直交曲線座標での性質
直交曲線座標での性質
直交座標\(x_{i}\)と直交曲線座標\(q_{i}\)で以下が成り立つ。
直交座標\(x_{i}\)と直交曲線座標\(q_{i}\)で以下が成り立つ。
(1)
\[ h_{i}\boldsymbol{\nabla}q_{i}=\frac{1}{h_{i}}\frac{\partial\boldsymbol{r}}{\partial q_{i}} \](2)
\[ \frac{\partial q_{i}}{\partial x_{j}}=\frac{1}{h_{i}}\frac{\partial x_{j}}{\partial q_{i}} \](3)
\[ \frac{\partial^{2}q_{i}}{\partial x_{k}\partial x_{j}}=\sum_{l}\frac{1}{h_{l}^{2}}\frac{\partial x_{k}}{\partial q_{l}}\left(\frac{1}{h_{i}^{2}}\frac{\partial^{2}x_{j}}{\partial q_{l}\partial q_{i}}-\frac{2}{h_{i}^{3}}\frac{\partial h_{i}}{\partial q_{l}}\frac{\partial x_{j}}{\partial q_{i}}\right) \]-
\(h_{i}\)はスケール因子(1)
\begin{align*} \delta_{ij} & =\frac{\partial q_{i}}{\partial q_{j}}\\ & =\frac{\partial q_{i}}{\partial x_{k}}\frac{\partial x_{k}}{\partial q_{j}}\\ & =\sum_{k,l}\boldsymbol{e}_{k}\frac{\partial q_{i}}{\partial x_{k}}\cdot\boldsymbol{e}_{l}\frac{\partial x_{l}}{\partial q_{j}}\\ & =\boldsymbol{\nabla}\left(q_{i}\right)\cdot\frac{\partial\boldsymbol{r}}{\partial q_{j}} \end{align*} これより\(\boldsymbol{\nabla}\left(q_{i}\right)\)と\(\frac{\partial\boldsymbol{r}}{\partial q_{i}}\)は平行となるので、\begin{align*} 1 & =\boldsymbol{\nabla}\left(q_{i}\right)\cdot\frac{\partial\boldsymbol{r}}{\partial q_{i}}\\ & =\frac{\left\Vert \boldsymbol{\nabla}\left(q_{i}\right)\right\Vert }{\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{i}}\right\Vert }\frac{\partial\boldsymbol{r}}{\partial q_{i}}\cdot\frac{\partial\boldsymbol{r}}{\partial q_{i}}\\ & =\left\Vert \boldsymbol{\nabla}\left(q_{i}\right)\right\Vert h_{i} \end{align*} となる。故に、
\begin{align*} h_{i}\boldsymbol{\nabla}q_{i} & =h_{i}\frac{\left\Vert \boldsymbol{\nabla}\left(q_{i}\right)\right\Vert }{\left\Vert \frac{\partial\boldsymbol{r}}{\partial q_{i}}\right\Vert }\frac{\partial\boldsymbol{r}}{\partial q_{i}}\\ & =\frac{1}{h_{i}}\frac{\partial\boldsymbol{r}}{\partial q_{i}} \end{align*}
(2)
\begin{align*} \frac{\partial q_{i}}{\partial x_{j}} & =\sum_{k}\boldsymbol{e}_{j}\cdot\boldsymbol{e}_{k}\frac{\partial q_{i}}{\partial x_{k}}\\ & =\boldsymbol{e}_{j}\cdot\boldsymbol{\nabla}q_{i}\\ & =\boldsymbol{e}_{j}\cdot\frac{1}{h_{i}^{\;2}}\frac{\partial\boldsymbol{r}}{\partial q_{i}}\\ & =\sum_{k}\boldsymbol{e}_{j}\cdot\boldsymbol{e}_{k}\frac{1}{h_{i}^{\;2}}\frac{\partial x_{k}}{\partial q_{i}}\\ & =\frac{1}{h_{i}^{\;2}}\frac{\partial x_{j}}{\partial q_{i}} \end{align*}(3)
\begin{align*} \frac{\partial^{2}q_{i}}{\partial x_{k}\partial x_{j}} & =\frac{\partial}{\partial x_{k}}\frac{\partial q_{i}}{\partial x_{j}}\\ & =\sum_{l}\frac{\partial q_{l}}{\partial x_{k}}\frac{\partial}{\partial q_{l}}\frac{1}{h_{i}^{2}}\frac{\partial x_{j}}{\partial q_{i}}\\ & =\sum_{l}\frac{1}{h_{l}^{2}}\frac{\partial x_{k}}{\partial q_{l}}\frac{\partial}{\partial q_{l}}\frac{1}{h_{i}^{2}}\frac{\partial x_{j}}{\partial q_{i}}\\ & =\sum_{l}\frac{1}{h_{l}^{2}}\frac{\partial x_{k}}{\partial q_{l}}\left(\frac{1}{h_{i}^{2}}\frac{\partial^{2}x_{j}}{\partial q_{l}\partial q_{i}}-\frac{2}{h_{i}^{3}}\frac{\partial h_{i}}{\partial q_{l}}\frac{\partial x_{j}}{\partial q_{i}}\right) \end{align*}
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直交曲線座標でのナブラ演算子・回転・発散・ラプラシアン
\[
\boldsymbol{\nabla}\cdot\boldsymbol{A}=\frac{1}{h}\sum_{i}\frac{\partial}{\partial q_{i}}\frac{A_{i}h}{h_{i}}
\]
アインシュタインの和の既約
直交曲線座標での単位基底ベクトルの回転・発散
\[
\boldsymbol{\nabla}\cdot\boldsymbol{u}_{i}=\frac{1}{hh_{i}}\frac{\partial}{\partial q_{i}}h-\frac{1}{h_{i}^{2}}\frac{\partial}{\partial q_{i}}h_{i}
\]
ストークスの定理とガウスの発散定理
\[
\iiint_{V}\boldsymbol{\nabla}\cdot\boldsymbol{A}dV=\iint_{S}\boldsymbol{A}\cdot d\boldsymbol{S}
\]