From 5a4eb1d13110048b3714754817b3f38d7a55970b Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 14 Jun 2023 20:25:38 +0200
Subject: Improve knowledge base

---
 source/know/concept/spherical-coordinates/index.md | 31 ++++++++++------------
 1 file changed, 14 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 7f6d111..1607b61 100644
--- a/source/know/concept/spherical-coordinates/index.md
+++ b/source/know/concept/spherical-coordinates/index.md
@@ -22,8 +22,8 @@ Note that this is the standard notation among physicists,
 but mathematicians often switch the definitions of $$\theta$$ and $$\varphi$$,
 while still writing $$(r, \theta, \varphi)$$.
 
-Cartesian coordinates $$(x, y, z)$$ and the spherical system
-$$(r, \theta, \varphi)$$ are related by:
+[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$
+and the spherical system $$(r, \theta, \varphi)$$ are related by:
 
 $$\begin{aligned}
     \boxed{
@@ -114,8 +114,6 @@ using the standard formulae for orthogonal curvilinear coordinates.
 
 
 
-
-
 ## Differential elements
 
 For line integrals,
@@ -169,7 +167,7 @@ $$\begin{aligned}
 $$\begin{aligned}
     \boxed{
         \nabla \cdot \vb{V}
-        = \pdv{V_r}{r} + \frac{2}{r} V_r
+        = \pdv{V_r}{r} + \frac{2 V_r}{r}
         + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_\theta}{r \tan{\theta}}
         + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi}
     }
@@ -216,15 +214,15 @@ $$\begin{aligned}
         \begin{aligned}
             \nabla (\nabla \cdot \vb{V})
             &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\theta}{r}{\theta} + \frac{1}{r \sin{\theta}} \mpdv{V_\varphi}{\varphi}{r}
-            + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta}
             \\
-            &\qquad\qquad - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
+            &\qquad\qquad + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta}
+            - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
             + \frac{1}{r \tan{\theta}} \pdv{V_\theta}{r} - \frac{2 V_r}{r^2} - \frac{V_\theta}{r^2 \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r} \mpdv{V_r}{\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
-            + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta}
+            + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi}
             \\
-            &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
+            &\qquad\qquad + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
             - \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r \sin{\theta}} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\theta}{\varphi}{\theta}
@@ -275,23 +273,22 @@ $$\begin{aligned}
         \begin{aligned}
             \nabla^2 \vb{V}
             &= \quad\: \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_r}{\varphi}
-            + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta}
             \\
-            &\qquad\qquad - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
+            &\qquad\qquad + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta}
+            - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
             - \frac{2 V_r}{r^2} - \frac{2 V_\theta}{r^2 \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( \pdvn{2}{V_\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
-            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r}
+            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi}
             \\
-            &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
+            &\qquad\qquad + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
             - \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\theta}
-            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi} + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi}
+            + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi}
             \\
-            &\qquad\qquad + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
-            + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\varphi}{\theta}
-            - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg)
+            &\qquad\qquad + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
+            + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\varphi}{\theta} - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
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