymbol{\not}{\mathrel}{symbols}{"36}
+// It's thus treated like a \mathrel, but defined by a symbol that has zero
+// width but extends to the right.  We use \rlap to get that spacing.
+// For MathML we write U+0338 here. buildMathML.js will then do the overlay.
+
+defineMacro("\\not", '\\html@mathml{\\mathrel{\\mathrlap\\@not}}{\\char"338}'); // Negated symbols from base/fontmath.ltx:
+// \def\neq{\not=} \let\ne=\neq
+// \DeclareRobustCommand
+//   \notin{\mathrel{\m@th\mathpalette\c@ncel\in}}
+// \def\c@ncel#1#2{\m@th\ooalign{$\hfil#1\mkern1mu/\hfil$\crcr$#1#2$}}
+
+defineMacro("\\neq", "\\html@mathml{\\mathrel{\\not=}}{\\mathrel{\\char`≠}}");
+defineMacro("\\ne", "\\neq");
+defineMacro("\u2260", "\\neq");
+defineMacro("\\notin", "\\html@mathml{\\mathrel{{\\in}\\mathllap{/\\mskip1mu}}}" + "{\\mathrel{\\char`∉}}");
+defineMacro("\u2209", "\\notin"); // Unicode stacked relations
+
+defineMacro("\u2258", "\\html@mathml{" + "\\mathrel{=\\kern{-1em}\\raisebox{0.4em}{$\\scriptsize\\frown$}}" + "}{\\mathrel{\\char`\u2258}}");
+defineMacro("\u2259", "\\html@mathml{\\stackrel{\\tiny\\wedge}{=}}{\\mathrel{\\char`\u2258}}");
+defineMacro("\u225A", "\\html@mathml{\\stackrel{\\tiny\\vee}{=}}{\\mathrel{\\char`\u225A}}");
+defineMacro("\u225B", "\\html@mathml{\\stackrel{\\scriptsize\\star}{=}}" + "{\\mathrel{\\char`\u225B}}");
+defineMacro("\u225D", "\\html@mathml{\\stackrel{\\tiny\\mathrm{def}}{=}}" + "{\\mathrel{\\char`\u225D}}");
+defineMacro("\u225E", "\\html@mathml{\\stackrel{\\tiny\\mathrm{m}}{=}}" + "{\\mathrel{\\char`\u225E}}");
+defineMacro("\u225F", "\\html@mathml{\\stackrel{\\tiny?}{=}}{\\mathrel{\\char`\u225F}}"); // Misc Unicode
+
+defineMacro("\u27C2", "\\perp");
+defineMacro("\u203C", "\\mathclose{!\\mkern-0.8mu!}");
+defineMacro("\u220C", "\\notni");
+defineMacro("\u231C", "\\ulcorner");
+defineMacro("\u231D", "\\urcorner");
+defineMacro("\u231E", "\\llcorner");
+defineMacro("\u231F", "\\lrcorner");
+defineMacro("\u00A9", "\\copyright");
+defineMacro("\u00AE", "\\textregistered");
+defineMacro("\uFE0F", "\\textregistered"); // The KaTeX fonts have corners at codepoints that don't match Unicode.
+// For MathML purposes, use the Unicode code point.
+
+defineMacro("\\ulcorner", "\\html@mathml{\\@ulcorner}{\\mathop{\\char\"231c}}");
+defineMacro("\\urcorner", "\\html@mathml{\\@urcorner}{\\mathop{\\char\"231d}}");
+defineMacro("\\llcorner", "\\html@mathml{\\@llcorner}{\\mathop{\\char\"231e}}");
+defineMacro("\\lrcorner", "\\html@mathml{\\@lrcorner}{\\mathop{\\char\"231f}}"); //////////////////////////////////////////////////////////////////////
+// LaTeX_2ε
+// \vdots{\vbox{\baselineskip4\p@  \lineskiplimit\z@
+// \kern6\p@\hbox{.}\hbox{.}\hbox{.}}}
+// We'll call \varvdots, which gets a glyph from symbols.js.
+// The zero-width rule gets us an equivalent to the vertical 6pt kern.
+
+defineMacro("\\vdots", "\\mathord{\\varvdots\\rule{0pt}{15pt}}");
+defineMacro("\u22ee", "\\vdots"); //////////////////////////////////////////////////////////////////////
+// amsmath.sty
+// http://mirrors.concertpass.com/tex-archive/macros/latex/required/amsmath/amsmath.pdf
+// Italic Greek capital letters.  AMS defines these with \DeclareMathSymbol,
+// but they are equivalent to \mathit{\Letter}.
+
+defineMacro("\\varGamma", "\\mathit{\\Gamma}");
+defineMacro("\\varDelta", "\\mathit{\\Delta}");
+defineMacro("\\varTheta", "\\mathit{\\Theta}");
+defineMacro("\\varLambda", "\\mathit{\\Lambda}");
+defineMacro("\\varXi", "\\mathit{\\Xi}");
+defineMacro("\\varPi", "\\mathit{\\Pi}");
+defineMacro("\\varSigma", "\\mathit{\\Sigma}");
+defineMacro("\\varUpsilon", "\\mathit{\\Upsilon}");
+defineMacro("\\varPhi", "\\mathit{\\Phi}");
+defineMacro("\\varPsi", "\\mathit{\\Psi}");
+defineMacro("\\varOmega", "\\mathit{\\Omega}"); //\newcommand{\substack}[1]{\subarray{c}#1\endsubarray}
+
+defineMacro("\\substack", "\\begin{subarray}{c}#1\\end{subarray}"); // \renewcommand{\colon}{\nobreak\mskip2mu\mathpunct{}\nonscript
+// \mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
+
+defineMacro("\\colon", "\\nobreak\\mskip2mu\\mathpunct{}" + "\\mathchoice{\\mkern-3mu}{\\mkern-3mu}{}{}{:}\\mskip6mu"); // \newcommand{\boxed}[1]{\fbox{\m@th$\displaystyle#1$}}
+
+defineMacro("\\boxed", "\\fbox{$\\displaystyle{#1}$}"); // \def\iff{\DOTSB\;\Longleftrightarrow\;}
+// \def\implies{\DOTSB\;\Longrightarrow\;}
+// \def\impliedby{\DOTSB\;\Longleftarrow\;}
+
+defineMacro("\\iff", "\\DOTSB\\;\\Longleftrightarrow\\;");
+defineMacro("\\implies", "\\DOTSB\\;\\Longrightarrow\\;");
+defineMacro("\\impliedby", "\\DOTSB\\;\\Longleftarrow\\;"); // AMSMath's automatic \dots, based on \mdots@@ macro.
+
+const dotsByToken = {
+  ',': '\\dotsc',
+  '\\not': '\\dotsb',
+  // \keybin@ checks for the following:
+  '+': '\\dotsb',
+  '=': '\\dotsb',
+  '<': '\\dotsb',
+  '>': '\\dotsb',
+  '-': '\\dotsb',
+  '*': '\\dotsb',
+  ':': '\\dotsb',
+  // Symbols whose definition starts with \DOTSB:
+  '\\DOTSB': '\\dotsb',
+  '\\coprod': '\\dotsb',
+  '\\bigvee': '\\dotsb',
+  '\\bigwedge': '\\dotsb',
+  '\\biguplus': '\\dotsb',
+  '\\bigcap': '\\dotsb',
+  '\\bigcup': '\\dotsb',
+  '\\prod': '\\dotsb',
+  '\\sum': '\\dotsb',
+  '\\bigotimes': '\\dotsb',
+  '\\bigoplus': '\\dotsb',
+  '\\bigodot': '\\dotsb',
+  '\\bigsqcup': '\\dotsb',
+  '\\And': '\\dotsb',
+  '\\longrightarrow': '\\dotsb',
+  '\\Longrightarrow': '\\dotsb',
+  '\\longleftarrow': '\\dotsb',
+  '\\Longleftarrow': '\\dotsb',
+  '\\longleftrightarrow': '\\dotsb',
+  '\\Longleftrightarrow': '\\dotsb',
+  '\\mapsto': '\\dotsb',
+  '\\longmapsto': '\\dotsb',
+  '\\hookrightarrow': '\\dotsb',
+  '\\doteq': '\\dotsb',
+  // Symbols whose definition starts with \mathbin:
+  '\\mathbin': '\\dotsb',
+  // Symbols whose definition starts with \mathrel:
+  '\\mathrel': '\\dotsb',
+  '\\relbar': '\\dotsb',
+  '\\Relbar': '\\dotsb',
+  '\\xrightarrow': '\\dotsb',
+  '\\xleftarrow': '\\dotsb',
+  // Symbols whose definition starts with \DOTSI:
+  '\\DOTSI': '\\dotsi',
+  '\\int': '\\dotsi',
+  '\\oint': '\\dotsi',
+  '\\iint': '\\dotsi',
+  '\\iiint': '\\dotsi',
+  '\\iiiint': '\\dotsi',
+  '\\idotsint': '\\dotsi',
+  // Symbols whose definition starts with \DOTSX:
+  '\\DOTSX': '\\dotsx'
+};
+defineMacro("\\dots", function (context) {
+  // TODO: If used in text mode, should expand to \textellipsis.
+  // However, in KaTeX, \textellipsis and \ldots behave the same
+  // (in text mode), and it's unlikely we'd see any of the math commands
+  // that affect the behavior of \dots when in text mode.  So fine for now
+  // (until we support \ifmmode ... \else ... \fi).
+  let thedots = '\\dotso';
+  const next = context.expandAfterFuture().text;
+
+  if (next in dotsByToken) {
+    thedots = dotsByToken[next];
+  } else if (next.substr(0, 4) === '\\not') {
+    thedots = '\\dotsb';
+  } else if (next in symbols.math) {
+    if (utils.contains(['bin', 'rel'], symbols.math[next].group)) {
+      thedots = '\\dotsb';
+    }
+  }
+
+  return thedots;
+});
+const spaceAfterDots = {
+  // \rightdelim@ checks for the following:
+  ')': true,
+  ']': true,
+  '\\rbrack': true,
+  '\\}': true,
+  '\\rbrace': true,
+  '\\rangle': true,
+  '\\rceil': true,
+  '\\rfloor': true,
+  '\\rgroup': true,
+  '\\rmoustache': true,
+  '\\right': true,
+  '\\bigr': true,
+  '\\biggr': true,
+  '\\Bigr': true,
+  '\\Biggr': true,
+  // \extra@ also tests for the following:
+  '$': true,
+  // \extrap@ checks for the following:
+  ';': true,
+  '.': true,
+  ',': true
+};
+defineMacro("\\dotso", function (context) {
+  const next = context.future().text;
+
+  if (next in spaceAfterDots) {
+    return "\\ldots\\,";
+  } else {
+    return "\\ldots";
+  }
+});
+defineMacro("\\dotsc", function (context) {
+  const next = context.future().text; // \dotsc uses \extra@ but not \extrap@, instead specially checking for
+  // ';' and '.', but doesn't check for ','.
+
+  if (next in spaceAfterDots && next !== ',') {
+    return "\\ldots\\,";
+  } else {
+    return "\\ldots";
+  }
+});
+defineMacro("\\cdots", function (context) {
+  const next = context.future().text;
+
+  if (next in spaceAfterDots) {
+    return "\\@cdots\\,";
+  } else {
+    return "\\@cdots";
+  }
+});
+defineMacro("\\dotsb", "\\cdots");
+defineMacro("\\dotsm", "\\cdots");
+defineMacro("\\dotsi", "\\!\\cdots"); // amsmath doesn't actually define \dotsx, but \dots followed by a macro
+// starting with \DOTSX implies \dotso, and then \extra@ detects this case
+// and forces the added `\,`.
+
+defineMacro("\\dotsx", "\\ldots\\,"); // \let\DOTSI\relax
+// \let\DOTSB\relax
+// \let\DOTSX\relax
+
+defineMacro("\\DOTSI", "\\relax");
+defineMacro("\\DOTSB", "\\relax");
+defineMacro("\\DOTSX", "\\relax"); // Spacing, based on amsmath.sty's override of LaTeX defaults
+// \DeclareRobustCommand{\tmspace}[3]{%
+//   \ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
+
+defineMacro("\\tmspace", "\\TextOrMath{\\kern#1#3}{\\mskip#1#2}\\relax"); // \renewcommand{\,}{\tmspace+\thinmuskip{.1667em}}
+// TODO: math mode should use \thinmuskip
+
+defineMacro("\\,", "\\tmspace+{3mu}{.1667em}"); // \let\thinspace\,
+
+defineMacro("\\thinspace", "\\,"); // \def\>{\mskip\medmuskip}
+// \renewcommand{\:}{\tmspace+\medmuskip{.2222em}}
+// TODO: \> and math mode of \: should use \medmuskip = 4mu plus 2mu minus 4mu
+
+defineMacro("\\>", "\\mskip{4mu}");
+defineMacro("\\:", "\\tmspace+{4mu}{.2222em}"); // \let\medspace\:
+
+defineMacro("\\medspace", "\\:"); // \renewcommand{\;}{\tmspace+\thickmuskip{.2777em}}
+// TODO: math mode should use \thickmuskip = 5mu plus 5mu
+
+defineMacro("\\;", "\\tmspace+{5mu}{.2777em}"); // \let\thickspace\;
+
+defineMacro("\\thickspace", "\\;"); // \renewcommand{\!}{\tmspace-\thinmuskip{.1667em}}
+// TODO: math mode should use \thinmuskip
+
+defineMacro("\\!", "\\tmspace-{3mu}{.1667em}"); // \let\negthinspace\!
+
+defineMacro("\\negthinspace", "\\!"); // \newcommand{\negmedspace}{\tmspace-\medmuskip{.2222em}}
+// TODO: math mode should use \medmuskip
+
+defineMacro("\\negmedspace", "\\tmspace-{4mu}{.2222em}"); // \newcommand{\negthickspace}{\tmspace-\thickmuskip{.2777em}}
+// TODO: math mode should use \thickmuskip
+
+defineMacro("\\negthickspace", "\\tmspace-{5mu}{.277em}"); // \def\enspace{\kern.5em }
+
+defineMacro("\\enspace", "\\kern.5em "); // \def\enskip{\hskip.5em\relax}
+
+defineMacro("\\enskip", "\\hskip.5em\\relax"); // \def\quad{\hskip1em\relax}
+
+defineMacro("\\quad", "\\hskip1em\\relax"); // \def\qquad{\hskip2em\relax}
+
+defineMacro("\\qquad", "\\hskip2em\\relax"); // \tag@in@display form of \tag
+
+defineMacro("\\tag", "\\@ifstar\\tag@literal\\tag@paren");
+defineMacro("\\tag@paren", "\\tag@literal{({#1})}");
+defineMacro("\\tag@literal", context => {
+  if (context.macros.get("\\df@tag")) {
+    throw new ParseError("Multiple \\tag");
+  }
+
+  return "\\gdef\\df@tag{\\text{#1}}";
+}); // \renewcommand{\bmod}{\nonscript\mskip-\medmuskip\mkern5mu\mathbin
+//   {\operator@font mod}\penalty900
+//   \mkern5mu\nonscript\mskip-\medmuskip}
+// \newcommand{\pod}[1]{\allowbreak
+//   \if@display\mkern18mu\else\mkern8mu\fi(#1)}
+// \renewcommand{\pmod}[1]{\pod{{\operator@font mod}\mkern6mu#1}}
+// \newcommand{\mod}[1]{\allowbreak\if@display\mkern18mu
+//   \else\mkern12mu\fi{\operator@font mod}\,\,#1}
+// TODO: math mode should use \medmuskip = 4mu plus 2mu minus 4mu
+
+defineMacro("\\bmod", "\\mathchoice{\\mskip1mu}{\\mskip1mu}{\\mskip5mu}{\\mskip5mu}" + "\\mathbin{\\rm mod}" + "\\mathchoice{\\mskip1mu}{\\mskip1mu}{\\mskip5mu}{\\mskip5mu}");
+defineMacro("\\pod", "\\allowbreak" + "\\mathchoice{\\mkern18mu}{\\mkern8mu}{\\mkern8mu}{\\mkern8mu}(#1)");
+defineMacro("\\pmod", "\\pod{{\\rm mod}\\mkern6mu#1}");
+defineMacro("\\mod", "\\allowbreak" + "\\mathchoice{\\mkern18mu}{\\mkern12mu}{\\mkern12mu}{\\mkern12mu}" + "{\\rm mod}\\,\\,#1"); // \pmb    --   A simulation of bold.
+// The version in ambsy.sty works by typesetting three copies of the argument
+// with small offsets. We use two copies. We omit the vertical offset because
+// of rendering problems that makeVList encounters in Safari.
+
+defineMacro("\\pmb", "\\html@mathml{" + "\\@binrel{#1}{\\mathrlap{#1}\\kern0.5px#1}}" + "{\\mathbf{#1}}"); //////////////////////////////////////////////////////////////////////
+// LaTeX source2e
+// \\ defaults to \newline, but changes to \cr within array environment
+
+defineMacro("\\\\", "\\newline"); // \def\TeX{T\kern-.1667em\lower.5ex\hbox{E}\kern-.125emX\@}
+// TODO: Doesn't normally work in math mode because \@ fails.  KaTeX doesn't
+// support \@ yet, so that's omitted, and we add \text so that the result
+// doesn't look funny in math mode.
+
+defineMacro("\\TeX", "\\textrm{\\html@mathml{" + "T\\kern-.1667em\\raisebox{-.5ex}{E}\\kern-.125emX" + "}{TeX}}"); // \DeclareRobustCommand{\LaTeX}{L\kern-.36em%
+//         {\sbox\z@ T%
+//          \vbox to\ht\z@{\hbox{\check@mathfonts
+//                               \fontsize\sf@size\z@
+//                               \math@fontsfalse\selectfont
+//                               A}%
+//                         \vss}%
+//         }%
+//         \kern-.15em%
+//         \TeX}
+// This code aligns the top of the A with the T (from the perspective of TeX's
+// boxes, though visually the A appears to extend above slightly).
+// We compute the corresponding \raisebox when A is rendered in \normalsize
+// \scriptstyle, which has a scale factor of 0.7 (see Options.js).
+
+const latexRaiseA = metricMap['Main-Regular']["T".charCodeAt(0)][1] - 0.7 * metricMap['Main-Regular']["A".charCodeAt(0)][1] + "em";
+defineMacro("\\LaTeX", "\\textrm{\\html@mathml{" + `L\\kern-.36em\\raisebox{${latexRaiseA}}{\\scriptstyle A}` + "\\kern-.15em\\TeX}{LaTeX}}"); // New KaTeX logo based on tweaking LaTeX logo
+
+defineMacro("\\KaTeX", "\\textrm{\\html@mathml{" + `K\\kern-.17em\\raisebox{${latexRaiseA}}{\\scriptstyle A}` + "\\kern-.15em\\TeX}{KaTeX}}"); // \DeclareRobustCommand\hspace{\@ifstar\@hspacer\@hspace}
+// \def\@hspace#1{\hskip  #1\relax}
+// \def\@hspacer#1{\vrule \@width\z@\nobreak
+//                 \hskip #1\hskip \z@skip}
+
+defineMacro("\\hspace", "\\@ifstar\\@hspacer\\@hspace");
+defineMacro("\\@hspace", "\\hskip #1\\relax");
+defineMacro("\\@hspacer", "\\rule{0pt}{0pt}\\hskip #1\\relax"); //////////////////////////////////////////////////////////////////////
+// mathtools.sty
+//\providecommand\ordinarycolon{:}
+
+defineMacro("\\ordinarycolon", ":"); //\def\vcentcolon{\mathrel{\mathop\ordinarycolon}}
+//TODO(edemaine): Not yet centered. Fix via \raisebox or #726
+
+defineMacro("\\vcentcolon", "\\mathrel{\\mathop\\ordinarycolon}"); // \providecommand*\dblcolon{\vcentcolon\mathrel{\mkern-.9mu}\vcentcolon}
+
+defineMacro("\\dblcolon", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-.9mu}\\vcentcolon}}" + "{\\mathop{\\char\"2237}}"); // \providecommand*\coloneqq{\vcentcolon\mathrel{\mkern-1.2mu}=}
+
+defineMacro("\\coloneqq", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}=}}" + "{\\mathop{\\char\"2254}}"); // ≔
+// \providecommand*\Coloneqq{\dblcolon\mathrel{\mkern-1.2mu}=}
+
+defineMacro("\\Coloneqq", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}=}}" + "{\\mathop{\\char\"2237\\char\"3d}}"); // \providecommand*\coloneq{\vcentcolon\mathrel{\mkern-1.2mu}\mathrel{-}}
+
+defineMacro("\\coloneq", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\mathrel{-}}}" + "{\\mathop{\\char\"3a\\char\"2212}}"); // \providecommand*\Coloneq{\dblcolon\mathrel{\mkern-1.2mu}\mathrel{-}}
+
+defineMacro("\\Coloneq", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\mathrel{-}}}" + "{\\mathop{\\char\"2237\\char\"2212}}"); // \providecommand*\eqqcolon{=\mathrel{\mkern-1.2mu}\vcentcolon}
+
+defineMacro("\\eqqcolon", "\\html@mathml{" + "\\mathrel{=\\mathrel{\\mkern-1.2mu}\\vcentcolon}}" + "{\\mathop{\\char\"2255}}"); // ≕
+// \providecommand*\Eqqcolon{=\mathrel{\mkern-1.2mu}\dblcolon}
+
+defineMacro("\\Eqqcolon", "\\html@mathml{" + "\\mathrel{=\\mathrel{\\mkern-1.2mu}\\dblcolon}}" + "{\\mathop{\\char\"3d\\char\"2237}}"); // \providecommand*\eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\vcentcolon}
+
+defineMacro("\\eqcolon", "\\html@mathml{" + "\\mathrel{\\mathrel{-}\\mathrel{\\mkern-1.2mu}\\vcentcolon}}" + "{\\mathop{\\char\"2239}}"); // \providecommand*\Eqcolon{\mathrel{-}\mathrel{\mkern-1.2mu}\dblcolon}
+
+defineMacro("\\Eqcolon", "\\html@mathml{" + "\\mathrel{\\mathrel{-}\\mathrel{\\mkern-1.2mu}\\dblcolon}}" + "{\\mathop{\\char\"2212\\char\"2237}}"); // \providecommand*\colonapprox{\vcentcolon\mathrel{\mkern-1.2mu}\approx}
+
+defineMacro("\\colonapprox", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\approx}}" + "{\\mathop{\\char\"3a\\char\"2248}}"); // \providecommand*\Colonapprox{\dblcolon\mathrel{\mkern-1.2mu}\approx}
+
+defineMacro("\\Colonapprox", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\approx}}" + "{\\mathop{\\char\"2237\\char\"2248}}"); // \providecommand*\colonsim{\vcentcolon\mathrel{\mkern-1.2mu}\sim}
+
+defineMacro("\\colonsim", "\\html@mathml{" + "\\mathrel{\\vcentcolon\\mathrel{\\mkern-1.2mu}\\sim}}" + "{\\mathop{\\char\"3a\\char\"223c}}"); // \providecommand*\Colonsim{\dblcolon\mathrel{\mkern-1.2mu}\sim}
+
+defineMacro("\\Colonsim", "\\html@mathml{" + "\\mathrel{\\dblcolon\\mathrel{\\mkern-1.2mu}\\sim}}" + "{\\mathop{\\char\"2237\\char\"223c}}"); // Some Unicode characters are implemented with macros to mathtools functions.
+
+defineMacro("\u2237", "\\dblcolon"); // ::
+
+defineMacro("\u2239", "\\eqcolon"); // -:
+
+defineMacro("\u2254", "\\coloneqq"); // :=
+
+defineMacro("\u2255", "\\eqqcolon"); // =:
+
+defineMacro("\u2A74", "\\Coloneqq"); // ::=
+//////////////////////////////////////////////////////////////////////
+// colonequals.sty
+// Alternate names for mathtools's macros:
+
+defineMacro("\\ratio", "\\vcentcolon");
+defineMacro("\\coloncolon", "\\dblcolon");
+defineMacro("\\colonequals", "\\coloneqq");
+defineMacro("\\coloncolonequals", "\\Coloneqq");
+defineMacro("\\equalscolon", "\\eqqcolon");
+defineMacro("\\equalscoloncolon", "\\Eqqcolon");
+defineMacro("\\colonminus", "\\coloneq");
+defineMacro("\\coloncolonminus", "\\Coloneq");
+defineMacro("\\minuscolon", "\\eqcolon");
+defineMacro("\\minuscoloncolon", "\\Eqcolon"); // \colonapprox name is same in mathtools and colonequals.
+
+defineMacro("\\coloncolonapprox", "\\Colonapprox"); // \colonsim name is same in mathtools and colonequals.
+
+defineMacro("\\coloncolonsim", "\\Colonsim"); // Additional macros, implemented by analogy with mathtools definitions:
+
+defineMacro("\\simcolon", "\\mathrel{\\sim\\mathrel{\\mkern-1.2mu}\\vcentcolon}");
+defineMacro("\\simcoloncolon", "\\mathrel{\\sim\\mathrel{\\mkern-1.2mu}\\dblcolon}");
+defineMacro("\\approxcolon", "\\mathrel{\\approx\\mathrel{\\mkern-1.2mu}\\vcentcolon}");
+defineMacro("\\approxcoloncolon", "\\mathrel{\\approx\\mathrel{\\mkern-1.2mu}\\dblcolon}"); // Present in newtxmath, pxfonts and txfonts
+
+defineMacro("\\notni", "\\html@mathml{\\not\\ni}{\\mathrel{\\char`\u220C}}");
+defineMacro("\\limsup", "\\DOTSB\\operatorname*{lim\\,sup}");
+defineMacro("\\liminf", "\\DOTSB\\operatorname*{lim\\,inf}"); //////////////////////////////////////////////////////////////////////
+// MathML alternates for KaTeX glyphs in the Unicode private area
+
+defineMacro("\\gvertneqq", "\\html@mathml{\\@gvertneqq}{\u2269}");
+defineMacro("\\lvertneqq", "\\html@mathml{\\@lvertneqq}{\u2268}");
+defineMacro("\\ngeqq", "\\html@mathml{\\@ngeqq}{\u2271}");
+defineMacro("\\ngeqslant", "\\html@mathml{\\@ngeqslant}{\u2271}");
+defineMacro("\\nleqq", "\\html@mathml{\\@nleqq}{\u2270}");
+defineMacro("\\nleqslant", "\\html@mathml{\\@nleqslant}{\u2270}");
+defineMacro("\\nshortmid", "\\html@mathml{\\@nshortmid}{∤}");
+defineMacro("\\nshortparallel", "\\html@mathml{\\@nshortparallel}{∦}");
+defineMacro("\\nsubseteqq", "\\html@mathml{\\@nsubseteqq}{\u2288}");
+defineMacro("\\nsupseteqq", "\\html@mathml{\\@nsupseteqq}{\u2289}");
+defineMacro("\\varsubsetneq", "\\html@mathml{\\@varsubsetneq}{⊊}");
+defineMacro("\\varsubsetneqq", "\\html@mathml{\\@varsubsetneqq}{⫋}");
+defineMacro("\\varsupsetneq", "\\html@mathml{\\@varsupsetneq}{⊋}");
+defineMacro("\\varsupsetneqq", "\\html@mathml{\\@varsupsetneqq}{⫌}");
+defineMacro("\\imath", "\\html@mathml{\\@imath}{\u0131}");
+defineMacro("\\jmath", "\\html@mathml{\\@jmath}{\u0237}"); //////////////////////////////////////////////////////////////////////
+// stmaryrd and semantic
+// The stmaryrd and semantic packages render the next four items by calling a
+// glyph. Those glyphs do not exist in the KaTeX fonts. Hence the macros.
+
+defineMacro("\\llbracket", "\\html@mathml{" + "\\mathopen{[\\mkern-3.2mu[}}" + "{\\mathopen{\\char`\u27e6}}");
+defineMacro("\\rrbracket", "\\html@mathml{" + "\\mathclose{]\\mkern-3.2mu]}}" + "{\\mathclose{\\char`\u27e7}}");
+defineMacro("\u27e6", "\\llbracket"); // blackboard bold [
+
+defineMacro("\u27e7", "\\rrbracket"); // blackboard bold ]
+
+defineMacro("\\lBrace", "\\html@mathml{" + "\\mathopen{\\{\\mkern-3.2mu[}}" + "{\\mathopen{\\char`\u2983}}");
+defineMacro("\\rBrace", "\\html@mathml{" + "\\mathclose{]\\mkern-3.2mu\\}}}" + "{\\mathclose{\\char`\u2984}}");
+defineMacro("\u2983", "\\lBrace"); // blackboard bold {
+
+defineMacro("\u2984", "\\rBrace"); // blackboard bold }
+// TODO: Create variable sized versions of the last two items. I believe that
+// will require new font glyphs.
+// The stmaryrd function `\minuso` provides a "Plimsoll" symbol that
+// superimposes the characters \circ and \mathminus. Used in chemistry.
+
+defineMacro("\\minuso", "\\mathbin{\\html@mathml{" + "{\\mathrlap{\\mathchoice{\\kern{0.145em}}{\\kern{0.145em}}" + "{\\kern{0.1015em}}{\\kern{0.0725em}}\\circ}{-}}}" + "{\\char`⦵}}");
+defineMacro("⦵", "\\minuso"); //////////////////////////////////////////////////////////////////////
+// texvc.sty
+// The texvc package contains macros available in mediawiki pages.
+// We omit the functions deprecated at
+// https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Deprecated_syntax
+// We also omit texvc's \O, which conflicts with \text{\O}
+
+defineMacro("\\darr", "\\downarrow");
+defineMacro("\\dArr", "\\Downarrow");
+defineMacro("\\Darr", "\\Downarrow");
+defineMacro("\\lang", "\\langle");
+defineMacro("\\rang", "\\rangle");
+defineMacro("\\uarr", "\\uparrow");
+defineMacro("\\uArr", "\\Uparrow");
+defineMacro("\\Uarr", "\\Uparrow");
+defineMacro("\\N", "\\mathbb{N}");
+defineMacro("\\R", "\\mathbb{R}");
+defineMacro("\\Z", "\\mathbb{Z}");
+defineMacro("\\alef", "\\aleph");
+defineMacro("\\alefsym", "\\aleph");
+defineMacro("\\Alpha", "\\mathrm{A}");
+defineMacro("\\Beta", "\\mathrm{B}");
+defineMacro("\\bull", "\\bullet");
+defineMacro("\\Chi", "\\mathrm{X}");
+defineMacro("\\clubs", "\\clubsuit");
+defineMacro("\\cnums", "\\mathbb{C}");
+defineMacro("\\Complex", "\\mathbb{C}");
+defineMacro("\\Dagger", "\\ddagger");
+defineMacro("\\diamonds", "\\diamondsuit");
+defineMacro("\\empty", "\\emptyset");
+defineMacro("\\Epsilon", "\\mathrm{E}");
+defineMacro("\\Eta", "\\mathrm{H}");
+defineMacro("\\exist", "\\exists");
+defineMacro("\\harr", "\\leftrightarrow");
+defineMacro("\\hArr", "\\Leftrightarrow");
+defineMacro("\\Harr", "\\Leftrightarrow");
+defineMacro("\\hearts", "\\heartsuit");
+defineMacro("\\image", "\\Im");
+defineMacro("\\infin", "\\infty");
+defineMacro("\\Iota", "\\mathrm{I}");
+defineMacro("\\isin", "\\in");
+defineMacro("\\Kappa", "\\mathrm{K}");
+defineMacro("\\larr", "\\leftarrow");
+defineMacro("\\lArr", "\\Leftarrow");
+defineMacro("\\Larr", "\\Leftarrow");
+defineMacro("\\lrarr", "\\leftrightarrow");
+defineMacro("\\lrArr", "\\Leftrightarrow");
+defineMacro("\\Lrarr", "\\Leftrightarrow");
+defineMacro("\\Mu", "\\mathrm{M}");
+defineMacro("\\natnums", "\\mathbb{N}");
+defineMacro("\\Nu", "\\mathrm{N}");
+defineMacro("\\Omicron", "\\mathrm{O}");
+defineMacro("\\plusmn", "\\pm");
+defineMacro("\\rarr", "\\rightarrow");
+defineMacro("\\rArr", "\\Rightarrow");
+defineMacro("\\Rarr", "\\Rightarrow");
+defineMacro("\\real", "\\Re");
+defineMacro("\\reals", "\\mathbb{R}");
+defineMacro("\\Reals", "\\mathbb{R}");
+defineMacro("\\Rho", "\\mathrm{P}");
+defineMacro("\\sdot", "\\cdot");
+defineMacro("\\sect", "\\S");
+defineMacro("\\spades", "\\spadesuit");
+defineMacro("\\sub", "\\subset");
+defineMacro("\\sube", "\\subseteq");
+defineMacro("\\supe", "\\supseteq");
+defineMacro("\\Tau", "\\mathrm{T}");
+defineMacro("\\thetasym", "\\vartheta"); // TODO: defineMacro("\\varcoppa", "\\\mbox{\\coppa}");
+
+defineMacro("\\weierp", "\\wp");
+defineMacro("\\Zeta", "\\mathrm{Z}"); //////////////////////////////////////////////////////////////////////
+// statmath.sty
+// https://ctan.math.illinois.edu/macros/latex/contrib/statmath/statmath.pdf
+
+defineMacro("\\argmin", "\\DOTSB\\operatorname*{arg\\,min}");
+defineMacro("\\argmax", "\\DOTSB\\operatorname*{arg\\,max}");
+defineMacro("\\plim", "\\DOTSB\\mathop{\\operatorname{plim}}\\limits"); //////////////////////////////////////////////////////////////////////
+// braket.sty
+// http://ctan.math.washington.edu/tex-archive/macros/latex/contrib/braket/braket.pdf
+
+defineMacro("\\bra", "\\mathinner{\\langle{#1}|}");
+defineMacro("\\ket", "\\mathinner{|{#1}\\rangle}");
+defineMacro("\\braket", "\\mathinner{\\langle{#1}\\rangle}");
+defineMacro("\\Bra", "\\left\\langle#1\\right|");
+defineMacro("\\Ket", "\\left|#1\\right\\rangle"); // Custom Khan Academy colors, should be moved to an optional package
+
+defineMacro("\\blue", "\\textcolor{##6495ed}{#1}");
+defineMacro("\\orange", "\\textcolor{##ffa500}{#1}");
+defineMacro("\\pink", "\\textcolor{##ff00af}{#1}");
+defineMacro("\\red", "\\textcolor{##df0030}{#1}");
+defineMacro("\\green", "\\textcolor{##28ae7b}{#1}");
+defineMacro("\\gray", "\\textcolor{gray}{#1}");
+defineMacro("\\purple", "\\textcolor{##9d38bd}{#1}");
+defineMacro("\\blueA", "\\textcolor{##ccfaff}{#1}");
+defineMacro("\\blueB", "\\textcolor{##80f6ff}{#1}");
+defineMacro("\\blueC", "\\textcolor{##63d9ea}{#1}");
+defineMacro("\\blueD", "\\textcolor{##11accd}{#1}");
+defineMacro("\\blueE", "\\textcolor{##0c7f99}{#1}");
+defineMacro("\\tealA", "\\textcolor{##94fff5}{#1}");
+defineMacro("\\tealB", "\\textcolor{##26edd5}{#1}");
+defineMacro("\\tealC", "\\textcolor{##01d1c1}{#1}");
+defineMacro("\\tealD", "\\textcolor{##01a995}{#1}");
+defineMacro("\\tealE", "\\textcolor{##208170}{#1}");
+defineMacro("\\greenA", "\\textcolor{##b6ffb0}{#1}");
+defineMacro("\\greenB", "\\textcolor{##8af281}{#1}");
+defineMacro("\\greenC", "\\textcolor{##74cf70}{#1}");
+defineMacro("\\greenD", "\\textcolor{##1fab54}{#1}");
+defineMacro("\\greenE", "\\textcolor{##0d923f}{#1}");
+defineMacro("\\goldA", "\\textcolor{##ffd0a9}{#1}");
+defineMacro("\\goldB", "\\textcolor{##ffbb71}{#1}");
+defineMacro("\\goldC", "\\textcolor{##ff9c39}{#1}");
+defineMacro("\\goldD", "\\textcolor{##e07d10}{#1}");
+defineMacro("\\goldE", "\\textcolor{##a75a05}{#1}");
+defineMacro("\\redA", "\\textcolor{##fca9a9}{#1}");
+defineMacro("\\redB", "\\textcolor{##ff8482}{#1}");
+defineMacro("\\redC", "\\textcolor{##f9685d}{#1}");
+defineMacro("\\redD", "\\textcolor{##e84d39}{#1}");
+defineMacro("\\redE", "\\textcolor{##bc2612}{#1}");
+defineMacro("\\maroonA", "\\textcolor{##ffbde0}{#1}");
+defineMacro("\\maroonB", "\\textcolor{##ff92c6}{#1}");
+defineMacro("\\maroonC", "\\textcolor{##ed5fa6}{#1}");
+defineMacro("\\maroonD", "\\textcolor{##ca337c}{#1}");
+defineMacro("\\maroonE", "\\textcolor{##9e034e}{#1}");
+defineMacro("\\purpleA", "\\textcolor{##ddd7ff}{#1}");
+defineMacro("\\purpleB", "\\textcolor{##c6b9fc}{#1}");
+defineMacro("\\purpleC", "\\textcolor{##aa87ff}{#1}");
+defineMacro("\\purpleD", "\\textcolor{##7854ab}{#1}");
+defineMacro("\\purpleE", "\\textcolor{##543b78}{#1}");
+defineMacro("\\mintA", "\\textcolor{##f5f9e8}{#1}");
+defineMacro("\\mintB", "\\textcolor{##edf2df}{#1}");
+defineMacro("\\mintC", "\\textcolor{##e0e5cc}{#1}");
+defineMacro("\\grayA", "\\textcolor{##f6f7f7}{#1}");
+defineMacro("\\grayB", "\\textcolor{##f0f1f2}{#1}");
+defineMacro("\\grayC", "\\textcolor{##e3e5e6}{#1}");
+defineMacro("\\grayD", "\\textcolor{##d6d8da}{#1}");
+defineMacro("\\grayE", "\\textcolor{##babec2}{#1}");
+defineMacro("\\grayF", "\\textcolor{##888d93}{#1}");
+defineMacro("\\grayG", "\\textcolor{##626569}{#1}");
+defineMacro("\\grayH", "\\textcolor{##3b3e40}{#1}");
+defineMacro("\\grayI", "\\textcolor{##21242c}{#1}");
+defineMacro("\\kaBlue", "\\textcolor{##314453}{#1}");
+defineMacro("\\kaGreen", "\\textcolor{##71B307}{#1}");
+
+/**
+ * This file contains the “gullet” where macros are expanded
+ * until only non-macro tokens remain.
+ */
+// List of commands that act like macros but aren't defined as a macro,
+// function, or symbol.  Used in `isDefined`.
+const implicitCommands = {
+  "\\relax": true,
+  // MacroExpander.js
+  "^": true,
+  // Parser.js
+  "_": true,
+  // Parser.js
+  "\\limits": true,
+  // Parser.js
+  "\\nolimits": true // Parser.js
+
+};
+class MacroExpander {
+  constructor(input, settings, mode) {
+    this.settings = void 0;
+    this.expansionCount = void 0;
+    this.lexer = void 0;
+    this.macros = void 0;
+    this.stack = void 0;
+    this.mode = void 0;
+    this.settings = settings;
+    this.expansionCount = 0;
+    this.feed(input); // Make new global namespace
+
+    this.macros = new Namespace(builtinMacros, settings.macros);
+    this.mode = mode;
+    this.stack = []; // contains tokens in REVERSE order
+  }
+  /**
+   * Feed a new input string to the same MacroExpander
+   * (with existing macros etc.).
+   */
+
+
+  feed(input) {
+    this.lexer = new Lexer(input, this.settings);
+  }
+  /**
+   * Switches between "text" and "math" modes.
+   */
+
+
+  switchMode(newMode) {
+    this.mode = newMode;
+  }
+  /**
+   * Start a new group nesting within all namespaces.
+   */
+
+
+  beginGroup() {
+    this.macros.beginGroup();
+  }
+  /**
+   * End current group nesting within all namespaces.
+   */
+
+
+  endGroup() {
+    this.macros.endGroup();
+  }
+  /**
+   * Returns the topmost token on the stack, without expanding it.
+   * Similar in behavior to TeX's `\futurelet`.
+   */
+
+
+  future() {
+    if (this.stack.length === 0) {
+      this.pushToken(this.lexer.lex());
+    }
+
+    return this.stack[this.stack.length - 1];
+  }
+  /**
+   * Remove and return the next unexpanded token.
+   */
+
+
+  popToken() {
+    this.future(); // ensure non-empty stack
+
+    return this.stack.pop();
+  }
+  /**
+   * Add a given token to the token stack.  In particular, this get be used
+   * to put back a token returned from one of the other methods.
+   */
+
+
+  pushToken(token) {
+    this.stack.push(token);
+  }
+  /**
+   * Append an array of tokens to the token stack.
+   */
+
+
+  pushTokens(tokens) {
+    this.stack.push(...tokens);
+  }
+  /**
+   * Consume all following space tokens, without expansion.
+   */
+
+
+  consumeSpaces() {
+    for (;;) {
+      const token = this.future();
+
+      if (token.text === " ") {
+        this.stack.pop();
+      } else {
+        break;
+      }
+    }
+  }
+  /**
+   * Consume the specified number of arguments from the token stream,
+   * and return the resulting array of arguments.
+   */
+
+
+  consumeArgs(numArgs) {
+    const args = []; // obtain arguments, either single token or balanced {…} group
+
+    for (let i = 0; i < numArgs; ++i) {
+      this.consumeSpaces(); // ignore spaces before each argument
+
+      const startOfArg = this.popToken();
+
+      if (startOfArg.text === "{") {
+        const arg = [];
+        let depth = 1;
+
+        while (depth !== 0) {
+          const tok = this.popToken();
+          arg.push(tok);
+
+          if (tok.text === "{") {
+            ++depth;
+          } else if (tok.text === "}") {
+            --depth;
+          } else if (tok.text === "EOF") {
+            throw new ParseError("End of input in macro argument", startOfArg);
+          }
+        }
+
+        arg.pop(); // remove last }
+
+        arg.reverse(); // like above, to fit in with stack order
+
+        args[i] = arg;
+      } else if (startOfArg.text === "EOF") {
+        throw new ParseError("End of input expecting macro argument");
+      } else {
+        args[i] = [startOfArg];
+      }
+    }
+
+    return args;
+  }
+  /**
+   * Expand the next token only once if possible.
+   *
+   * If the token is expanded, the resulting tokens will be pushed onto
+   * the stack in reverse order and will be returned as an array,
+   * also in reverse order.
+   *
+   * If not, the next token will be returned without removing it
+   * from the stack.  This case can be detected by a `Token` return value
+   * instead of an `Array` return value.
+   *
+   * In either case, the next token will be on the top of the stack,
+   * or the stack will be empty.
+   *
+   * Used to implement `expandAfterFuture` and `expandNextToken`.
+   *
+   * At the moment, macro expansion doesn't handle delimited macros,
+   * i.e. things like those defined by \def\foo#1\end{…}.
+   * See the TeX book page 202ff. for details on how those should behave.
+   *
+   * If expandableOnly, only expandable tokens are expanded and
+   * an undefined control sequence results in an error.
+   */
+
+
+  expandOnce(expandableOnly) {
+    const topToken = this.popToken();
+    const name = topToken.text;
+    const expansion = !topToken.noexpand ? this._getExpansion(name) : null;
+
+    if (expansion == null || expandableOnly && expansion.unexpandable) {
+      if (expandableOnly && expansion == null && name[0] === "\\" && !this.isDefined(name)) {
+        throw new ParseError("Undefined control sequence: " + name);
+      }
+
+      this.pushToken(topToken);
+      return topToken;
+    }
+
+    this.expansionCount++;
+
+    if (this.expansionCount > this.settings.maxExpand) {
+      throw new ParseError("Too many expansions: infinite loop or " + "need to increase maxExpand setting");
+    }
+
+    let tokens = expansion.tokens;
+
+    if (expansion.numArgs) {
+      const args = this.consumeArgs(expansion.numArgs); // paste arguments in place of the placeholders
+
+      tokens = tokens.slice(); // make a shallow copy
+
+      for (let i = tokens.length - 1; i >= 0; --i) {
+        let tok = tokens[i];
+
+        if (tok.text === "#") {
+          if (i === 0) {
+            throw new ParseError("Incomplete placeholder at end of macro body", tok);
+          }
+
+          tok = tokens[--i]; // next token on stack
+
+          if (tok.text === "#") {
+            // ## → #
+            tokens.splice(i + 1, 1); // drop first #
+          } else if (/^[1-9]$/.test(tok.text)) {
+            // replace the placeholder with the indicated argument
+            tokens.splice(i, 2, ...args[+tok.text - 1]);
+          } else {
+            throw new ParseError("Not a valid argument number", tok);
+          }
+        }
+      }
+    } // Concatenate expansion onto top of stack.
+
+
+    this.pushTokens(tokens);
+    return tokens;
+  }
+  /**
+   * Expand the next token only once (if possible), and return the resulting
+   * top token on the stack (without removing anything from the stack).
+   * Similar in behavior to TeX's `\expandafter\futurelet`.
+   * Equivalent to expandOnce() followed by future().
+   */
+
+
+  expandAfterFuture() {
+    this.expandOnce();
+    return this.future();
+  }
+  /**
+   * Recursively expand first token, then return first non-expandable token.
+   */
+
+
+  expandNextToken() {
+    for (;;) {
+      const expanded = this.expandOnce(); // expandOnce returns Token if and only if it's fully expanded.
+
+      if (expanded instanceof Token) {
+        // \relax stops the expansion, but shouldn't get returned (a
+        // null return value couldn't get implemented as a function).
+        // the token after \noexpand is interpreted as if its meaning
+        // were ‘\relax’
+        if (expanded.text === "\\relax" || expanded.treatAsRelax) {
+          this.stack.pop();
+        } else {
+          return this.stack.pop(); // === expanded
+        }
+      }
+    } // Flow unable to figure out that this pathway is impossible.
+    // https://github.com/facebook/flow/issues/4808
+
+
+    throw new Error(); // eslint-disable-line no-unreachable
+  }
+  /**
+   * Fully expand the given macro name and return the resulting list of
+   * tokens, or return `undefined` if no such macro is defined.
+   */
+
+
+  expandMacro(name) {
+    return this.macros.has(name) ? this.expandTokens([new Token(name)]) : undefined;
+  }
+  /**
+   * Fully expand the given token stream and return the resulting list of tokens
+   */
+
+
+  expandTokens(tokens) {
+    const output = [];
+    const oldStackLength = this.stack.length;
+    this.pushTokens(tokens);
+
+    while (this.stack.length > oldStackLength) {
+      const expanded = this.expandOnce(true); // expand only expandable tokens
+      // expandOnce returns Token if and only if it's fully expanded.
+
+      if (expanded instanceof Token) {
+        if (expanded.treatAsRelax) {
+          // the expansion of \noexpand is the token itself
+          expanded.noexpand = false;
+          expanded.treatAsRelax = false;
+        }
+
+        output.push(this.stack.pop());
+      }
+    }
+
+    return output;
+  }
+  /**
+   * Fully expand the given macro name and return the result as a string,
+   * or return `undefined` if no such macro is defined.
+   */
+
+
+  expandMacroAsText(name) {
+    const tokens = this.expandMacro(name);
+
+    if (tokens) {
+      return tokens.map(token => token.text).join("");
+    } else {
+      return tokens;
+    }
+  }
+  /**
+   * Returns the expanded macro as a reversed array of tokens and a macro
+   * argument count.  Or returns `null` if no such macro.
+   */
+
+
+  _getExpansion(name) {
+    const definition = this.macros.get(name);
+
+    if (definition == null) {
+      // mainly checking for undefined here
+      return definition;
+    }
+
+    const expansion = typeof definition === "function" ? definition(this) : definition;
+
+    if (typeof expansion === "string") {
+      let numArgs = 0;
+
+      if (expansion.indexOf("#") !== -1) {
+        const stripped = expansion.replace(/##/g, "");
+
+        while (stripped.indexOf("#" + (numArgs + 1)) !== -1) {
+          ++numArgs;
+        }
+      }
+
+      const bodyLexer = new Lexer(expansion, this.settings);
+      const tokens = [];
+      let tok = bodyLexer.lex();
+
+      while (tok.text !== "EOF") {
+        tokens.push(tok);
+        tok = bodyLexer.lex();
+      }
+
+      tokens.reverse(); // to fit in with stack using push and pop
+
+      const expanded = {
+        tokens,
+        numArgs
+      };
+      return expanded;
+    }
+
+    return expansion;
+  }
+  /**
+   * Determine whether a command is currently "defined" (has some
+   * functionality), meaning that it's a macro (in the current group),
+   * a function, a symbol, or one of the special commands listed in
+   * `implicitCommands`.
+   */
+
+
+  isDefined(name) {
+    return this.macros.has(name) || functions.hasOwnProperty(name) || symbols.math.hasOwnProperty(name) || symbols.text.hasOwnProperty(name) || implicitCommands.hasOwnProperty(name);
+  }
+  /**
+   * Determine whether a command is expandable.
+   */
+
+
+  isExpandable(name) {
+    const macro = this.macros.get(name);
+    return macro != null ? typeof macro === "string" || typeof macro === "function" || !macro.unexpandable // TODO(ylem): #2085
+    : functions.hasOwnProperty(name)
+    /* && !functions[name].primitive*/
+    ;
+  }
+
+}
+
+/* eslint no-constant-condition:0 */
+
+const unicodeAccents = {
+  "́": {
+    "text": "\\'",
+    "math": "\\acute"
+  },
+  "̀": {
+    "text": "\\`",
+    "math": "\\grave"
+  },
+  "̈": {
+    "text": "\\\"",
+    "math": "\\ddot"
+  },
+  "̃": {
+    "text": "\\~",
+    "math": "\\tilde"
+  },
+  "̄": {
+    "text": "\\=",
+    "math": "\\bar"
+  },
+  "̆": {
+    "text": "\\u",
+    "math": "\\breve"
+  },
+  "̌": {
+    "text": "\\v",
+    "math": "\\check"
+  },
+  "̂": {
+    "text": "\\^",
+    "math": "\\hat"
+  },
+  "̇": {
+    "text": "\\.",
+    "math": "\\dot"
+  },
+  "̊": {
+    "text": "\\r",
+    "math": "\\mathring"
+  },
+  "̋": {
+    "text": "\\H"
+  }
+};
+const unicodeSymbols = {
+  "á": "á",
+  "à": "à",
+  "ä": "ä",
+  "ǟ": "ǟ",
+  "ã": "ã",
+  "ā": "ā",
+  "ă": "ă",
+  "ắ": "ắ",
+  "ằ": "ằ",
+  "ẵ": "ẵ",
+  "ǎ": "ǎ",
+  "â": "â",
+  "ấ": "ấ",
+  "ầ": "ầ",
+  "ẫ": "ẫ",
+  "ȧ": "ȧ",
+  "ǡ": "ǡ",
+  "å": "å",
+  "ǻ": "ǻ",
+  "ḃ": "ḃ",
+  "ć": "ć",
+  "č": "č",
+  "ĉ": "ĉ",
+  "ċ": "ċ",
+  "ď": "ď",
+  "ḋ": "ḋ",
+  "é": "é",
+  "è": "è",
+  "ë": "ë",
+  "ẽ": "ẽ",
+  "ē": "ē",
+  "ḗ": "ḗ",
+  "ḕ": "ḕ",
+  "ĕ": "ĕ",
+  "ě": "ě",
+  "ê": "ê",
+  "ế": "ế",
+  "ề": "ề",
+  "ễ": "ễ",
+  "ė": "ė",
+  "ḟ": "ḟ",
+  "ǵ": "ǵ",
+  "ḡ": "ḡ",
+  "ğ": "ğ",
+  "ǧ": "ǧ",
+  "ĝ": "ĝ",
+  "ġ": "ġ",
+  "ḧ": "ḧ",
+  "ȟ": "ȟ",
+  "ĥ": "ĥ",
+  "ḣ": "ḣ",
+  "í": "í",
+  "ì": "ì",
+  "ï": "ï",
+  "ḯ": "ḯ",
+  "ĩ": "ĩ",
+  "ī": "ī",
+  "ĭ": "ĭ",
+  "ǐ": "ǐ",
+  "î": "î",
+  "ǰ": "ǰ",
+  "ĵ": "ĵ",
+  "ḱ": "ḱ",
+  "ǩ": "ǩ",
+  "ĺ": "ĺ",
+  "ľ": "ľ",
+  "ḿ": "ḿ",
+  "ṁ": "ṁ",
+  "ń": "ń",
+  "ǹ": "ǹ",
+  "ñ": "ñ",
+  "ň": "ň",
+  "ṅ": "ṅ",
+  "ó": "ó",
+  "ò": "ò",
+  "ö": "ö",
+  "ȫ": "ȫ",
+  "õ": "õ",
+  "ṍ": "ṍ",
+  "ṏ": "ṏ",
+  "ȭ": "ȭ",
+  "ō": "ō",
+  "ṓ": "ṓ",
+  "ṑ": "ṑ",
+  "ŏ": "ŏ",
+  "ǒ": "ǒ",
+  "ô": "ô",
+  "ố": "ố",
+  "ồ": "ồ",
+  "ỗ": "ỗ",
+  "ȯ": "ȯ",
+  "ȱ": "ȱ",
+  "ő": "ő",
+  "ṕ": "ṕ",
+  "ṗ": "ṗ",
+  "ŕ": "ŕ",
+  "ř": "ř",
+  "ṙ": "ṙ",
+  "ś": "ś",
+  "ṥ": "ṥ",
+  "š": "š",
+  "ṧ": "ṧ",
+  "ŝ": "ŝ",
+  "ṡ": "ṡ",
+  "ẗ": "ẗ",
+  "ť": "ť",
+  "ṫ": "ṫ",
+  "ú": "ú",
+  "ù": "ù",
+  "ü": "ü",
+  "ǘ": "ǘ",
+  "ǜ": "ǜ",
+  "ǖ": "ǖ",
+  "ǚ": "ǚ",
+  "ũ": "ũ",
+  "ṹ": "ṹ",
+  "ū": "ū",
+  "ṻ": "ṻ",
+  "ŭ": "ŭ",
+  "ǔ": "ǔ",
+  "û": "û",
+  "ů": "ů",
+  "ű": "ű",
+  "ṽ": "ṽ",
+  "ẃ": "ẃ",
+  "ẁ": "ẁ",
+  "ẅ": "ẅ",
+  "ŵ": "ŵ",
+  "ẇ": "ẇ",
+  "ẘ": "ẘ",
+  "ẍ": "ẍ",
+  "ẋ": "ẋ",
+  "ý": "ý",
+  "ỳ": "ỳ",
+  "ÿ": "ÿ",
+  "ỹ": "ỹ",
+  "ȳ": "ȳ",
+  "ŷ": "ŷ",
+  "ẏ": "ẏ",
+  "ẙ": "ẙ",
+  "ź": "ź",
+  "ž": "ž",
+  "ẑ": "ẑ",
+  "ż": "ż",
+  "Á": "Á",
+  "À": "À",
+  "Ä": "Ä",
+  "Ǟ": "Ǟ",
+  "Ã": "Ã",
+  "Ā": "Ā",
+  "Ă": "Ă",
+  "Ắ": "Ắ",
+  "Ằ": "Ằ",
+  "Ẵ": "Ẵ",
+  "Ǎ": "Ǎ",
+  "Â": "Â",
+  "Ấ": "Ấ",
+  "Ầ": "Ầ",
+  "Ẫ": "Ẫ",
+  "Ȧ": "Ȧ",
+  "Ǡ": "Ǡ",
+  "Å": "Å",
+  "Ǻ": "Ǻ",
+  "Ḃ": "Ḃ",
+  "Ć": "Ć",
+  "Č": "Č",
+  "Ĉ": "Ĉ",
+  "Ċ": "Ċ",
+  "Ď": "Ď",
+  "Ḋ": "Ḋ",
+  "É": "É",
+  "È": "È",
+  "Ë": "Ë",
+  "Ẽ": "Ẽ",
+  "Ē": "Ē",
+  "Ḗ": "Ḗ",
+  "Ḕ": "Ḕ",
+  "Ĕ": "Ĕ",
+  "Ě": "Ě",
+  "Ê": "Ê",
+  "Ế": "Ế",
+  "Ề": "Ề",
+  "Ễ": "Ễ",
+  "Ė": "Ė",
+  "Ḟ": "Ḟ",
+  "Ǵ": "Ǵ",
+  "Ḡ": "Ḡ",
+  "Ğ": "Ğ",
+  "Ǧ": "Ǧ",
+  "Ĝ": "Ĝ",
+  "Ġ": "Ġ",
+  "Ḧ": "Ḧ",
+  "Ȟ": "Ȟ",
+  "Ĥ": "Ĥ",
+  "Ḣ": "Ḣ",
+  "Í": "Í",
+  "Ì": "Ì",
+  "Ï": "Ï",
+  "Ḯ": "Ḯ",
+  "Ĩ": "Ĩ",
+  "Ī": "Ī",
+  "Ĭ": "Ĭ",
+  "Ǐ": "Ǐ",
+  "Î": "Î",
+  "İ": "İ",
+  "Ĵ": "Ĵ",
+  "Ḱ": "Ḱ",
+  "Ǩ": "Ǩ",
+  "Ĺ": "Ĺ",
+  "Ľ": "Ľ",
+  "Ḿ": "Ḿ",
+  "Ṁ": "Ṁ",
+  "Ń": "Ń",
+  "Ǹ": "Ǹ",
+  "Ñ": "Ñ",
+  "Ň": "Ň",
+  "Ṅ": "Ṅ",
+  "Ó": "Ó",
+  "Ò": "Ò",
+  "Ö": "Ö",
+  "Ȫ": "Ȫ",
+  "Õ": "Õ",
+  "Ṍ": "Ṍ",
+  "Ṏ": "Ṏ",
+  "Ȭ": "Ȭ",
+  "Ō": "Ō",
+  "Ṓ": "Ṓ",
+  "Ṑ": "Ṑ",
+  "Ŏ": "Ŏ",
+  "Ǒ": "Ǒ",
+  "Ô": "Ô",
+  "Ố": "Ố",
+  "Ồ": "Ồ",
+  "Ỗ": "Ỗ",
+  "Ȯ": "Ȯ",
+  "Ȱ": "Ȱ",
+  "Ő": "Ő",
+  "Ṕ": "Ṕ",
+  "Ṗ": "Ṗ",
+  "Ŕ": "Ŕ",
+  "Ř": "Ř",
+  "Ṙ": "Ṙ",
+  "Ś": "Ś",
+  "Ṥ": "Ṥ",
+  "Š": "Š",
+  "Ṧ": "Ṧ",
+  "Ŝ": "Ŝ",
+  "Ṡ": "Ṡ",
+  "Ť": "Ť",
+  "Ṫ": "Ṫ",
+  "Ú": "Ú",
+  "Ù": "Ù",
+  "Ü": "Ü",
+  "Ǘ": "Ǘ",
+  "Ǜ": "Ǜ",
+  "Ǖ": "Ǖ",
+  "Ǚ": "Ǚ",
+  "Ũ": "Ũ",
+  "Ṹ": "Ṹ",
+  "Ū": "Ū",
+  "Ṻ": "Ṻ",
+  "Ŭ": "Ŭ",
+  "Ǔ": "Ǔ",
+  "Û": "Û",
+  "Ů": "Ů",
+  "Ű": "Ű",
+  "Ṽ": "Ṽ",
+  "Ẃ": "Ẃ",
+  "Ẁ": "Ẁ",
+  "Ẅ": "Ẅ",
+  "Ŵ": "Ŵ",
+  "Ẇ": "Ẇ",
+  "Ẍ": "Ẍ",
+  "Ẋ": "Ẋ",
+  "Ý": "Ý",
+  "Ỳ": "Ỳ",
+  "Ÿ": "Ÿ",
+  "Ỹ": "Ỹ",
+  "Ȳ": "Ȳ",
+  "Ŷ": "Ŷ",
+  "Ẏ": "Ẏ",
+  "Ź": "Ź",
+  "Ž": "Ž",
+  "Ẑ": "Ẑ",
+  "Ż": "Ż",
+  "ά": "ά",
+  "ὰ": "ὰ",
+  "ᾱ": "ᾱ",
+  "ᾰ": "ᾰ",
+  "έ": "έ",
+  "ὲ": "ὲ",
+  "ή": "ή",
+  "ὴ": "ὴ",
+  "ί": "ί",
+  "ὶ": "ὶ",
+  "ϊ": "ϊ",
+  "ΐ": "ΐ",
+  "ῒ": "ῒ",
+  "ῑ": "ῑ",
+  "ῐ": "ῐ",
+  "ό": "ό",
+  "ὸ": "ὸ",
+  "ύ": "ύ",
+  "ὺ": "ὺ",
+  "ϋ": "ϋ",
+  "ΰ": "ΰ",
+  "ῢ": "ῢ",
+  "ῡ": "ῡ",
+  "ῠ": "ῠ",
+  "ώ": "ώ",
+  "ὼ": "ὼ",
+  "Ύ": "Ύ",
+  "Ὺ": "Ὺ",
+  "Ϋ": "Ϋ",
+  "Ῡ": "Ῡ",
+  "Ῠ": "Ῠ",
+  "Ώ": "Ώ",
+  "Ὼ": "Ὼ"
+};
+
+/**
+ * This file contains the parser used to parse out a TeX expression from the
+ * input. Since TeX isn't context-free, standard parsers don't work particularly
+ * well.
+ *
+ * The strategy of this parser is as such:
+ *
+ * The main functions (the `.parse...` ones) take a position in the current
+ * parse string to parse tokens from. The lexer (found in Lexer.js, stored at
+ * this.gullet.lexer) also supports pulling out tokens at arbitrary places. When
+ * individual tokens are needed at a position, the lexer is called to pull out a
+ * token, which is then used.
+ *
+ * The parser has a property called "mode" indicating the mode that
+ * the parser is currently in. Currently it has to be one of "math" or
+ * "text", which denotes whether the current environment is a math-y
+ * one or a text-y one (e.g. inside \text). Currently, this serves to
+ * limit the functions which can be used in text mode.
+ *
+ * The main functions then return an object which contains the useful data that
+ * was parsed at its given point, and a new position at the end of the parsed
+ * data. The main functions can call each other and continue the parsing by
+ * using the returned position as a new starting point.
+ *
+ * There are also extra `.handle...` functions, which pull out some reused
+ * functionality into self-contained functions.
+ *
+ * The functions return ParseNodes.
+ */
+class Parser {
+  constructor(input, settings) {
+    this.mode = void 0;
+    this.gullet = void 0;
+    this.settings = void 0;
+    this.leftrightDepth = void 0;
+    this.nextToken = void 0;
+    // Start in math mode
+    this.mode = "math"; // Create a new macro expander (gullet) and (indirectly via that) also a
+    // new lexer (mouth) for this parser (stomach, in the language of TeX)
+
+    this.gullet = new MacroExpander(input, settings, this.mode); // Store the settings for use in parsing
+
+    this.settings = settings; // Count leftright depth (for \middle errors)
+
+    this.leftrightDepth = 0;
+  }
+  /**
+   * Checks a result to make sure it has the right type, and throws an
+   * appropriate error otherwise.
+   */
+
+
+  expect(text, consume) {
+    if (consume === void 0) {
+      consume = true;
+    }
+
+    if (this.fetch().text !== text) {
+      throw new ParseError(`Expected '${text}', got '${this.fetch().text}'`, this.fetch());
+    }
+
+    if (consume) {
+      this.consume();
+    }
+  }
+  /**
+   * Discards the current lookahead token, considering it consumed.
+   */
+
+
+  consume() {
+    this.nextToken = null;
+  }
+  /**
+   * Return the current lookahead token, or if there isn't one (at the
+   * beginning, or if the previous lookahead token was consume()d),
+   * fetch the next token as the new lookahead token and return it.
+   */
+
+
+  fetch() {
+    if (this.nextToken == null) {
+      this.nextToken = this.gullet.expandNextToken();
+    }
+
+    return this.nextToken;
+  }
+  /**
+   * Switches between "text" and "math" modes.
+   */
+
+
+  switchMode(newMode) {
+    this.mode = newMode;
+    this.gullet.switchMode(newMode);
+  }
+  /**
+   * Main parsing function, which parses an entire input.
+   */
+
+
+  parse() {
+    if (!this.settings.globalGroup) {
+      // Create a group namespace for the math expression.
+      // (LaTeX creates a new group for every $...$, $$...$$, \[...\].)
+      this.gullet.beginGroup();
+    } // Use old \color behavior (same as LaTeX's \textcolor) if requested.
+    // We do this within the group for the math expression, so it doesn't
+    // pollute settings.macros.
+
+
+    if (this.settings.colorIsTextColor) {
+      this.gullet.macros.set("\\color", "\\textcolor");
+    } // Try to parse the input
+
+
+    const parse = this.parseExpression(false); // If we succeeded, make sure there's an EOF at the end
+
+    this.expect("EOF"); // End the group namespace for the expression
+
+    if (!this.settings.globalGroup) {
+      this.gullet.endGroup();
+    }
+
+    return parse;
+  }
+
+  parseExpression(breakOnInfix, breakOnTokenText) {
+    const body = []; // Keep adding atoms to the body until we can't parse any more atoms (either
+    // we reached the end, a }, or a \right)
+
+    while (true) {
+      // Ignore spaces in math mode
+      if (this.mode === "math") {
+        this.consumeSpaces();
+      }
+
+      const lex = this.fetch();
+
+      if (Parser.endOfExpression.indexOf(lex.text) !== -1) {
+        break;
+      }
+
+      if (breakOnTokenText && lex.text === breakOnTokenText) {
+        break;
+      }
+
+      if (breakOnInfix && functions[lex.text] && functions[lex.text].infix) {
+        break;
+      }
+
+      const atom = this.parseAtom(breakOnTokenText);
+
+      if (!atom) {
+        break;
+      } else if (atom.type === "internal") {
+        continue;
+      }
+
+      body.push(atom);
+    }
+
+    if (this.mode === "text") {
+      this.formLigatures(body);
+    }
+
+    return this.handleInfixNodes(body);
+  }
+  /**
+   * Rewrites infix operators such as \over with corresponding commands such
+   * as \frac.
+   *
+   * There can only be one infix operator per group.  If there's more than one
+   * then the expression is ambiguous.  This can be resolved by adding {}.
+   */
+
+
+  handleInfixNodes(body) {
+    let overIndex = -1;
+    let funcName;
+
+    for (let i = 0; i < body.length; i++) {
+      if (body[i].type === "infix") {
+        if (overIndex !== -1) {
+          throw new ParseError("only one infix operator per group", body[i].token);
+        }
+
+        overIndex = i;
+        funcName = body[i].replaceWith;
+      }
+    }
+
+    if (overIndex !== -1 && funcName) {
+      let numerNode;
+      let denomNode;
+      const numerBody = body.slice(0, overIndex);
+      const denomBody = body.slice(overIndex + 1);
+
+      if (numerBody.length === 1 && numerBody[0].type === "ordgroup") {
+        numerNode = numerBody[0];
+      } else {
+        numerNode = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: numerBody
+        };
+      }
+
+      if (denomBody.length === 1 && denomBody[0].type === "ordgroup") {
+        denomNode = denomBody[0];
+      } else {
+        denomNode = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: denomBody
+        };
+      }
+
+      let node;
+
+      if (funcName === "\\\\abovefrac") {
+        node = this.callFunction(funcName, [numerNode, body[overIndex], denomNode], []);
+      } else {
+        node = this.callFunction(funcName, [numerNode, denomNode], []);
+      }
+
+      return [node];
+    } else {
+      return body;
+    }
+  } // The greediness of a superscript or subscript
+
+
+  /**
+   * Handle a subscript or superscript with nice errors.
+   */
+  handleSupSubscript(name) {
+    const symbolToken = this.fetch();
+    const symbol = symbolToken.text;
+    this.consume();
+    const group = this.parseGroup(name, false, Parser.SUPSUB_GREEDINESS, undefined, undefined, true); // ignore spaces before sup/subscript argument
+
+    if (!group) {
+      throw new ParseError("Expected group after '" + symbol + "'", symbolToken);
+    }
+
+    return group;
+  }
+  /**
+   * Converts the textual input of an unsupported command into a text node
+   * contained within a color node whose color is determined by errorColor
+   */
+
+
+  formatUnsupportedCmd(text) {
+    const textordArray = [];
+
+    for (let i = 0; i < text.length; i++) {
+      textordArray.push({
+        type: "textord",
+        mode: "text",
+        text: text[i]
+      });
+    }
+
+    const textNode = {
+      type: "text",
+      mode: this.mode,
+      body: textordArray
+    };
+    const colorNode = {
+      type: "color",
+      mode: this.mode,
+      color: this.settings.errorColor,
+      body: [textNode]
+    };
+    return colorNode;
+  }
+  /**
+   * Parses a group with optional super/subscripts.
+   */
+
+
+  parseAtom(breakOnTokenText) {
+    // The body of an atom is an implicit group, so that things like
+    // \left(x\right)^2 work correctly.
+    const base = this.parseGroup("atom", false, null, breakOnTokenText); // In text mode, we don't have superscripts or subscripts
+
+    if (this.mode === "text") {
+      return base;
+    } // Note that base may be empty (i.e. null) at this point.
+
+
+    let superscript;
+    let subscript;
+
+    while (true) {
+      // Guaranteed in math mode, so eat any spaces first.
+      this.consumeSpaces(); // Lex the first token
+
+      const lex = this.fetch();
+
+      if (lex.text === "\\limits" || lex.text === "\\nolimits") {
+        // We got a limit control
+        if (base && base.type === "op") {
+          const limits = lex.text === "\\limits";
+          base.limits = limits;
+          base.alwaysHandleSupSub = true;
+        } else if (base && base.type === "operatorname" && base.alwaysHandleSupSub) {
+          const limits = lex.text === "\\limits";
+          base.limits = limits;
+        } else {
+          throw new ParseError("Limit controls must follow a math operator", lex);
+        }
+
+        this.consume();
+      } else if (lex.text === "^") {
+        // We got a superscript start
+        if (superscript) {
+          throw new ParseError("Double superscript", lex);
+        }
+
+        superscript = this.handleSupSubscript("superscript");
+      } else if (lex.text === "_") {
+        // We got a subscript start
+        if (subscript) {
+          throw new ParseError("Double subscript", lex);
+        }
+
+        subscript = this.handleSupSubscript("subscript");
+      } else if (lex.text === "'") {
+        // We got a prime
+        if (superscript) {
+          throw new ParseError("Double superscript", lex);
+        }
+
+        const prime = {
+          type: "textord",
+          mode: this.mode,
+          text: "\\prime"
+        }; // Many primes can be grouped together, so we handle this here
+
+        const primes = [prime];
+        this.consume(); // Keep lexing tokens until we get something that's not a prime
+
+        while (this.fetch().text === "'") {
+          // For each one, add another prime to the list
+          primes.push(prime);
+          this.consume();
+        } // If there's a superscript following the primes, combine that
+        // superscript in with the primes.
+
+
+        if (this.fetch().text === "^") {
+          primes.push(this.handleSupSubscript("superscript"));
+        } // Put everything into an ordgroup as the superscript
+
+
+        superscript = {
+          type: "ordgroup",
+          mode: this.mode,
+          body: primes
+        };
+      } else {
+        // If it wasn't ^, _, or ', stop parsing super/subscripts
+        break;
+      }
+    } // Base must be set if superscript or subscript are set per logic above,
+    // but need to check here for type check to pass.
+
+
+    if (superscript || subscript) {
+      // If we got either a superscript or subscript, create a supsub
+      return {
+        type: "supsub",
+        mode: this.mode,
+        base: base,
+        sup: superscript,
+        sub: subscript
+      };
+    } else {
+      // Otherwise return the original body
+      return base;
+    }
+  }
+  /**
+   * Parses an entire function, including its base and all of its arguments.
+   */
+
+
+  parseFunction(breakOnTokenText, name, // For error reporting.
+  greediness) {
+    const token = this.fetch();
+    const func = token.text;
+    const funcData = functions[func];
+
+    if (!funcData) {
+      return null;
+    }
+
+    this.consume(); // consume command token
+
+    if (greediness != null && funcData.greediness <= greediness) {
+      throw new ParseError("Got function '" + func + "' with no arguments" + (name ? " as " + name : ""), token);
+    } else if (this.mode === "text" && !funcData.allowedInText) {
+      throw new ParseError("Can't use function '" + func + "' in text mode", token);
+    } else if (this.mode === "math" && funcData.allowedInMath === false) {
+      throw new ParseError("Can't use function '" + func + "' in math mode", token);
+    }
+
+    const _this$parseArguments = this.parseArguments(func, funcData),
+          args = _this$parseArguments.args,
+          optArgs = _this$parseArguments.optArgs;
+
+    return this.callFunction(func, args, optArgs, token, breakOnTokenText);
+  }
+  /**
+   * Call a function handler with a suitable context and arguments.
+   */
+
+
+  callFunction(name, args, optArgs, token, breakOnTokenText) {
+    const context = {
+      funcName: name,
+      parser: this,
+      token,
+      breakOnTokenText
+    };
+    const func = functions[name];
+
+    if (func && func.handler) {
+      return func.handler(context, args, optArgs);
+    } else {
+      throw new ParseError(`No function handler for ${name}`);
+    }
+  }
+  /**
+   * Parses the arguments of a function or environment
+   */
+
+
+  parseArguments(func, // Should look like "\name" or "\begin{name}".
+  funcData) {
+    const totalArgs = funcData.numArgs + funcData.numOptionalArgs;
+
+    if (totalArgs === 0) {
+      return {
+        args: [],
+        optArgs: []
+      };
+    }
+
+    const baseGreediness = funcData.greediness;
+    const args = [];
+    const optArgs = [];
+
+    for (let i = 0; i < totalArgs; i++) {
+      const argType = funcData.argTypes && funcData.argTypes[i];
+      const isOptional = i < funcData.numOptionalArgs; // Ignore spaces between arguments.  As the TeXbook says:
+      // "After you have said ‘\def\row#1#2{...}’, you are allowed to
+      //  put spaces between the arguments (e.g., ‘\row x n’), because
+      //  TeX doesn’t use single spaces as undelimited arguments."
+
+      const consumeSpaces = i > 0 && !isOptional || // Also consume leading spaces in math mode, as parseSymbol
+      // won't know what to do with them.  This can only happen with
+      // macros, e.g. \frac\foo\foo where \foo expands to a space symbol.
+      // In LaTeX, the \foo's get treated as (blank) arguments.
+      // In KaTeX, for now, both spaces will get consumed.
+      // TODO(edemaine)
+      i === 0 && !isOptional && this.mode === "math";
+      const arg = this.parseGroupOfType(`argument to '${func}'`, argType, isOptional, baseGreediness, consumeSpaces);
+
+      if (!arg) {
+        if (isOptional) {
+          optArgs.push(null);
+          continue;
+        }
+
+        throw new ParseError(`Expected group after '${func}'`, this.fetch());
+      }
+
+      (isOptional ? optArgs : args).push(arg);
+    }
+
+    return {
+      args,
+      optArgs
+    };
+  }
+  /**
+   * Parses a group when the mode is changing.
+   */
+
+
+  parseGroupOfType(name, type, optional, greediness, consumeSpaces) {
+    switch (type) {
+      case "color":
+        if (consumeSpaces) {
+          this.consumeSpaces();
+        }
+
+        return this.parseColorGroup(optional);
+
+      case "size":
+        if (consumeSpaces) {
+          this.consumeSpaces();
+        }
+
+        return this.parseSizeGroup(optional);
+
+      case "url":
+        return this.parseUrlGroup(optional, consumeSpaces);
+
+      case "math":
+      case "text":
+        return this.parseGroup(name, optional, greediness, undefined, type, consumeSpaces);
+
+      case "hbox":
+        {
+          // hbox argument type wraps the argument in the equivalent of
+          // \hbox, which is like \text but switching to \textstyle size.
+          const group = this.parseGroup(name, optional, greediness, undefined, "text", consumeSpaces);
+
+          if (!group) {
+            return group;
+          }
+
+          const styledGroup = {
+            type: "styling",
+            mode: group.mode,
+            body: [group],
+            style: "text" // simulate \textstyle
+
+          };
+          return styledGroup;
+        }
+
+      case "raw":
+        {
+          if (consumeSpaces) {
+            this.consumeSpaces();
+          }
+
+          if (optional && this.fetch().text === "{") {
+            return null;
+          }
+
+          const token = this.parseStringGroup("raw", optional, true);
+
+          if (token) {
+            return {
+              type: "raw",
+              mode: "text",
+              string: token.text
+            };
+          } else {
+            throw new ParseError("Expected raw group", this.fetch());
+          }
+        }
+
+      case "original":
+      case null:
+      case undefined:
+        return this.parseGroup(name, optional, greediness, undefined, undefined, consumeSpaces);
+
+      default:
+        throw new ParseError("Unknown group type as " + name, this.fetch());
+    }
+  }
+  /**
+   * Discard any space tokens, fetching the next non-space token.
+   */
+
+
+  consumeSpaces() {
+    while (this.fetch().text === " ") {
+      this.consume();
+    }
+  }
+  /**
+   * Parses a group, essentially returning the string formed by the
+   * brace-enclosed tokens plus some position information.
+   */
+
+
+  parseStringGroup(modeName, // Used to describe the mode in error messages.
+  optional, raw) {
+    const groupBegin = optional ? "[" : "{";
+    const groupEnd = optional ? "]" : "}";
+    const beginToken = this.fetch();
+
+    if (beginToken.text !== groupBegin) {
+      if (optional) {
+        return null;
+      } else if (raw && beginToken.text !== "EOF" && /[^{}[\]]/.test(beginToken.text)) {
+        this.consume();
+        return beginToken;
+      }
+    }
+
+    const outerMode = this.mode;
+    this.mode = "text";
+    this.expect(groupBegin);
+    let str = "";
+    const firstToken = this.fetch();
+    let nested = 0; // allow nested braces in raw string group
+
+    let lastToken = firstToken;
+    let nextToken;
+
+    while ((nextToken = this.fetch()).text !== groupEnd || raw && nested > 0) {
+      switch (nextToken.text) {
+        case "EOF":
+          throw new ParseError("Unexpected end of input in " + modeName, firstToken.range(lastToken, str));
+
+        case groupBegin:
+          nested++;
+          break;
+
+        case groupEnd:
+          nested--;
+          break;
+      }
+
+      lastToken = nextToken;
+      str += lastToken.text;
+      this.consume();
+    }
+
+    this.expect(groupEnd);
+    this.mode = outerMode;
+    return firstToken.range(lastToken, str);
+  }
+  /**
+   * Parses a regex-delimited group: the largest sequence of tokens
+   * whose concatenated strings match `regex`. Returns the string
+   * formed by the tokens plus some position information.
+   */
+
+
+  parseRegexGroup(regex, modeName) {
+    const outerMode = this.mode;
+    this.mode = "text";
+    const firstToken = this.fetch();
+    let lastToken = firstToken;
+    let str = "";
+    let nextToken;
+
+    while ((nextToken = this.fetch()).text !== "EOF" && regex.test(str + nextToken.text)) {
+      lastToken = nextToken;
+      str += lastToken.text;
+      this.consume();
+    }
+
+    if (str === "") {
+      throw new ParseError("Invalid " + modeName + ": '" + firstToken.text + "'", firstToken);
+    }
+
+    this.mode = outerMode;
+    return firstToken.range(lastToken, str);
+  }
+  /**
+   * Parses a color description.
+   */
+
+
+  parseColorGroup(optional) {
+    const res = this.parseStringGroup("color", optional);
+
+    if (!res) {
+      return null;
+    }
+
+    const match = /^(#[a-f0-9]{3}|#?[a-f0-9]{6}|[a-z]+)$/i.exec(res.text);
+
+    if (!match) {
+      throw new ParseError("Invalid color: '" + res.text + "'", res);
+    }
+
+    let color = match[0];
+
+    if (/^[0-9a-f]{6}$/i.test(color)) {
+      // We allow a 6-digit HTML color spec without a leading "#".
+      // This follows the xcolor package's HTML color model.
+      // Predefined color names are all missed by this RegEx pattern.
+      color = "#" + color;
+    }
+
+    return {
+      type: "color-token",
+      mode: this.mode,
+      color
+    };
+  }
+  /**
+   * Parses a size specification, consisting of magnitude and unit.
+   */
+
+
+  parseSizeGroup(optional) {
+    let res;
+    let isBlank = false;
+
+    if (!optional && this.fetch().text !== "{") {
+      res = this.parseRegexGroup(/^[-+]? *(?:$|\d+|\d+\.\d*|\.\d*) *[a-z]{0,2} *$/, "size");
+    } else {
+      res = this.parseStringGroup("size", optional);
+    }
+
+    if (!res) {
+      return null;
+    }
+
+    if (!optional && res.text.length === 0) {
+      // Because we've tested for what is !optional, this block won't
+      // affect \kern, \hspace, etc. It will capture the mandatory arguments
+      // to \genfrac and \above.
+      res.text = "0pt"; // Enable \above{}
+
+      isBlank = true; // This is here specifically for \genfrac
+    }
+
+    const match = /([-+]?) *(\d+(?:\.\d*)?|\.\d+) *([a-z]{2})/.exec(res.text);
+
+    if (!match) {
+      throw new ParseError("Invalid size: '" + res.text + "'", res);
+    }
+
+    const data = {
+      number: +(match[1] + match[2]),
+      // sign + magnitude, cast to number
+      unit: match[3]
+    };
+
+    if (!validUnit(data)) {
+      throw new ParseError("Invalid unit: '" + data.unit + "'", res);
+    }
+
+    return {
+      type: "size",
+      mode: this.mode,
+      value: data,
+      isBlank
+    };
+  }
+  /**
+   * Parses an URL, checking escaped letters and allowed protocols,
+   * and setting the catcode of % as an active character (as in \hyperref).
+   */
+
+
+  parseUrlGroup(optional, consumeSpaces) {
+    this.gullet.lexer.setCatcode("%", 13); // active character
+
+    const res = this.parseStringGroup("url", optional, true); // get raw string
+
+    this.gullet.lexer.setCatcode("%", 14); // comment character
+
+    if (!res) {
+      return null;
+    } // hyperref package allows backslashes alone in href, but doesn't
+    // generate valid links in such cases; we interpret this as
+    // "undefined" behaviour, and keep them as-is. Some browser will
+    // replace backslashes with forward slashes.
+
+
+    const url = res.text.replace(/\\([#$%&~_^{}])/g, '$1');
+    return {
+      type: "url",
+      mode: this.mode,
+      url
+    };
+  }
+  /**
+   * If `optional` is false or absent, this parses an ordinary group,
+   * which is either a single nucleus (like "x") or an expression
+   * in braces (like "{x+y}") or an implicit group, a group that starts
+   * at the current position, and ends right before a higher explicit
+   * group ends, or at EOF.
+   * If `optional` is true, it parses either a bracket-delimited expression
+   * (like "[x+y]") or returns null to indicate the absence of a
+   * bracket-enclosed group.
+   * If `mode` is present, switches to that mode while parsing the group,
+   * and switches back after.
+   */
+
+
+  parseGroup(name, // For error reporting.
+  optional, greediness, breakOnTokenText, mode, consumeSpaces) {
+    // Switch to specified mode
+    const outerMode = this.mode;
+
+    if (mode) {
+      this.switchMode(mode);
+    } // Consume spaces if requested, crucially *after* we switch modes,
+    // so that the next non-space token is parsed in the correct mode.
+
+
+    if (consumeSpaces) {
+      this.consumeSpaces();
+    } // Get first token
+
+
+    const firstToken = this.fetch();
+    const text = firstToken.text;
+    let result; // Try to parse an open brace or \begingroup
+
+    if (optional ? text === "[" : text === "{" || text === "\\begingroup") {
+      this.consume();
+      const groupEnd = Parser.endOfGroup[text]; // Start a new group namespace
+
+      this.gullet.beginGroup(); // If we get a brace, parse an expression
+
+      const expression = this.parseExpression(false, groupEnd);
+      const lastToken = this.fetch(); // Check that we got a matching closing brace
+
+      this.expect(groupEnd); // End group namespace
+
+      this.gullet.endGroup();
+      result = {
+        type: "ordgroup",
+        mode: this.mode,
+        loc: SourceLocation.range(firstToken, lastToken),
+        body: expression,
+        // A group formed by \begingroup...\endgroup is a semi-simple group
+        // which doesn't affect spacing in math mode, i.e., is transparent.
+        // https://tex.stackexchange.com/questions/1930/when-should-one-
+        // use-begingroup-instead-of-bgroup
+        semisimple: text === "\\begingroup" || undefined
+      };
+    } else if (optional) {
+      // Return nothing for an optional group
+      result = null;
+    } else {
+      // If there exists a function with this name, parse the function.
+      // Otherwise, just return a nucleus
+      result = this.parseFunction(breakOnTokenText, name, greediness) || this.parseSymbol();
+
+      if (result == null && text[0] === "\\" && !implicitCommands.hasOwnProperty(text)) {
+        if (this.settings.throwOnError) {
+          throw new ParseError("Undefined control sequence: " + text, firstToken);
+        }
+
+        result = this.formatUnsupportedCmd(text);
+        this.consume();
+      }
+    } // Switch mode back
+
+
+    if (mode) {
+      this.switchMode(outerMode);
+    }
+
+    return result;
+  }
+  /**
+   * Form ligature-like combinations of characters for text mode.
+   * This includes inputs like "--", "---", "``" and "''".
+   * The result will simply replace multiple textord nodes with a single
+   * character in each value by a single textord node having multiple
+   * characters in its value.  The representation is still ASCII source.
+   * The group will be modified in place.
+   */
+
+
+  formLigatures(group) {
+    let n = group.length - 1;
+
+    for (let i = 0; i < n; ++i) {
+      const a = group[i]; // $FlowFixMe: Not every node type has a `text` property.
+
+      const v = a.text;
+
+      if (v === "-" && group[i + 1].text === "-") {
+        if (i + 1 < n && group[i + 2].text === "-") {
+          group.splice(i, 3, {
+            type: "textord",
+            mode: "text",
+            loc: SourceLocation.range(a, group[i + 2]),
+            text: "---"
+          });
+          n -= 2;
+        } else {
+          group.splice(i, 2, {
+            type: "textord",
+            mode: "text",
+            loc: SourceLocation.range(a, group[i + 1]),
+            text: "--"
+          });
+          n -= 1;
+        }
+      }
+
+      if ((v === "'" || v === "`") && group[i + 1].text === v) {
+        group.splice(i, 2, {
+          type: "textord",
+          mode: "text",
+          loc: SourceLocation.range(a, group[i + 1]),
+          text: v + v
+        });
+        n -= 1;
+      }
+    }
+  }
+  /**
+   * Parse a single symbol out of the string. Here, we handle single character
+   * symbols and special functions like \verb.
+   */
+
+
+  parseSymbol() {
+    const nucleus = this.fetch();
+    let text = nucleus.text;
+
+    if (/^\\verb[^a-zA-Z]/.test(text)) {
+      this.consume();
+      let arg = text.slice(5);
+      const star = arg.charAt(0) === "*";
+
+      if (star) {
+        arg = arg.slice(1);
+      } // Lexer's tokenRegex is constructed to always have matching
+      // first/last characters.
+
+
+      if (arg.length < 2 || arg.charAt(0) !== arg.slice(-1)) {
+        throw new ParseError(`\\verb assertion failed --
+                    please report what input caused this bug`);
+      }
+
+      arg = arg.slice(1, -1); // remove first and last char
+
+      return {
+        type: "verb",
+        mode: "text",
+        body: arg,
+        star
+      };
+    } // At this point, we should have a symbol, possibly with accents.
+    // First expand any accented base symbol according to unicodeSymbols.
+
+
+    if (unicodeSymbols.hasOwnProperty(text[0]) && !symbols[this.mode][text[0]]) {
+      // This behavior is not strict (XeTeX-compatible) in math mode.
+      if (this.settings.strict && this.mode === "math") {
+        this.settings.reportNonstrict("unicodeTextInMathMode", `Accented Unicode text character "${text[0]}" used in ` + `math mode`, nucleus);
+      }
+
+      text = unicodeSymbols[text[0]] + text.substr(1);
+    } // Strip off any combining characters
+
+
+    const match = combiningDiacriticalMarksEndRegex.exec(text);
+
+    if (match) {
+      text = text.substring(0, match.index);
+
+      if (text === 'i') {
+        text = '\u0131'; // dotless i, in math and text mode
+      } else if (text === 'j') {
+        text = '\u0237'; // dotless j, in math and text mode
+      }
+    } // Recognize base symbol
+
+
+    let symbol;
+
+    if (symbols[this.mode][text]) {
+      if (this.settings.strict && this.mode === 'math' && extraLatin.indexOf(text) >= 0) {
+        this.settings.reportNonstrict("unicodeTextInMathMode", `Latin-1/Unicode text character "${text[0]}" used in ` + `math mode`, nucleus);
+      }
+
+      const group = symbols[this.mode][text].group;
+      const loc = SourceLocation.range(nucleus);
+      let s;
+
+      if (ATOMS.hasOwnProperty(group)) {
+        // $FlowFixMe
+        const family = group;
+        s = {
+          type: "atom",
+          mode: this.mode,
+          family,
+          loc,
+          text
+        };
+      } else {
+        // $FlowFixMe
+        s = {
+          type: group,
+          mode: this.mode,
+          loc,
+          text
+        };
+      }
+
+      symbol = s;
+    } else if (text.charCodeAt(0) >= 0x80) {
+      // no symbol for e.g. ^
+      if (this.settings.strict) {
+        if (!supportedCodepoint(text.charCodeAt(0))) {
+          this.settings.reportNonstrict("unknownSymbol", `Unrecognized Unicode character "${text[0]}"` + ` (${text.charCodeAt(0)})`, nucleus);
+        } else if (this.mode === "math") {
+          this.settings.reportNonstrict("unicodeTextInMathMode", `Unicode text character "${text[0]}" used in math mode`, nucleus);
+        }
+      } // All nonmathematical Unicode characters are rendered as if they
+      // are in text mode (wrapped in \text) because that's what it
+      // takes to render them in LaTeX.  Setting `mode: this.mode` is
+      // another natural choice (the user requested math mode), but
+      // this makes it more difficult for getCharacterMetrics() to
+      // distinguish Unicode characters without metrics and those for
+      // which we want to simulate the letter M.
+
+
+      symbol = {
+        type: "textord",
+        mode: "text",
+        loc: SourceLocation.range(nucleus),
+        text
+      };
+    } else {
+      return null; // EOF, ^, _, {, }, etc.
+    }
+
+    this.consume(); // Transform combining characters into accents
+
+    if (match) {
+      for (let i = 0; i < match[0].length; i++) {
+        const accent = match[0][i];
+
+        if (!unicodeAccents[accent]) {
+          throw new ParseError(`Unknown accent ' ${accent}'`, nucleus);
+        }
+
+        const command = unicodeAccents[accent][this.mode];
+
+        if (!command) {
+          throw new ParseError(`Accent ${accent} unsupported in ${this.mode} mode`, nucleus);
+        }
+
+        symbol = {
+          type: "accent",
+          mode: this.mode,
+          loc: SourceLocation.range(nucleus),
+          label: command,
+          isStretchy: false,
+          isShifty: true,
+          base: symbol
+        };
+      }
+    }
+
+    return symbol;
+  }
+
+}
+Parser.endOfExpression = ["}", "\\endgroup", "\\end", "\\right", "&"];
+Parser.endOfGroup = {
+  "[": "]",
+  "{": "}",
+  "\\begingroup": "\\endgroup"
+  /**
+   * Parses an "expression", which is a list of atoms.
+   *
+   * `breakOnInfix`: Should the parsing stop when we hit infix nodes? This
+   *                 happens when functions have higher precendence han infix
+   *                 nodes in implicit parses.
+   *
+   * `breakOnTokenText`: The text of the token that the expression should end
+   *                     with, or `null` if something else should end the
+   *                     expression.
+   */
+
+};
+Parser.SUPSUB_GREEDINESS = 1;
+
+/**
+ * Provides a single function for parsing an expression using a Parser
+ * TODO(emily): Remove this
+ */
+
+/**
+ * Parses an expression using a Parser, then returns the parsed result.
+ */
+const parseTree = function parseTree(toParse, settings) {
+  if (!(typeof toParse === 'string' || toParse instanceof String)) {
+    throw new TypeError('KaTeX can only parse string typed expression');
+  }
+
+  const parser = new Parser(toParse, settings); // Blank out any \df@tag to avoid spurious "Duplicate \tag" errors
+
+  delete parser.gullet.macros.current["\\df@tag"];
+  let tree = parser.parse(); // If the input used \tag, it will set the \df@tag macro to the tag.
+  // In this case, we separately parse the tag and wrap the tree.
+
+  if (parser.gullet.macros.get("\\df@tag")) {
+    if (!settings.displayMode) {
+      throw new ParseError("\\tag works only in display equations");
+    }
+
+    parser.gullet.feed("\\df@tag");
+    tree = [{
+      type: "tag",
+      mode: "text",
+      body: tree,
+      tag: parser.parse()
+    }];
+  }
+
+  return tree;
+};
+
+/* eslint no-console:0 */
+
+/**
+ * Parse and build an expression, and place that expression in the DOM node
+ * given.
+ */
+let render = function render(expression, baseNode, options) {
+  baseNode.textContent = "";
+  const node = renderToDomTree(expression, options).toNode();
+  baseNode.appendChild(node);
+}; // KaTeX's styles don't work properly in quirks mode. Print out an error, and
+// disable rendering.
+
+
+if (typeof document !== "undefined") {
+  if (document.compatMode !== "CSS1Compat") {
+    typeof console !== "undefined" && console.warn("Warning: KaTeX doesn't work in quirks mode. Make sure your " + "website has a suitable doctype.");
+
+    render = function render() {
+      throw new ParseError("KaTeX doesn't work in quirks mode.");
+    };
+  }
+}
+/**
+ * Parse and build an expression, and return the markup for that.
+ */
+
+
+const renderToString = function renderToString(expression, options) {
+  const markup = renderToDomTree(expression, options).toMarkup();
+  return markup;
+};
+/**
+ * Parse an expression and return the parse tree.
+ */
+
+
+const generateParseTree = function generateParseTree(expression, options) {
+  const settings = new Settings(options);
+  return parseTree(expression, settings);
+};
+/**
+ * If the given error is a KaTeX ParseError and options.throwOnError is false,
+ * renders the invalid LaTeX as a span with hover title giving the KaTeX
+ * error message.  Otherwise, simply throws the error.
+ */
+
+
+const renderError = function renderError(error, expression, options) {
+  if (options.throwOnError || !(error instanceof ParseError)) {
+    throw error;
+  }
+
+  const node = buildCommon.makeSpan(["katex-error"], [new SymbolNode(expression)]);
+  node.setAttribute("title", error.toString());
+  node.setAttribute("style", `color:${options.errorColor}`);
+  return node;
+};
+/**
+ * Generates and returns the katex build tree. This is used for advanced
+ * use cases (like rendering to custom output).
+ */
+
+
+const renderToDomTree = function renderToDomTree(expression, options) {
+  const settings = new Settings(options);
+
+  try {
+    const tree = parseTree(expression, settings);
+    return buildTree(tree, expression, settings);
+  } catch (error) {
+    return renderError(error, expression, settings);
+  }
+};
+/**
+ * Generates and returns the katex build tree, with just HTML (no MathML).
+ * This is used for advanced use cases (like rendering to custom output).
+ */
+
+
+const renderToHTMLTree = function renderToHTMLTree(expression, options) {
+  const settings = new Settings(options);
+
+  try {
+    const tree = parseTree(expression, settings);
+    return buildHTMLTree(tree, expression, settings);
+  } catch (error) {
+    return renderError(error, expression, settings);
+  }
+};
+
+var katex = {
+  /**
+   * Current KaTeX version
+   */
+  version: "0.12.0",
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML combination, and adds
+   * it as a child to the specified DOM node.
+   */
+  render,
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML combination string,
+   * for sending to the client.
+   */
+  renderToString,
+
+  /**
+   * KaTeX error, usually during parsing.
+   */
+  ParseError,
+
+  /**
+   * Parses the given LaTeX into KaTeX's internal parse tree structure,
+   * without rendering to HTML or MathML.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __parse: generateParseTree,
+
+  /**
+   * Renders the given LaTeX into an HTML+MathML internal DOM tree
+   * representation, without flattening that representation to a string.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __renderToDomTree: renderToDomTree,
+
+  /**
+   * Renders the given LaTeX into an HTML internal DOM tree representation,
+   * without MathML and without flattening that representation to a string.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __renderToHTMLTree: renderToHTMLTree,
+
+  /**
+   * extends internal font metrics object with a new object
+   * each key in the new object represents a font name
+  */
+  __setFontMetrics: setFontMetrics,
+
+  /**
+   * adds a new symbol to builtin symbols table
+   */
+  __defineSymbol: defineSymbol,
+
+  /**
+   * adds a new macro to builtin macro list
+   */
+  __defineMacro: defineMacro,
+
+  /**
+   * Expose the dom tree node types, which can be useful for type checking nodes.
+   *
+   * NOTE: This method is not currently recommended for public use.
+   * The internal tree representation is unstable and is very likely
+   * to change. Use at your own risk.
+   */
+  __domTree: {
+    Span,
+    Anchor,
+    SymbolNode,
+    SvgNode,
+    PathNode,
+    LineNode
+  }
+};
+
+export default katex;
diff --git a/layouts/partials/docs/inject/content-after.html b/layouts/partials/docs/inject/content-after.html
index a03d168a..4309b441 100644
--- a/layouts/partials/docs/inject/content-after.html
+++ b/layouts/partials/docs/inject/content-after.html
@@ -1,3 +1,3 @@
 <script>window.onload = function() {
     window.scrollTo(0,document.body.scrollHeight);
-};</script>
\ No newline at end of file
+};</script>
diff --git a/layouts/shortcodes/katex.html b/layouts/shortcodes/katex.html
index 8bcdb791..a1c2a798 100644
--- a/layouts/shortcodes/katex.html
+++ b/layouts/shortcodes/katex.html
@@ -2,47 +2,49 @@
 <!-- Include katext only first time -->
 <link rel="stylesheet" href="{{ "katex/katex.min.css" | relURL }}" />
 <script defer src="{{ "katex/katex.min.js" | relURL }}"></script>
-<script defer src="{{ "katex/auto-render.min.js" | relURL }}"></script>
-<script>document.addEventListener("DOMContentLoaded", function() {
-  renderMathInElement(document.body, {
-    delimiters: [
-    {left: "$$", right: "$$", display: true},
-    {left: "\\[", right: "\\]", display: true},
-    {left: "$", right: "$", display: false},
-    {left: "\\(", right: "\\)", display: false} ],
-    macros: {
-      // Algebra-Precal
-      "\\lcm": "\\mathop{\\mathrm{lcm}}",
-      "\\sen": '\\text{sen}\\,',
-      // Calculus
-      "\\dd": "\\mathop{\\mathrm{d} #1}",
-      "\\abs": "\\lvert #1 \\rvert",
-      // Physics
-      "\\dd": "\\text{d}",
-      "\\cross" : "\\times",
-      "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
-      "\curl": "\\nabla \\cross #1",
-      "\\vu": "\\mathbf{\\hat{#1}}",
-      "\\dot": "\\boldsymbol\\cdot",
-      "\\vdot" : "\\dot",
-      "\\div": "\\nabla \\dot #1",
-      "\\grad" : "\\nabla",
-      "\\dv" : "\\frac{d\\,#1}{d\\,#2}",
-      "\\gr" : "\\gamma",
-      "\\vec" : "\\mathbf{#1}",
-      "\\tagl" : "\\tag{#1}",
-      "\\eqref" : "(#1)",
-      "\\laplacian" : "\\mathbf{\\nabla ^2}",
-      "\\label" : "{}",
-      "\\negadelta" : "\\nabla",
-      }
-  });
- });</script>
+<script defer src="{{ "katex/auto-render.min.js" | relURL }}" onload="loadKatex();"></script>
+
 {{- .Page.Scratch.Set "katex" true -}}
 {{- end -}}
 
-<span {{- with .Get "class" }} class="{{ . }}"{{ end }}>
+<span {{- with .Get "class" }} class="{{ . }}" {{ end }}>
   {{ cond (in .Params "display") "\\[" "\\(" -}}
   {{- trim .Inner "\n" -}}
   {{- cond (in .Params "display") "\\]" "\\)" }}
 </span>
+<script>
+  function loadKatex() {
+    renderMathInElement(document.body, {
+      delimiters: [
+        { left: "$$", right: "$$", display: true },
+        { left: "\\[", right: "\\]", display: true },
+        { left: "$", right: "$", display: false },
+        { left: "\\(", right: "\\)", display: false }],
+      macros: {
+        // Algebra-Precal
+        "\\lcm": "\\mathop{\\mathrm{lcm}}",
+        "\\sen": '\\text{sen}\\,',
+        // Calculus
+        "\\dd": "\\mathop{\\mathrm{d} #1}",
+        "\\abs": "\\lvert #1 \\rvert",
+        // Physics
+        "\\dd": "\\text{d}",
+        "\\cross": "\\times",
+        "\\pdv": "\\frac{\\partial #1}{\\partial #2}",
+        "\\curl": "\\nabla \\cross #1",
+        "\\vu": "\\mathbf{\\hat{#1}}",
+        "\\vdot": "\\dot",
+        "\\div": "\\nabla \\cdot #1",
+        "\\grad": "\\nabla",
+        "\\dv": "\\frac{d\\,#1}{d\\,#2}",
+        "\\gr": "\\gamma",
+        "\\vec": "\\mathbf{#1}",
+        "\\tagl": "\\tag{#1}",
+        "\\eqref": "(#1)",
+        "\\laplacian": "\\mathbf{\\nabla ^2}",
+        "\\label": "{}",
+        "\\negadelta": "\\nabla",
+      }
+    });
+  };
+</script>
\ No newline at end of file
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