Deploy

app/globals.css

@@ -104,7 +104,12 @@
   }
 
   article {
-    counter-reset: fig axiom theorem proposition criterion lemma corollary definition example;
+    counter-reset: fig axiom theorem proposition criterion lemma corollary definition example equation;
+  }
+
+  span[id]::before {
+    counter-increment: equation;
+    content: counter(equation);
   }
 
   pre {

content/notes/math/complex_analysis.mdx

@@ -11,13 +11,13 @@ showToc: true
 An $\varepsilon$ neighbourhood of a complex number $z_0 \in\mathbb{C}$, also called an $\varepsilon$-ball, is an open subset of all points lying inside a circle centered at $z_0$ and with radius $\varepsilon > 0$
 
 $$
-  B_\varepsilon (z_0) = \Set{ z \in\mathbb{C} : \lvert z - z_0 \rvert < \varepsilon }
+  B_\varepsilon (z_0) = \Set{z \in\mathbb{C} : \lvert z - z_0 \rvert < \varepsilon}
 $$
 
 The closed $\varepsilon$ neighbourhood of $z_0$ is the closure of $B_\varepsilon$, including the boundary points 
 
 $$
-  \overline{B}_\varepsilon (z_0) = \Set{ z \in\mathbb{C} : \lvert z - z_0 \rvert < \varepsilon }
+  \overline{B}_\varepsilon (z_0) = \Set{z \in\mathbb{C} : \lvert z - z_0 \rvert < \varepsilon}
 $$
 
 ## Riemann sphere

content/notes/math/differential_geometry.mdx

@@ -2626,6 +2626,83 @@ $$
 
 # Geodesics
 
+## Christoffel symbols
+
+### Covariant derivative along a curve
+
+Let $c:[a,b]\to M$ be a smooth parametrized curve in a smooth $n$-manifold $M$. Recall that a vector field along the curve $c$ in $M$ is a function
+
+$$
+  V:[a,b]\to\bigsqcup_{t\in[a,b]} T_{c(t)} M
+$$
+
+such that $V(t) \in T_{c(t)}M$. Such a vector field $V(t)$ is smooth if for any smooth function $f:M\to\R$ on $M$, the composition $V(t)f$ is smooth as a function of $t$. The vector space of all smooth vector fields along the curve $c$ is denoted $\Gamma(TM|_{c(t)})$.
+
+For a smooth vector field $V(t) = \sum_{i=1}^n v^i(t) \frac{\partial}{\partial x^i}$ along a smooth curve $c(t)$ in $\R^n$, the derivative $\frac{\mathrm{d}V}{\mathrm{d}t} = \sum_{i=1}^n \dot{v}^i (t) \frac{\partial}{\partial x^i}$ satisfies the following properties
+1. $\frac{\mathrm{d}V}{\mathrm{d}t}$ is $\R$-linear in $V$
+2. **Leibniz rule:** for any smooth function $f$ on $[a,b]$
+$$
+  \frac{\mathrm{d}(fV)}{\mathrm{d}t} = \frac{\mathrm{d}f}{\mathrm{d}t}V + f\frac{\mathrm{d}V}{\mathrm{d}t}
+$$
+3. if $V$ is induced from a smooth vector field $\tilde{V}$ on $\R^n$, in the sense that $V(t) = \tilde{V}_{c(t)}$ and $\mathrm{D}$ is the directional derivative in $\R^n$, then
+$$
+  \frac{\mathrm{d}V}{\mathrm{d}t} = \mathrm{D}_{c'(t)}\tilde{V}
+$$
+
+It turns out that to every connection $\nabla$ on a manifold $M$ one can associate a way of differentiating vector fields along a curve having the same properties as $\frac{\mathrm{d}V}{\mathrm{d}t}$.
+
+<MathBox title='Covariant derivative of vector fields along curves' boxType='theorem'>
+Let $M$ be a smooth manifold with an affine connection $\nabla$ and $c:[a,b]\to M$ a smooth curve in $M$. Then there is a unique map
+
+$$
+  \frac{\mathrm{D}}{\mathrm{d}t}: \Gamma(TM|_{c(t)})\to\Gamma(TM|_{c(t)})
+$$
+
+such that for $V\in\Gamma(TM|_{c(t)})$
+1. $\frac{\mathrm{D}V}{\mathrm{d}t}$ is $\R$-linear in $V$
+2. **Leibniz rule:** for any smooth function $f:[a,b]\to\R$
+$$
+  \frac{\mathrm{D}(fV)}{\mathrm{d}t} = \frac{\mathrm{d}f}{\mathrm{d}t}V + f\frac{\mathrm{D}V}{\mathrm{d}t}
+$$
+3. **Compatibility with the connection:** if $V$ is induced from a smooth vector field $\tilde{V}$ on $M$, in the sense that $V(t) = \tilde{V}_{c(t)}$, then
+$$
+  \frac{\mathrm{D}V}{\mathrm{d}t}(t) = \nabla_{c'(t)} \tilde{V} 
+$$
+
+We call $\frac{\mathrm{D}V}{\mathrm{d}t}$ the *covariant derivative* (associated to $\nabla$) of the vector field $V$ along the curve $c$ in $M$.
+
+<details>
+<summary>Proof</summary>
+
+To prove uniqueness, suppose such a covariant derivative $\frac{\mathrm{D}}{\mathrm{d}t}$ exists. On an open set $U$ with a frame $\Set{\mathbf{e}_i}_{i=1}^n$, a vector field $V(t)$ along $c$ can written as a linear combination
+
+$$
+  V(t) = \sum_{i=1}^n v^i (t)\mathbf{e}_{i,c(t)}
+$$
+
+Then
+
+$$
+\begin{align*}
+  \frac{\mathrm{D}}{\mathrm{d}t} =& \sum_{i=1}^n \frac{\mathrm{D}}{\mathrm{d}t} (v^i(t)\mathbf{e}_{i,c(t)}) &\text{[by (1)]} \\
+  =& \sum_{i=1}^n \frac{\mathrm{d}v^i}{\mathrm{d}t}\mathbf{e}_{i,c(t)} + v^i\frac{\mathrm{D}\mathbf{e}_{i,c(t)}}{\mathrm{d}t} &\text{[by (2)]} \\
+  =& \sum_{i=1}^n \frac{\mathrm{d}v^i}{\mathrm{d}t}\mathbf{e}_{i,c(t)} + v^i \nabla_{c'(t)}\mathbf{e}_{i,c(t)} &\text{[by (3)]}\tag{\label{eq-1}}
+\end{align*}
+$$
+
+This formula proves the uniqueness of $\frac{\mathrm{D}}{\mathrm{d}t}$ if it exists.
+
+As for existence, we define $\frac{\mathrm{D}V}{\mathrm{d}t}$ for a curve $c$ in a framed open set $U$ by the formula above. It is easily verified that $\frac{\mathrm{D}V}{\mathrm{d}t}$ satisfiess the three properties of the theorem. Hence, $\frac{\mathrm{D}}{\mathrm{d}t}$ exists for curves in $U$. If $\Set{\bar{\mathbf{e}}_i}_{i=1}^n$ is another frame on $U$, then $V(t)$ is a linear combination $\sum_{i=1}^n \bar{v}^i (t) \bar{\mathbf{e}}_{i,c(t)}$ and the covariant derivative $\frac{\bar{\mathrm{D}}V}{\mathrm{d}t}$ defined by
+
+$$
+  \frac{\bar{\mathrm{D}}V}{\mathrm{d}t} = \sum_{i=1}^n \frac{\mathrm{d}\bar{v}^i}{\mathrm{d}t}\mathbf{e}_{i,c(t)} + \bar{v}^i \nabla_{c'(t)}\mathbf{e}_{i,c(t)}
+$$
+
+also satifes the three properties of the theorem. By the uniqueness of the covariant derivative, $\frac{\mathrm{D}V}{\mathrm{d}t} = \frac{\bar{\mathrm{D}}V}{\mathrm{d}t}$. This proves that the covariant derivative $\frac{\mathrm{D}V}{\mathrm{d}t}$ is independent of the frame. By covering $M$ with framed open sets, $\eqref{1}$ then defines a covariant derivative $\frac{\mathrm{D}V}{\mathrm{d}t}$ for the curve $c(t)$ in $M$.
+
+</details>
+</MathBox>
+
 # Partial derivative (basis vector)
 
 Partial derivatives represent basis vectors

utils/mdxParse.tsx

@@ -25,7 +25,14 @@ export const rehypePlugins = [
 	rehypeAutolinkHeadings, 
 	[rehypeImgSize, {dir: 'public'}],
 	rehypePrettyCode,
-	rehypeKatex,
+	[rehypeKatex, {
+		trust: (context) => ['\\htmlId', '\\href'].includes(context.command),
+		macros: {
+			"\\eqref": "\\href{###1}{(\\text{#1})}",
+			"\\ref": "\\href{###1}{\\text{#1}}",
+			"\\label": "\\htmlId{#1}{}"
+		}
+	}]
 	//[rehypeMathjax, {
 	//	loader: {
   //    load: ['[custom]/xypic.js'],