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.vscode/keybindings.json

@@ -1,7 +1,23 @@
 [
+  {
+    "key": "altgr+4",
+    "command": "editor.action.insertSnippet",
+    "args": {
+      "snippet": "$ $"
+    },
+    "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'"
+  },
   {
     "key": "ctrl+1",
     "command": "editor.action.insertSnippet",
+    "args": {
+      "snippet": "\\mathbf{}"
+    },
+    "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'"
+  },
+  {
+    "key": "ctrl+shift+1",
+    "command": "editor.action.insertSnippet",
     "args": {
       "snippet": "\\boldsymbol{}"
     },
@@ -32,11 +48,19 @@
     "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'"
   },
   {
-    "key": "ctrl+|",
+    "key": "ctrl+5",
     "command": "editor.action.insertSnippet",
     "args": {
       "snippet": "<details>\n<summary>Proof</summary>\n\n</details>"
     },
     "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'"
+  },
+  {
+    "key": "ctrl+6",
+    "command": "editor.action.insertSnippet",
+    "args": {
+      "snippet": "```math\n\n```"
+    },
+    "when": "config.workspaceKeybindings.latex.enabled && editorTextFocus && editorLangId == 'mdx'"
   }
 ]

README.md

@@ -1,4 +1,9 @@
 ## Run Typescript with Node
 
 - Add `"type": "module"` to `package.json`
-- Run .ts files with `tsx script.ts` 
+- Run .ts files with `tsx script.ts` 
+
+## VS Code regex replacements
+
+\$\$((.|\n)*?)\$\$
+```math\n$1\n```

content/notes/math/functional_analysis.mdx

@@ -23,7 +23,7 @@ Let $(X, d)$ be a metric space. The open ball about a point $x\in X$ with radius
 
 $$
   B_\epsilon (x) := \Set{ y \in X | d(x, y) < \epsilon }
-$$ 
+$$
 </MathBox>
 
 <MathBox title='Open and closed sets' boxType='definition'>

content/notes/math/logic.mdx

@@ -90,7 +90,7 @@ Arguments in subjective logic are called subjected opinions, commonly shortened
 
 $$
   \omega_X^A = (b_X, u_X, a_X)
-$$ 
+$$
 
 where
 - $b_X:\mathcal{R}(\Omega)\to[0,1]$ is a belief mass distribution over the possible state values of $X$, defined on the reduced power set $\mathcal{R}(\Omega) := \mathcal{P}(\Omega)\setminus\Set{\Omega, \emptyset}$

content/notes/math/real_analysis.mdx

@@ -729,7 +729,7 @@ Let $f, g: I \subseteq \R \to \R$ be differentiable at $x_0 \in I$. Then $f \cdo
 
 $$
   (f \cdot g)'\left(x_0\right) = f\left(x_0\right) g'\left(x_0\right) + f'\left(x_0\right) g\left(x_0\right)
-$$ 
+$$
 
 <details>
 <summary>Proof</summary>

content/notes/math/statistics.mdx

@@ -898,7 +898,7 @@ Suppose $C(\mathbf{x})$ is a $1 - \alpha$ level confidence set for $\theta$. Con
 
 $$
   H_0:\theta = \theta_0 \textrm{ versus }H_1:\theta\neq\theta_0
-$$ 
+$$
 
 The following test has significance level $\alpha$ for the hypothesis: Reject $H_0$ if and only if $\theta_0\notin C(\mathbf{x})$.
 

content/notes/math/stochastic_processes.mdx

@@ -3306,7 +3306,7 @@ Recall that the positive part of $x\in\R$ is
 
 $$
   x^+ = x\vee 0 = \begin{cases} x,\quad x > 0 \\ 0,\quad x \leq 0 \end{cases}
-$$ 
+$$
 
 <MathBox title='' boxType='corollary'>
 Suppose $\mathbf{X}$ is a sub-martingale. For $t\in T$, 

content/notes/math/tensor_analysis.mdx

@@ -2242,7 +2242,7 @@ $$
   &= \left[ \partial_a \left( \Gamma_{bc}^i \right) - \partial_b \left( \Gamma_{ac}^d \right) + \Gamma_{bc}^i \Gamma_{ai}^d - \Gamma_{ac}^j \Gamma_{bj}^d \right] \boldsymbol{\partial}_d \\
   &= R_{cab}^d \boldsymbol{\partial}_d
 \end{align*}
-$$ 
+$$
 
 ## Symmetries
 

content/notes/physics/electromagnetism.mdx

@@ -2033,7 +2033,7 @@ $$
   k = \frac{\omega}{c} \\
   E_x = cB_y \quad E_y = - cB_x
 \end{gather*}
-$$ 
+$$
 
 implying that the waves travel at speed $c$ and are nondispersive. Assuming vacuum in the transmission line, the field amplitudes correspond to the static fields for an infinite line charge and and infinite straigth current
 

content/notes/physics/quantum_mechanics.mdx

@@ -54,7 +54,7 @@ The Hermitian conjugate of an operator $\hat{A}$ is defined as
 
 $$
   \langle \varphi | \hat{A} \psi\rangle = \langle \hat{A}^\dagger \varphi | \psi \rangle
-$$ 
+$$
 
 or alternatively 
 

content/notes/physics/spacetime_physics.mdx

@@ -8,9 +8,9 @@ showToc: true
 
 | Form | Multivector expansion | Terms |
 | --- | :-: | --- |
-| Graded | $M = \alpha + v + \mathbf{F} + wI + \beta I$ | $$\begin{aligned} \alpha&:\textrm{ scalar} \\ v&:\textrm{ vector} \\ \mathbf{F}&:\textrm{ bivector} \\ wI&:\textrm{ pseudovector} \\ \beta I&:\textrm{ pseudoscalar} \end{aligned}$$ |
-| Complex | $M = \zeta + z + \mathbf{F}$ | $$\begin{aligned} \zeta = \alpha + \beta I&:\textrm{ complex scalar} \\ z = v + wI&:\textrm{ complex vector} \\ \mathbf{F} = \mathbf{f}e^{\varphi I}&:\textrm{ complex bivector} \end{aligned}$$ |
-| Relative | $M = \left[ \left(\alpha + \vec{E} \right) + \left( \delta + \mathbf{p} \right)\gamma_0 \right] + \left[\left( \beta + \vec{B} \right) + \left( \omega + \mathbf{a} \right)\gamma_0 \right]I$ | $$\begin{aligned} \alpha&:\textrm{ proper scalar} \\ \vec{E}&:\textrm{ relative polar 3-vector part of }\mathbf{F} \\ \delta&:\textrm{ relative scalar part of }v \\ \mathbf{p}&:\textrm{ relative polar 3-vector part of }v \\ \beta I&:\textrm{ proper pseudoscalar} \\ \vec{B}&:\textrm{ relative axial 3-vector part of }\mathbf{F} \\ \omega I&:\textrm{ relative pseudoscalar part of }wI \\ \mathbf{a}&:\textrm{ relative axial 3-vector part of }wI \end{aligned}$$ |
+| Graded | $M = \alpha + v + \mathbf{F} + wI + \beta I$ | $$\begin{aligned} \alpha&:\textrm{ scalar} \\ v&:\textrm{ vector} \\ \mathbf{F}&:\textrm{ bivector} \\ wI&:\textrm{ pseudovector} \\ \beta I&:\textrm{ pseudoscalar} \end{aligned}$$|
+| Complex | $M = \zeta + z + \mathbf{F}$ | $$\begin{aligned} \zeta = \alpha + \beta I&:\textrm{ complex scalar} \\ z = v + wI&:\textrm{ complex vector} \\ \mathbf{F} = \mathbf{f}e^{\varphi I}&:\textrm{ complex bivector} \end{aligned}$$|
+| Relative | $M = \left[ \left(\alpha + \vec{E} \right) + \left( \delta + \mathbf{p} \right)\gamma_0 \right] + \left[\left( \beta + \vec{B} \right) + \left( \omega + \mathbf{a} \right)\gamma_0 \right]I$ | $$\begin{aligned} \alpha&:\textrm{ proper scalar} \\ \vec{E}&:\textrm{ relative polar 3-vector part of }\mathbf{F} \\ \delta&:\textrm{ relative scalar part of }v \\ \mathbf{p}&:\textrm{ relative polar 3-vector part of }v \\ \beta I&:\textrm{ proper pseudoscalar} \\ \vec{B}&:\textrm{ relative axial 3-vector part of }\mathbf{F} \\ \omega I&:\textrm{ relative pseudoscalar part of }wI \\ \mathbf{a}&:\textrm{ relative axial 3-vector part of }wI \end{aligned}$$|
 
 ## Spacetime
 The principle of invariant light speed, which toghether with the principle of relativity form the postulates of special relativity, predicates that the scalar time $t$ and vector spatial coordinates $\mathbf{x} := \left(x_1, x_2, x_3\right)$ form an invariant interval $(ct)^2 - \mathbf{x}^2$ under coordinate transformations. This motivates the concept of spacetime in which physical quantities are described by 4-vectors $x = (ct, \mathbf{x}) = (ct, x_1, x_2, x_3)\in\R^{1,3}$ combining the scaled time $ct$ with the spatial component $\mathbf{x}$. For compatibility with the principle of invariant light speed, the squared norm of a 4-vector $x\in\R^{1,3}$ is defined to equal the invariant interval, i.e. $|x|^2 = (ct)^2 - \mathbf{x}^2$. This invariant interval can be generalized into a bilinear form, $\eta: \R^{1,3}\to\R^{1,3}\to\mathbf{R}$ with $\eta = \mathrm{diag}(1, -1, -1, -1)$, called the Minkowski metric (inner product). For two 4-vectors $u = (u_0, u_1, u_2, u_3), v = (v_0, v_1, v_2, v_3) \in\R^{1,3}$ this metric has the form
@@ -137,7 +137,7 @@ where $R := mn$ defines a rotor satisfying $R\tilde{R} = \tilde{R} = 1$.
 | Grade | Orthonormal basis | Blade type | Geometry |
 | :-: | :-: | --- | --- |
 | 4 | $\color{red}{\gamma_{0123}} = I$ | pseudoscalars | 4-volumes |
-| 3 | $$\begin{aligned}\color{red}{\gamma_{123}} &= \gamma_0 I \\ \color{blue}{\gamma_{230}} &= \gamma_1 I \\ \color{blue}{\gamma_{310}} &= \gamma_2 I \\ \color{blue}{\gamma_{120}} &= \gamma_3 I\end{aligned}$$ | pseudovectors | 3-volumes |
+| 3 | $$\begin{aligned}\color{red}{\gamma_{123}} &= \gamma_0 I \\ \color{blue}{\gamma_{230}} &= \gamma_1 I \\ \color{blue}{\gamma_{310}} &= \gamma_2 I \\ \color{blue}{\gamma_{120}} &= \gamma_3 I\end{aligned}$$| pseudovectors | 3-volumes |
 | 2 | $\color{red}{\gamma_{10}}\;\color{red}{\gamma_{20}}\;\color{red}{\gamma_{30}}\;\color{blue}{\gamma_{23}}\;\color{blue}{\gamma_{31}}\;\color{blue}{\gamma_{12}}$ | bivectors | planes |
 | 1 | $\color{blue}{\gamma_{0}}\;\color{red}{\gamma_{1}}\;\color{red}{\gamma_{2}}\;\color{red}{\gamma_{3}}$ | vectors | lines |
 | 0 | $\color{blue}{1}$ | scalars | points |
@@ -254,7 +254,7 @@ Any trivector $\mathfrak{F}$ can be written as a product of a vector and the pse
 
 $$
   \mathfrak{F} = f_0\gamma_{123} + f_1\gamma_{230} + f_2\gamma_{310} + f_3\gamma_{120} = \left(\sum_{\mu=0}^3 f_\mu \gamma_\mu\right) I = fI
-$$ 
+$$
 
 where $f$ is a vector. The expanded form of a multivector can thus be reformulated as
 

content/notes/physics/special_relativity.mdx

@@ -221,7 +221,7 @@ where $\beta = \frac{v}{c}$ and
 
 $$
   \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1 - \beta^2}}
-$$ 
+$$
 
 is the Lorentz factor. In matrix form, the Lorentz boost is written as