From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/spherical-coordinates/index.md | 34 +++++++++++-----------
 1 file changed, 17 insertions(+), 17 deletions(-)

(limited to 'source/know/concept/spherical-coordinates')

diff --git a/source/know/concept/spherical-coordinates/index.md b/source/know/concept/spherical-coordinates/index.md
index bc01c95..9df5d51 100644
--- a/source/know/concept/spherical-coordinates/index.md
+++ b/source/know/concept/spherical-coordinates/index.md
@@ -10,16 +10,16 @@ layout: "concept"
 
 **Spherical coordinates** are an extension of polar coordinates to 3D.
 The position of a given point in space is described by
-three coordinates $(r, \theta, \varphi)$, defined as:
+three coordinates $$(r, \theta, \varphi)$$, defined as:
 
-*   $r$: the **radius** or **radial distance**: distance to the origin.
-*   $\theta$: the **elevation**, **polar angle** or **colatitude**:
-    angle to the positive $z$-axis, or **zenith**, i.e. the "north pole".
-*   $\varphi$: the **azimuth**, **azimuthal angle** or **longitude**:
-    angle from the positive $x$-axis, typically in the counter-clockwise sense.
+*   $$r$$: the **radius** or **radial distance**: distance to the origin.
+*   $$\theta$$: the **elevation**, **polar angle** or **colatitude**:
+    angle to the positive $$z$$-axis, or **zenith**, i.e. the "north pole".
+*   $$\varphi$$: the **azimuth**, **azimuthal angle** or **longitude**:
+    angle from the positive $$x$$-axis, typically in the counter-clockwise sense.
 
-Cartesian coordinates $(x, y, z)$ and the spherical system
-$(r, \theta, \varphi)$ are related by:
+Cartesian coordinates $$(x, y, z)$$ and the spherical system
+$$(r, \theta, \varphi)$$ are related by:
 
 $$\begin{aligned}
     \boxed{
@@ -31,8 +31,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Conversely, a point given in $(x, y, z)$
-can be converted to $(r, \theta, \varphi)$
+Conversely, a point given in $$(x, y, z)$$
+can be converted to $$(r, \theta, \varphi)$$
 using these formulae:
 
 $$\begin{aligned}
@@ -47,7 +47,7 @@ $$\begin{aligned}
 
 The spherical coordinate system is an orthogonal
 [curvilinear system](/know/concept/curvilinear-coordinates/),
-whose scale factors $h_r$, $h_\theta$ and $h_\varphi$ we want to find.
+whose scale factors $$h_r$$, $$h_\theta$$ and $$h_\varphi$$ we want to find.
 To do so, we calculate the differentials of the Cartesian coordinates:
 
 $$\begin{aligned}
@@ -58,7 +58,7 @@ $$\begin{aligned}
     \dd{z} &= \dd{r} \cos\theta - \dd{\theta} r \sin\theta
 \end{aligned}$$
 
-And then we calculate the line element $\dd{\ell}^2$,
+And then we calculate the line element $$\dd{\ell}^2$$,
 skipping many terms thanks to orthogonality:
 
 $$\begin{aligned}
@@ -74,7 +74,7 @@ $$\begin{aligned}
 
 Finally, we can simply read off
 the squares of the desired scale factors
-$h_r^2$, $h_\theta^2$ and $h_\varphi^2$:
+$$h_r^2$$, $$h_\theta^2$$ and $$h_\varphi^2$$:
 
 $$\begin{aligned}
     \boxed{
@@ -146,7 +146,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The differential element of volume $\dd{V}$
+The differential element of volume $$\dd{V}$$
 takes the following form:
 
 $$\begin{aligned}
@@ -163,7 +163,7 @@ $$\begin{aligned}
     = \int_0^{2\pi} \int_0^\pi \int_0^\infty f(r, \theta, \varphi) \: r^2 \sin\theta \dd{r} \dd{\theta} \dd{\varphi}
 \end{aligned}$$
 
-The isosurface elements are as follows, where $S_r$ is a surface at constant $r$, etc.:
+The isosurface elements are as follows, where $$S_r$$ is a surface at constant $$r$$, etc.:
 
 $$\begin{aligned}
     \boxed{
@@ -177,7 +177,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Similarly, the normal vector element $\dd{\vu{S}}$ for an arbitrary surface is given by:
+Similarly, the normal vector element $$\dd{\vu{S}}$$ for an arbitrary surface is given by:
 
 $$\begin{aligned}
     \boxed{
@@ -188,7 +188,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-And finally, the tangent vector element $\dd{\vu{\ell}}$ of a given curve is as follows:
+And finally, the tangent vector element $$\dd{\vu{\ell}}$$ of a given curve is as follows:
 
 $$\begin{aligned}
     \boxed{
-- 
cgit v1.2.3