# 正交坐標系

## 向量與積分

$ds^{2} = \sum_{i=1}^{n} \left( h_{i} dq_{i} \right)^{2}$

$h_{i}(\mathbf{q})\ \stackrel{\mathrm{def}}{=}\ \sqrt{g_{ii}(\mathbf{q})}$

$d\mathbf{r} = \sum_{i=1}^{n} h_{i} dq_{i} \mathbf{e}_{i}$

$\mathbf{A} \cdot \mathbf{B} = \sum_{i=1}^{n} A_{i} B_{i}$

$\int_{\mathbb{C}} \mathbf{F} \cdot d\mathbf{r} = \sum_{i=1}^{n} \int_{\mathbb{C}} F_{i} h_{i} dq_{i}$

$F_{i} \ \stackrel{\mathrm{def}}{=}\ \mathbf{e}_{i} \cdot \mathbf{F}$

$dA = ds_{i} ds_{j} = h_{i} h_{j} dq_{i} dq_{j},\qquad i\neq j$

$dV = ds_{i} ds_{j} ds_{k} = h_{i} h_{j} h_{k} dq_{i} dq_{j} dq_{k},\qquad i \neq j \neq k$

$\int_{\mathbb{S}} \mathbf{F} \cdot d\mathbf{A} = \int_{\mathbb{S}} F_{1} h_{2} h_{3} dq_{2} dq_{3} + \int_{\mathbb{S}} F_{2} h_{3} h_{1} dq_{3} dq_{1} + \int_{\mathbb{S}} F_{3} h_{1} h_{2} dq_{1} dq_{2}$

### 球坐標系實例

$x=r\sin\theta\cos\phi$
$y=r\sin\theta\sin\phi$
$z=r\cos\theta$

$dx=\sin\theta\cos\phi dr+r\cos\theta\cos\phi d\theta - r\sin\theta\sin\phi d\phi$
$dy=\sin\theta\sin\phi dr+r\cos\theta\sin\phi d\theta+r\sin\theta\cos\phi d\phi$
$dz=\cos\theta dr - r\sin\theta d\theta$

\begin{align}ds^{2} & = dx^{2}+dy^{2}+dz^{2} \\ & =dr^{2}+(rd\theta)^{2}+(r\sin\theta d\phi)^{2} \\\end{align}

$h_r=1$
$h_{\theta}=r$
$h_{\phi}=r\sin\theta$

$\int_{\mathbb{C}}\mathbf{F}\cdot d\mathbf{r}=\int_{\mathbb{C}}F_{r}\ dr+F_{\theta}\ rd\theta+F_{\phi}\ r\sin\theta d\phi$

$\int_{\mathbb{S}} \mathbf{F} \cdot d\mathbf{A} = \int_{\mathbb{S}} F_r\ r^2\sin\theta d\theta d\phi+ \int_{\mathbb{S}} F_{\theta}\ r\sin\theta dr d\phi + \int_{\mathbb{S}} F_{\phi}\ r dr d\theta$

## 三維微分算子

### 梯度導引

$\nabla \Phi \cdot\hat{\mathbf{n}}=\frac{d\phi}{ds}$

$\nabla \Phi = \hat{\mathbf{e}}_{1}\frac{1}{h_{1}} \frac{\partial \Phi}{\partial q_{1}} + \hat{\mathbf{e}}_{2}\frac{1}{h_{2}} \frac{\partial \Phi}{\partial q_{2}} + \hat{\mathbf{e}}_{3}\frac{1}{h_{3}} \frac{\partial \Phi}{\partial q_{3}}$

### 散度導引

$\nabla \cdot \mathbf{F} = \nabla \cdot (\hat{\mathbf{e}}_{1}F_1+\hat{\mathbf{e}}_{2}F_2+\hat{\mathbf{e}}_{3}F_3)$

$\nabla \cdot (\hat{\mathbf{e}}_1F_1)= \nabla \cdot \left[\left(\frac{\hat{\mathbf{e}}_1}{h_2 h_3}\right)\left(h_2 h_3 F_1\right)\right]$

\begin{align} \nabla \cdot (\hat{\mathbf{e}}_1F_1) & =(h_2 h_3 F_1)\nabla \cdot\left(\frac{\hat{\mathbf{e}}_1}{h_2 h_3}\right)+\left(\frac{\hat{\mathbf{e}}_1}{h_2 h_3}\right)\cdot \nabla(h_2 h_3 F_1) \\ & =(h_2 h_3 F_1)\nabla\cdot[(\nabla q_2)\times\nabla(q_3)] +\left(\frac{\hat{\mathbf{e}}_1}{h_2 h_3}\right)\cdot \nabla(h_2 h_3 F_1) \\ & =\left(\frac{\hat{\mathbf{e}}_1}{h_2 h_3}\right)\cdot \nabla(h_2 h_3 F_1) \\ & =\frac{1}{h_1 h_2 h_3} \frac{\partial}{\partial q_1}(F_1 h_2 h_3) \\ \end{align}

$\nabla \cdot \mathbf{F} = \frac{1}{h_{1} h_{2} h_{3}} \left[\frac{\partial}{\partial q_1}(F_1 h_2 h_3)+\frac{\partial}{\partial q_2}(F_2 h_3 h_1) + \frac{\partial}{\partial q_3}(F_3 h_1 h_2) \right]$

### 旋度導引

$\nabla \times \mathbf{F}=\nabla \times (\hat{\mathbf{e}}_{1}F_1+\hat{\mathbf{e}}_{2}F_2+\hat{\mathbf{e}}_{3}F_3)$

$\nabla \times (\hat{\mathbf{e}}_{1}F_1)=\nabla \times \left[\left(\frac{\hat{\mathbf{e}}_1}{h_1}\right)\left(h_1 F_1\right)\right]$

\begin{align} \nabla \times (\hat{\mathbf{e}}_{1}F_1) & =(h_1 F_1)\nabla\times\left(\frac{\hat{\mathbf{e}}_1}{h_1}\right) - \left(\frac{\hat{\mathbf{e}}_1}{h_1}\right)\times\nabla(h_1 F_1) \\ & =(h_1 F_1)\nabla\times(\nabla q_1) - \left(\frac{\hat{\mathbf{e}}_1}{h_1}\right)\times\left(\frac{\hat{\mathbf{e}}_{2}}{h_{2}} \frac{\partial}{\partial q_{2}}(h_1 F_1) + \frac{\hat{\mathbf{e}}_{3}}{h_{3}} \frac{\partial}{\partial q_{3}}(h_1 F_1)\right) \\ \end{align}

$\nabla \times (\hat{\mathbf{e}}_{1}F_1)=\frac{\hat{\mathbf{e}}_{2}}{h_1 h_{3}} \frac{\partial}{\partial q_{3}}(h_1 F_1) - \frac{\hat{\mathbf{e}}_{3}}{h_1 h_{2}} \frac{\partial}{\partial q_{2}}(h_1 F_1)$

\begin{align}\nabla \times \mathbf{F} & =\frac{\mathbf{e}_{1}}{h_{2} h_{3}} \left[\frac{\partial}{\partial q_{2}} \left( h_{3} F_{3} \right) - \frac{\partial}{\partial q_{3}} \left( h_{2} F_{2} \right)\right] + \frac{\mathbf{e}_{2}}{h_{3} h_{1}} \left[\frac{\partial}{\partial q_{3}} \left( h_{1} F_{1} \right) - \frac{\partial}{\partial q_{1}} \left( h_{3} F_{3} \right)\right] \\ & +\frac{\mathbf{e}_{3}}{h_{1} h_{2}} \left[\frac{\partial}{\partial q_{1}} \left( h_{2} F_{2} \right) - \frac{\partial}{\partial q_{2}} \left( h_{1} F_{1} \right)\right] \\ \end{align}

### 拉普拉斯算子

$\nabla^{2} \Phi = \frac{1}{h_{1} h_{2} h_{3}} \left[ \frac{\partial}{\partial q_{1}} \left( \frac{h_{2} h_{3}}{h_{1}} \frac{\partial \Phi}{\partial q_{1}} \right) + \frac{\partial}{\partial q_{2}} \left( \frac{h_{3} h_{1}}{h_{2}} \frac{\partial \Phi}{\partial q_{2}} \right) + \frac{\partial}{\partial q_{3}} \left(\frac{h_{1} h_{2}}{h_{3}} \frac{\partial \Phi}{\partial q_{3}} \right) \right]$

## 參考文獻

• Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill, pp. 164-182。
• Morse PM and Feshbach H. (1953) Methods of Theoretical Physics, McGraw-Hill, pp. 494-523, 655-666。
• Margenau H. and Murphy GM. (1956) The Mathematics of Physics and Chemistry, 2nd. ed., Van Nostrand, pp. 172-192。