Improve knowledge base

1 files changed, 19 insertions, 18 deletions

diff --git a/source/know/concept/spherical-coordinates/index.md b/source/know/concept/spherical-coordinates/index.md
index 01c5a61..7f6d111 100644
--- a/source/know/concept/spherical-coordinates/index.md
+++ b/source/know/concept/spherical-coordinates/index.md
@@ -170,7 +170,7 @@ $$\begin{aligned}
     \boxed{
         \nabla \cdot \vb{V}
         = \pdv{V_r}{r} + \frac{2}{r} V_r
-        + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{\cot{\theta}}{r} V_\theta
+        + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_\theta}{r \tan{\theta}}
         + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi}
     }
 \end{aligned}$$
@@ -179,7 +179,7 @@ $$\begin{aligned}
     \boxed{
         \begin{aligned}
             \nabla \times \vb{V}
-            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{\cot{\theta}}{r} V_\varphi
+            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{V_\varphi}{r \tan{\theta}}
             - \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi}
@@ -195,7 +195,7 @@ $$\begin{aligned}
     \boxed{
         \nabla^2 f
         = \pdvn{2}{f}{r} + \frac{2}{r} \pdv{f}{r}
-        + \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{\cot{\theta}}{r^2} \pdv{f}{\theta}
+        + \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{f}{\theta}
         + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{f}{\varphi}
     }
 \end{aligned}$$
@@ -219,12 +219,12 @@ $$\begin{aligned}
             + \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}
-            + \frac{\cot{\theta}}{r} \pdv{V_\theta}{r} - \frac{2}{r^2} V_r - \frac{\cot{\theta}}{r^2} V_\theta \bigg)
+            + \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}
             \\
-            &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta}
+            &\qquad\qquad + \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}
@@ -246,10 +246,10 @@ $$\begin{aligned}
             + \vu{e}_\theta \vu{e}_\varphi \frac{1}{r} \pdv{V_\varphi}{\theta}
             \\
             &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
-            + \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{\cot{\theta}}{r} V_\varphi \bigg)
+            + \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{V_\varphi}{r \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \vu{e}_\varphi
-            \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{\cot{\theta}}{r} V_\theta \bigg)
+            \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{V_\theta}{r \tan{\theta}}  \bigg)
         \end{aligned}
         }
 \end{aligned}$$
@@ -262,10 +262,10 @@ $$\begin{aligned}
             + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{U_\theta V_\theta}{r} - \frac{U_\varphi V_\varphi}{r} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( U_r \pdv{V_\theta}{r} + \frac{U_\theta}{r} \pdv{V_\theta}{\theta}
-            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{\cot{\theta}}{r} U_\varphi V_\varphi \bigg)
+            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{U_\varphi V_\varphi}{r \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\theta}{r} \pdv{V_\varphi}{\theta}
-            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{\cot{\theta}}{r} U_\varphi V_\theta \bigg)
+            + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{U_\varphi V_\theta}{r \tan{\theta}} \bigg)
         \end{aligned}
     }
 \end{aligned}$$
@@ -275,22 +275,22 @@ $$\begin{aligned}
         \begin{aligned}
             \nabla^2 \vb{V}
             &= \quad\: \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_r}{\varphi}
-            + \frac{2}{r} \pdv{V_r}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_r}{\theta}
+            + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta}
             \\
             &\qquad\qquad - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi}
-            - \frac{2}{r^2} V_r - \frac{2 \cot{\theta}}{r^2} V_\theta \bigg)
+            - \frac{2 V_r}{r^2} - \frac{2 V_\theta}{r^2 \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg( \pdvn{2}{V_\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta}
             + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r}
             \\
-            &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta}
+            &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta}
             - \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\theta}
             + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi} + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi}
             \\
             &\qquad\qquad + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi}
-            + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_\varphi}{\theta}
+            + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\varphi}{\theta}
             - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg)
         \end{aligned}
     }
@@ -302,19 +302,20 @@ $$\begin{aligned}
             \nabla \cdot \overline{\overline{\mathbf{T}}}
             &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\theta r}}{\theta} + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi r}}{\varphi}
             \\
-            &\qquad\qquad + \frac{2}{r} T_{rr} + \frac{\cot{\theta}}{r} T_{\theta r} - \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
+            &\qquad\qquad + \frac{2 T_{rr}}{r} + \frac{T_{\theta r}}{r \tan{\theta}}
+            - \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
             \\
             &\quad\: + \vu{e}_\theta \bigg(\pdv{T_{r \theta}}{r} + \frac{1}{r} \pdv{T_{\theta \theta}}{\theta}
             + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \theta}}{\varphi}
             \\
-            &\qquad\qquad + \frac{2}{r} T_{r \theta} + \frac{T_{\theta r}}{r}
-            + \frac{\cot{\theta}}{r} T_{\theta \theta} - \frac{\cot{\theta}}{r} T_{\varphi \varphi} \bigg)
+            &\qquad\qquad + \frac{2 T_{r \theta}}{r} + \frac{T_{\theta r}}{r}
+            + \frac{T_{\theta \theta}}{r \tan{\theta}} - \frac{T_{\varphi \varphi}}{r \tan{\theta}} \bigg)
             \\
             &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\theta \varphi}}{\theta}
             + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \varphi}}{\varphi}
             \\
-            &\qquad\qquad + \frac{2}{r} T_{r \varphi} + \frac{\cot{\theta}}{r} T_{\theta \varphi}
-            + \frac{T_{\varphi r}}{r} + \frac{\cot{\theta}}{r} T_{\varphi \theta} \bigg)
+            &\qquad\qquad + \frac{2 T_{r \varphi}}{r} + \frac{T_{\theta \varphi}}{r \tan{\theta}}
+            + \frac{T_{\varphi r}}{r} + \frac{T_{\varphi \theta}}{r \tan{\theta}} \bigg)
         \end{aligned}
     }
 \end{aligned}$$