直交曲線座標の公式メモ

直交曲線座標を $u_i\, (i=1,2,3)$ とする．

位置ベクトル $\mathbf{r}(u_1, u_2, u_3)$ に対して

$\begin{align} d\mathbf{r} &= \sum_{i=1}^3 h_i\, du_i\, \hat{\mathbf{e}}_i \\\\ \hat{\mathbf{e}}_i &= \frac{1}{h_i} \frac{\partial \mathbf{r}}{du_i} \\\\ h_i &= \left| \frac{\partial \mathbf{r}}{du_i} \right| \end{align}$

円柱座標 $(r, \theta, z)$ ： $(h_1, h_2, h_3)=(1, r, 1)$

球座標 $(r, \theta, \phi)$ ： $(h_1, h_2, h_3)=(1, r, r\sin\theta)$

直交曲線座標（右手系）の定義により

$\begin{align} \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j &= \delta_{ij} \\\\ \hat{\mathbf{e}}_1 \times \hat{\mathbf{e}}_2 = \hat{\mathbf{e}}_3,\; \hat{\mathbf{e}}_2 \times \hat{\mathbf{e}}_3 &= \hat{\mathbf{e}}_1,\; \hat{\mathbf{e}}_3 \times \hat{\mathbf{e}}_1 = \hat{\mathbf{e}}_2 \end{align}$

体積要素

$\begin{align} dV = h_1 h_2 h_3\, du_1\, du_2\, du_3 \end{align}$

勾配

$\begin{align} \pmb{\nabla} = \sum_{i=1}^3 \frac{\hat{\mathbf{e}}_i}{h_i} \frac{\partial}{\partial u_i} \end{align}$

発散

$\begin{align} \pmb{\nabla}\cdot \mathbf{A} = \frac{1}{h_1h_2h_3} \left[ \frac{\partial ( h_2 h_3 A_1) }{\partial u_1} + \frac{\partial (h_3 h_1 A_2)}{\partial u_2} + \frac{\partial (h_1 h_2 A_3)}{\partial u_3}\right] \end{align}$

回転

$\begin{align} \pmb{\nabla} \times \mathbf{A} &=\frac{1}{h_{1} h_{2} h_{3}}\left|\begin{array}{ccc} h_{1} \hat{\mathbf{e}}_{1} & h_{2} \hat{\mathbf{e}}_{2} & h_{3} \hat{\mathbf{e}}_{3} \\ \displaystyle \frac{\partial}{\partial u_{1}} & \displaystyle \frac{\partial}{\partial u_{2}} & \displaystyle \displaystyle \frac{\partial}{\partial u_{3}} \\ h_{1} A_{1} & h_{2} A_{2} & h_{3} A_{3} \end{array}\right| \\\\ &= \frac{\hat{\mathbf{e}}_1}{h_2 h_3}\left[ \frac{\partial (h_3 A_3)}{\partial u_2} -\frac{\partial (h_2 A_2) }{\partial u_3} \right] + \frac{\hat{\mathbf{e}}_2}{h_3 h_1} \left[ \frac{\partial (h_1 A_1)}{\partial u_3} - \frac{\partial (h_3 A_3) }{\partial u_1} \right] \\\\ &\quad + \frac{\hat{\mathbf{e}}_3}{h_1 h_2} \left[ \frac{\partial (h_2 A_2)}{\partial u_1} - \frac{\partial (h_1 A_1)}{\partial u_2} \right] \end{align}$

ラプラシアン

$\begin{align} \nabla^2 f = \frac{1}{h_1h_2h_3} \left[ \frac{\partial}{\partial u_1}\left(\frac{h_2h_3}{h_1}\frac{\partial f}{\partial u_1} \right) +\frac{\partial}{\partial u_2}\left(\frac{h_3h_1}{h_2}\frac{\partial f}{\partial u_2} \right) +\frac{\partial}{\partial u_3}\left(\frac{h_1h_2}{h_3}\frac{\partial f}{\partial u_3} \right)\right] \end{align}$

変形速度テンソル

$\begin{align} e_{11} &= \frac{1}{h_1} \frac{\partial a_1}{\partial u_1} + \frac{A_2}{h_1 h_2} \frac{\partial h_1}{\partial u_2} + \frac{A_3}{h_3 h_1} \frac{\partial h_1}{\partial u_3} \\\\ e_{22} &= \frac{1}{h_2} \frac{\partial a_2}{\partial u_2} + \frac{A_3}{h_2 h_3} \frac{\partial h_2}{\partial u_3} + \frac{A_1}{h_1 h_2} \frac{\partial h_2}{\partial u_1} \\\\ e_{33} &= \frac{1}{h_3} \frac{\partial a_3}{\partial u_3} + \frac{A_1}{h_3 h_1} \frac{\partial h_3}{\partial u_1} + \frac{A_2}{h_2 h_3} \frac{\partial h_3}{\partial u_2} \\\\ 2e_{ij} &= \frac{h_j}{h_i}\frac{\partial}{\partial u_i}\left(\frac{A_j}{h_j}\right) + \frac{h_i}{h_j}\frac{\partial}{\partial u_j}\left(\frac{A_i}{h_i}\right) \qquad (i\neq j) \end{align}$

ベクトルラプラシアン

$\begin{align} \nabla^2 \mathbf{A} = \pmb{\nabla}\pmb{\nabla}\cdot\mathbf{A} -\pmb{\nabla}\times(\pmb{\nabla}\times \mathbf{A}) \end{align}$

$\begin{align} \pmb{\nabla}\pmb{\nabla}\cdot\mathbf{A} &= \sum_{i=1}^3 \frac{\hat{\mathbf{e}}_i}{h_i}\frac{\partial}{\partial u_i} \left\{ \frac{1}{h_1h_2h_3} \left[ \frac{\partial (h_2 h_3 A_1)}{\partial u_1} + \frac{\partial (h_3 h_1 A_2)}{\partial u_2} + \frac{\partial (h_1 h_2 A_3)}{\partial u_3}\right] \right\} \end{align}$

$\begin{align} &\ \pmb{\nabla}\times(\pmb{\nabla}\times\mathbf{A}) \\ &= \frac{\hat{\mathbf{e}}_1}{h_2 h_3}\bigg\{ \frac{\partial }{\partial u_2} \left[ \frac{h_3}{h_1 h_2} \left( \frac{\partial (h_2 A_2)}{\partial u_1} - \frac{\partial (h_1 A_1)}{\partial u_2} \right)\right] -\frac{\partial }{\partial u_3}\left[ \frac{h_2}{h_3 h_1} \left( \frac{\partial (h_1 A_1)}{\partial u_3} - \frac{\partial (h_3 A_3) }{\partial u_1} \right) \right] \bigg\} \\\\ &\ + \frac{\hat{\mathbf{e}}_2}{h_3 h_1} \bigg\{ \frac{\partial}{\partial u_3} \left[ \frac{h_1}{h_2 h_3}\left( \frac{\partial (h_3 A_3)}{\partial u_2} -\frac{\partial (h_2 A_2) }{\partial u_3} \right) \right] - \frac{\partial}{\partial u_1} \left[ \frac{h_3}{h_1 h_2} \left( \frac{\partial (h_2 A_2)}{\partial u_1} - \frac{\partial (h_1 A_1)}{\partial u_2} \right) \right] \bigg\} \\\\ &\ + \frac{\hat{\mathbf{e}}_3}{h_1 h_2} \bigg\{ \frac{\partial}{\partial u_1} \left[ \frac{h_2}{h_3 h_1} \left( \frac{\partial (h_1 A_1)}{\partial u_3} - \frac{\partial (h_3 A_3) }{\partial u_1} \right) \right] - \frac{\partial}{\partial u_2} \left[ \frac{h_1}{h_2 h_3}\left( \frac{\partial (h_3 A_3)}{\partial u_2} -\frac{\partial (h_2 A_2) }{\partial u_3} \right) \right] \bigg\} \end{align}$