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Diffstat (limited to 'source/know/concept/spherical-coordinates/index.md')
| -rw-r--r-- | source/know/concept/spherical-coordinates/index.md | 31 |
1 files changed, 14 insertions, 17 deletions
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}$$ |