直交曲線座標でのナブラ演算子・回転・発散・ラプラシアン
直交曲線座標でのナブラ演算子・回転・発散・ラプラシアン
直交曲線座標\(q_{i}\)でのナブラ演算子・回転・発散・ラプラシアンは以下のように表される。
\(\boldsymbol{u}_{k}\)は直交曲線座標\(q_{i}\)での単位基底ベクトル、\(h_{k}\)はスケール因子で\(h=\prod_{i}h_{i}\)とする。
直交曲線座標\(q_{i}\)でのナブラ演算子・回転・発散・ラプラシアンは以下のように表される。
\(\boldsymbol{u}_{k}\)は直交曲線座標\(q_{i}\)での単位基底ベクトル、\(h_{k}\)はスケール因子で\(h=\prod_{i}h_{i}\)とする。
(1)ナブラ演算子
\[ \boldsymbol{\nabla}=\sum_{k}\boldsymbol{u}_{k}\frac{\partial}{h_{k}\partial q_{k}} \](2)回転
\begin{align*} \boldsymbol{\nabla}\times\boldsymbol{A} & =\sum_{i,j}\epsilon_{ijk}\frac{\boldsymbol{u}_{k}}{h_{i}h_{j}}\frac{\partial}{\partial q_{i}}\left(A_{j}h_{j}\right)\\ & =\frac{1}{h}\left|\begin{array}{ccc} h_{1}\boldsymbol{e}_{1} & h_{2}\boldsymbol{e}_{2} & h_{3}\boldsymbol{e}_{3}\\ \frac{\partial}{\partial q_{1}} & \frac{\partial}{\partial q_{2}} & \frac{\partial}{\partial q_{3}}\\ h_{1}A_{1} & h_{2}A_{2} & h_{3}A_{3} \end{array}\right| \end{align*}(3)発散
\[ \boldsymbol{\nabla}\cdot\boldsymbol{A}=\frac{1}{h}\sum_{i}\frac{\partial}{\partial q_{i}}\frac{A_{i}h}{h_{i}} \](4)ラプラシアン
\[ \Delta=\frac{1}{h}\sum_{i}\frac{\partial}{\partial q_{i}}\frac{h}{h_{i}^{2}}\frac{\partial}{\partial q_{i}} \]極座標
極座標\[ \begin{cases} x=r\sin\theta\cos\phi\\ y=r\sin\theta\sin\phi\\ z=r\cos\theta \end{cases} \] \[ \begin{cases} r=\sqrt{x^{2}+y^{2}+z^{2}}\\ \theta=\cos^{\bullet}\left(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\right)\\ \phi=\tan^{\bullet}\left(\frac{y}{x}\right) \end{cases} \] ではスケール因子は
\[ \begin{cases} h_{r}=1\\ h_{\theta}=r\\ h_{\phi}=r\sin\theta \end{cases} \] なので、
\[ \boldsymbol{\nabla}=\boldsymbol{e}_{r}\frac{\partial}{\partial r}+\boldsymbol{e}_{\theta}\frac{1}{r}\frac{\partial}{\partial\theta}+\boldsymbol{e}_{\phi}\frac{1}{r\sin\theta}\frac{\partial}{\partial\phi} \] \[ \boldsymbol{\nabla}\times\boldsymbol{A}=\frac{1}{r^{2}\sin\theta}\left|\begin{array}{ccc} \boldsymbol{e}_{r} & r\boldsymbol{e}_{\theta} & r\sin\theta\boldsymbol{e}_{\phi}\\ \frac{\partial}{\partial r} & \frac{\partial}{\partial\theta} & \frac{\partial}{\partial\phi}\\ A_{r} & rA_{\theta} & r\sin\theta A_{\phi} \end{array}\right| \] \begin{align*} \boldsymbol{\nabla}\cdot\boldsymbol{A} & =\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}A_{r}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta A_{\theta}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\phi}A_{\phi}\\ & =\frac{\partial A_{r}}{\partial r}+\frac{2A_{r}}{r^{2}}+\frac{1}{r}\frac{\partial A_{\theta}}{\partial\theta}+\frac{\cos\theta A_{\theta}}{r\sin\theta}+\frac{1}{r\sin\theta}\frac{\partial A_{\phi}}{\partial\phi} \end{align*} \begin{align*} \Delta & =\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial}{\partial r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\\ & =\frac{\partial^{2}}{\partial r^{2}}+\frac{2}{r}\frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{\cos\theta}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}} \end{align*} となる。
円柱座標
円柱座標\[ \begin{cases} x=r\cos\theta\\ y=r\sin\theta\\ z=z \end{cases} \] \[ \begin{cases} r=\sqrt{x^{2}+y^{2}}\\ \theta=\tan^{\bullet}\left(\frac{y}{x}\right)\\ z=z \end{cases} \] ではスケール因子は
\[ \begin{cases} h_{r}=1\\ h_{\theta}=r\\ h_{\phi}=1 \end{cases} \] なので、
\[ \boldsymbol{\nabla}=\boldsymbol{e}_{r}\frac{\partial}{\partial r}+\boldsymbol{e}_{\theta}\frac{1}{r}\frac{\partial}{\partial\theta}+\boldsymbol{e}_{z}\frac{\partial}{\partial z} \] \[ \boldsymbol{\nabla}\times\boldsymbol{A}=\frac{1}{r}\left|\begin{array}{ccc} \boldsymbol{e}_{r} & r\boldsymbol{e}_{\theta} & \boldsymbol{e}_{z}\\ \frac{\partial}{\partial r} & \frac{\partial}{\partial\theta} & \frac{\partial}{\partial z}\\ A_{r} & rA_{\theta} & A_{z} \end{array}\right| \] \begin{align*} \boldsymbol{\nabla}\cdot\boldsymbol{A} & =\frac{1}{r}\frac{\partial}{\partial r}\left(rA_{r}\right)+\frac{1}{r}\frac{\partial}{\partial\theta}A_{\theta}+\frac{\partial}{\partial z}A_{z}\\ & =\frac{\partial}{\partial r}A_{r}+\frac{1}{r}A_{r}+\frac{1}{r}\frac{\partial}{\partial\theta}A_{\theta}+\frac{\partial}{\partial z}A_{z} \end{align*} \begin{align*} \Delta & =\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{\partial^{2}}{\partial z^{2}}\\ & =\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}+\frac{\partial^{2}}{\partial z^{2}} \end{align*}
(1)
\begin{align*} \boldsymbol{\nabla} & =\sum_{i}\boldsymbol{e}_{i}\frac{\partial}{\partial x_{i}}\\ & =\sum_{i,j,k}\frac{\partial q_{j}}{\partial x_{i}}\frac{\partial\boldsymbol{r}}{\partial q_{j}}\frac{\partial q_{k}}{\partial x_{i}}\frac{\partial}{\partial q_{k}}\\ & =\sum_{i,j,k}\frac{\partial q_{j}}{\partial x_{i}}h_{j}\boldsymbol{u}_{j}\frac{\partial q_{k}}{\partial x_{i}}\frac{\partial}{\partial q_{k}}\\ & =\sum_{i,j,k}\frac{\partial x_{i}}{h_{j}\partial q_{j}}\boldsymbol{u}_{j}\frac{\partial q_{k}}{\partial x_{i}}\frac{\partial}{\partial q_{k}}\\ & =\sum_{k}\boldsymbol{u}_{k}\frac{\partial}{h_{k}\partial q_{k}} \end{align*}(2)
\begin{align*} \boldsymbol{\nabla}\times\boldsymbol{A} & =\sum_{i}\boldsymbol{\nabla}\times A_{i}\boldsymbol{u}_{i}\\ & =\sum_{i}\left(A_{i}\boldsymbol{\nabla}\times\boldsymbol{u}_{i}-\boldsymbol{u}_{i}\times\boldsymbol{\nabla}A_{i}\right)\\ & =\sum_{i}\left(A_{i}\left(-\boldsymbol{u}_{i}\times\frac{1}{h_{i}}\boldsymbol{\nabla}h_{i}\right)-\boldsymbol{u}_{i}\times\boldsymbol{\nabla}A_{i}\right)\cmt{\because\boldsymbol{\nabla}\times\boldsymbol{u}_{i}=-\boldsymbol{u}_{i}\times\frac{1}{h_{i}}\boldsymbol{\nabla}h_{i}}\\ & =-\sum_{i}\left(\boldsymbol{u}_{i}\times\frac{A_{i}}{h_{i}}\boldsymbol{\nabla}h_{i}+\boldsymbol{u}_{i}\times\boldsymbol{\nabla}A_{i}\right)\\ & =-\sum_{i,j}\left(\boldsymbol{u}_{i}\times\boldsymbol{u}_{j}\left(\frac{A_{i}}{h_{i}h_{j}}\frac{\partial}{\partial q_{j}}h_{i}+\frac{1}{h_{j}}\frac{\partial}{\partial q_{j}}A_{i}\right)\right)\\ & =-\sum_{i,j}\epsilon_{ijk}\frac{\boldsymbol{u}_{k}}{h_{i}h_{j}}\frac{\partial}{\partial q_{j}}\left(A_{i}h_{i}\right)\\ & =\sum_{i,j}\epsilon_{ijk}\frac{\boldsymbol{u}_{k}}{h_{i}h_{j}}\frac{\partial}{\partial q_{i}}\left(A_{j}h_{j}\right) \end{align*}(3)
\begin{align*} \boldsymbol{\nabla}\cdot\boldsymbol{A} & =\sum_{i}\boldsymbol{\nabla}\cdot A_{i}\boldsymbol{u}_{i}\\ & =\sum_{i}\left(A_{i}\boldsymbol{\nabla}\cdot\boldsymbol{u}_{i}+\boldsymbol{u}_{i}\cdot\boldsymbol{\nabla}A_{i}\right)\\ & =\sum_{i}\left(\frac{A_{i}}{hh_{i}}\frac{\partial}{\partial q_{i}}h-\frac{A_{i}}{h_{i}^{2}}\frac{\partial}{\partial q_{i}}h_{i}+\sum_{j}\boldsymbol{u}_{i}\cdot\boldsymbol{u}_{j}\frac{1}{h_{j}}\frac{\partial}{\partial q_{j}}A_{i}\right)\cmt{\because\boldsymbol{\nabla}\cdot\boldsymbol{u}_{i}=\frac{1}{hh_{i}}\frac{\partial}{\partial q_{i}}h-\frac{1}{h_{i}^{2}}\frac{\partial}{\partial q_{i}}h_{i}}\\ & =\sum_{i}\left(\frac{A_{i}}{hh_{i}}\frac{\partial}{\partial q_{i}}h-\frac{A_{i}}{h_{i}^{2}}\frac{\partial}{\partial q_{i}}h_{i}+\frac{1}{h_{i}}\frac{\partial}{\partial q_{i}}A_{i}\right)\\ & =\sum_{i}\left(\frac{1}{h}\frac{\partial}{\partial q_{i}}\frac{A_{i}h}{h_{i}}-\frac{\partial}{\partial q_{i}}\frac{A_{i}}{h_{i}}-\frac{A_{i}}{h_{i}^{2}}\frac{\partial}{\partial q_{i}}h_{i}+\frac{1}{h_{i}}\frac{\partial}{\partial q_{i}}A_{i}\right)\\ & =\frac{1}{h}\sum_{i}\frac{\partial}{\partial q_{i}}\frac{A_{i}h}{h_{i}} \end{align*}(4)
(3)より、\begin{align*} \Delta & =\nabla\cdot\nabla\\ & =\frac{1}{h}\sum_{i}\frac{\partial}{\partial q_{i}}\frac{h}{h_{i}}\nabla_{i}\\ & =\frac{1}{h}\sum_{i}\frac{\partial}{\partial q_{i}}\frac{h}{h_{i}^{2}}\frac{\partial}{\partial q_{i}}\cmt{\because\nabla_{k}=\frac{\partial}{h_{k}\partial q_{k}}} \end{align*}
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| タイトル | 直交曲線座標でのナブラ演算子・回転・発散・ラプラシアン |
| URL | https://www.nomuramath.com/hh0mb9w4/ |
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勾配の方向と方向微分
\[
\nabla_{\boldsymbol{v}}f\left(\boldsymbol{r}\right):=\boldsymbol{v}\cdot\boldsymbol{\nabla}f
\]
ストークスの定理とガウスの発散定理
\[
\iiint_{V}\boldsymbol{\nabla}\cdot\boldsymbol{A}dV=\iint_{S}\boldsymbol{A}\cdot d\boldsymbol{S}
\]
アインシュタインの和の既約
ナブラ演算子・勾配・発散・回転・ラプラシアンの定義
\[
\boldsymbol{\nabla}:=\boldsymbol{e}_{i}\partial_{i}
\]