From 7ec42764de400df4db629780f3c758f553ac5a93 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 9 Jun 2023 23:33:22 +0200 Subject: Improve knowledge base --- source/know/concept/spherical-coordinates/index.md | 37 +++++++++++----------- 1 file changed, 19 insertions(+), 18 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 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}$$ -- cgit v1.2.3