在圓柱和球坐標系中的del

下面是常用於正交曲線坐標系中的一些向量微積分公式。

## 注釋

- 本文對球坐標使用標準符號ISO 80000-2，它取代了ISO 31-11，（部分其他來源可能有著顛倒θ和φ的定義）：
- 極角表示為θ：它是在z軸與連接原點和目標點的徑向向量之間的角度。
- 方位角表示為φ：它是在x軸與徑向向量在xy面上的投影之間的角度。
- 函數atan2(y, x)可以用於替代數學函數arctan(y/x)。這是由於它的定義域和像的緣故，經典arctan函數的像為(−π/2, +π/2)，而atan2定義的像為(−π, π。

## 坐標轉換

從

直角｜圓柱｜球

到｜直角｜$\begin{align} x &= x \\ y &= y \\ z &= z \end{align}$｜$\begin{align} x &= \rho \cos\varphi \\ y &= \rho \sin\varphi \\ z &= z \end{align}$｜$\begin{align} x &= r \sin\theta \cos\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\theta \end{align}$

圓柱｜$\begin{align} \rho &= \sqrt{x^2 + y^2} \\ \varphi &= \arctan\left(\frac{y}{x}\right) \\ z &= z \end{align}$｜$\begin{align} \rho &= \rho \\ \varphi &= \varphi \\ z &= z \end{align}$｜$\begin{align} \rho &= r \sin\theta \\ \varphi &= \varphi \\ z &= r\cos\theta \end{align}$

球｜$\begin{align} r &= \sqrt{x^2 + y^2 + z^2} \\ \theta &= \arctan\left(\frac{\sqrt{x^2 + y^2}}{z}\right) \\ \varphi &= \arctan\left(\frac{y}{x}\right) \end{align}$｜$\begin{align} r &= \sqrt{\rho^2 + z^2} \\ \theta &= \arctan{\left(\frac{\rho}{z}\right)} \\ \varphi &= \varphi \end{align}$｜$\begin{align} r &= r \\ \theta &= \theta \\ \varphi &= \varphi \end{align}$

## 單位向量轉換

直角｜圓柱｜球

直角｜不適用｜$\begin{align} \hat{\mathbf x} &= \cos\varphi \hat{\boldsymbol \rho} - \sin\varphi \hat{\boldsymbol \varphi} \\ \hat{\mathbf y} &= \sin\varphi \hat{\boldsymbol \rho} + \cos\varphi \hat{\boldsymbol \varphi} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$｜$\begin{align} \hat{\mathbf x} &= \sin\theta \cos\varphi \hat{\mathbf r} + \cos\theta \cos\varphi \hat{\boldsymbol \theta} - \sin\varphi \hat{\boldsymbol \varphi} \\ \hat{\mathbf y} &= \sin\theta \sin\varphi \hat{\mathbf r} + \cos\theta \sin\varphi \hat{\boldsymbol \theta} + \cos\varphi \hat{\boldsymbol \varphi} \\ \hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta} \end{align}$

圓柱｜$\begin{align} \hat{\boldsymbol \rho} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\ \hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$｜不適用｜$\begin{align} \hat{\boldsymbol \rho} &= \sin\theta \hat{\mathbf r} + \cos\theta \hat{\boldsymbol \theta} \\ \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\ \hat{\mathbf z} &= \cos\theta \hat{\mathbf r} - \sin\theta \hat{\boldsymbol \theta} \end{align}$

球｜$\begin{align} \hat{\mathbf r} &= \frac{x \hat{\mathbf x} + y \hat{\mathbf y} + z \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2}} \\ \hat{\boldsymbol \theta} &= \frac{\left(x \hat{\mathbf x} + y \hat{\mathbf y}\right) z - \left(x^2 + y^2\right) \hat{\mathbf z}}{\sqrt{x^2 + y^2 + z^2} \sqrt{x^2 + y^2}} \\ \hat{\boldsymbol \varphi} &= \frac{-y \hat{\mathbf x} + x \hat{\mathbf y}}{\sqrt{x^2 + y^2}} \end{align}$｜$\begin{align} \hat{\mathbf r} &= \frac{\rho \hat{\boldsymbol \rho} + z \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\ \hat{\boldsymbol \theta} &= \frac{z \hat{\boldsymbol \rho} - \rho \hat{\mathbf z}}{\sqrt{\rho^2 + z^2}} \\ \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \end{align}$｜不適用

直角｜圓柱｜球

直角｜不適用｜$\begin{align} \hat{\mathbf x} &= \frac{x \hat{\boldsymbol \rho} - y \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\ \hat{\mathbf y} &= \frac{y \hat{\boldsymbol \rho} + x \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2}} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$｜$\begin{align} \hat{\mathbf x} &= \frac{x \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) - y \sqrt{x^2 + y^2 + z^2} \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2} \sqrt{x^2 + y^2 + z^2}} \\ \hat{\mathbf y} &= \frac{y \left(\sqrt{x^2 + y^2} \hat{\mathbf r} + z \hat{\boldsymbol \theta}\right) + x \sqrt{x^2 + y^2 + z^2} \hat{\boldsymbol \varphi}}{\sqrt{x^2 + y^2} \sqrt{x^2 + y^2 + z^2}} \\ \hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \sqrt{x^2 + y^2} \hat{\boldsymbol \theta}}{\sqrt{x^2 + y^2 + z^2}} \end{align}$

圓柱｜$\begin{align} \hat{\boldsymbol \rho} &= \cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y} \\ \hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y} \\ \hat{\mathbf z} &= \hat{\mathbf z} \end{align}$｜不適用｜$\begin{align} \hat{\boldsymbol \rho} &= \frac{\rho \hat{\mathbf r} + z \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}} \\ \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \\ \hat{\mathbf z} &= \frac{z \hat{\mathbf r} - \rho \hat{\boldsymbol \theta}}{\sqrt{\rho^2 + z^2}} \end{align}$

球｜$\begin{align} \hat{\mathbf r} &= \sin\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) + \cos\theta \hat{\mathbf z} \\ \hat{\boldsymbol \theta} &= \cos\theta \left(\cos\varphi \hat{\mathbf x} + \sin\varphi \hat{\mathbf y}\right) - \sin\theta \hat{\mathbf z} \\ \hat{\boldsymbol \varphi} &= -\sin\varphi \hat{\mathbf x} + \cos\varphi \hat{\mathbf y} \end{align}$｜$\begin{align} \hat{\mathbf r} &= \sin\theta \hat{\boldsymbol \rho} + \cos\theta \hat{\mathbf z} \\ \hat{\boldsymbol \theta} &= \cos\theta \hat{\boldsymbol \rho} - \sin\theta \hat{\mathbf z} \\ \hat{\boldsymbol \varphi} &= \hat{\boldsymbol \varphi} \end{align}$｜不適用

## Del公式

運算｜直角坐標 (x, y, z)｜圓柱坐標 (ρ, φ, z)｜球坐標 (r, θ, φ)，這裡的θ是極角而φ是方位角

向量場 A｜$A_x \hat{\mathbf x} + A_y \hat{\mathbf y} + A_z \hat{\mathbf z}$｜$A_\rho \hat{\boldsymbol \rho} + A_\varphi \hat{\boldsymbol \varphi} + A_z \hat{\mathbf z}$｜$A_r \hat{\mathbf r} + A_\theta \hat{\boldsymbol \theta} + A_\varphi \hat{\boldsymbol \varphi}$

梯度 ∇f｜${\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y} + {\partial f \over \partial z}\hat{\mathbf z}$｜${\partial f \over \partial \rho}\hat{\boldsymbol \rho} + {1 \over \rho}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi} + {\partial f \over \partial z}\hat{\mathbf z}$｜${\partial f \over \partial r}\hat{\mathbf r} + {1 \over r}{\partial f \over \partial \theta}\hat{\boldsymbol \theta} + {1 \over r\sin\theta}{\partial f \over \partial \varphi}\hat{\boldsymbol \varphi}$

散度 ∇ ⋅ A｜${\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}$｜${1 \over \rho}{\partial \left( \rho A_\rho \right) \over \partial \rho} + {1 \over \rho}{\partial A_\varphi \over \partial \varphi} + {\partial A_z \over \partial z}$｜${1 \over r^2}{\partial \left( r^2 A_r \right) \over \partial r} + {1 \over r\sin\theta}{\partial \over \partial \theta} \left( A_\theta\sin\theta \right) + {1 \over r\sin\theta}{\partial A_\varphi \over \partial \varphi}$

旋度 ∇ × A｜$\begin{align} \left(\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right) &\hat{\mathbf x} \\ + \left(\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right) &\hat{\mathbf y} \\ + \left(\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right) &\hat{\mathbf z} \end{align}$｜$\begin{align} \left( \frac{1}{\rho} \frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z} \right) &\hat{\boldsymbol \rho} \\ + \left( \frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho} \right) &\hat{\boldsymbol \varphi} \\ {}+ \frac{1}{\rho} \left( \frac{\partial \left(\rho A_\varphi\right)}{\partial \rho} - \frac{\partial A_\rho}{\partial \varphi} \right) &\hat{\mathbf z} \end{align}$｜$\begin{align} \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left(A_\varphi\sin\theta \right) - \frac{\partial A_\theta}{\partial \varphi} \right) &\hat{\mathbf r} \\ {}+ \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial A_r}{\partial \varphi} - \frac{\partial}{\partial r} \left( r A_\varphi \right) \right) &\hat{\boldsymbol \theta} \\ {}+ \frac{1}{r} \left( \frac{\partial}{\partial r} \left( r A_{\theta} \right) - \frac{\partial A_r}{\partial \theta} \right) &\hat{\boldsymbol \varphi} \end{align}$

拉普拉斯算子 ∇2f ≡ ∆f｜${\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}$｜${1 \over \rho}{\partial \over \partial \rho}\left(\rho {\partial f \over \partial \rho}\right) + {1 \over \rho^2}{\partial^2 f \over \partial \varphi^2} + {\partial^2 f \over \partial z^2}$｜${1 \over r^2}{\partial \over \partial r}\!\left(r^2 {\partial f \over \partial r}\right) \!+\!{1 \over r^2\!\sin\theta}{\partial \over \partial \theta}\!\left(\sin\theta {\partial f \over \partial \theta}\right) \!+\!{1 \over r^2\!\sin^2\theta}{\partial^2 f \over \partial \varphi^2}$

向量拉普拉斯算子 ∇2A ≡ ∆A｜$\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z}$｜$\begin{align} \mathopen{}\left(\nabla^2 A_\rho - \frac{A_\rho}{\rho^2} - \frac{2}{\rho^2} \frac{\partial A_\varphi}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \rho} \\ + \mathopen{}\left(\nabla^2 A_\varphi - \frac{A_\varphi}{\rho^2} + \frac{2}{\rho^2} \frac{\partial A_\rho}{\partial \varphi}\right)\mathclose{} &\hat{\boldsymbol \varphi} \\ {}+ \nabla^2 A_z &\hat{\mathbf z} \end{align}$｜$\begin{align} \left(\nabla^2 A_r - \frac{2 A_r}{r^2} - \frac{2}{r^2\sin\theta} \frac{\partial \left(A_\theta \sin\theta\right)}{\partial\theta} - \frac{2}{r^2\sin\theta}{\frac{\partial A_\varphi}{\partial \varphi}}\right) &\hat{\mathbf r} \\ + \left(\nabla^2 A_\theta - \frac{A_\theta}{r^2\sin^2\theta} + \frac{2}{r^2} \frac{\partial A_r}{\partial \theta} - \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\varphi}{\partial \varphi}\right) &\hat{\boldsymbol \theta} \\ + \left(\nabla^2 A_\varphi - \frac{A_\varphi}{r^2\sin^2\theta} + \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \varphi} + \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \varphi}\right) &\hat{\boldsymbol \varphi} \end{align}$

物質導數 (A ⋅ ∇)B｜$\mathbf{A} \cdot \nabla B_x \hat{\mathbf x} + \mathbf{A} \cdot \nabla B_y \hat{\mathbf y} + \mathbf{A} \cdot \nabla B_z \hat{\mathbf{z}}$｜$\begin{align} \left(A_\rho \frac{\partial B_\rho}{\partial \rho}+\frac{A_\varphi}{\rho}\frac{\partial B_\rho}{\partial \varphi}+A_z\frac{\partial B_\rho}{\partial z}-\frac{A_\varphi B_\varphi}{\rho}\right) &\hat{\boldsymbol \rho} \\ + \left(A_\rho \frac{\partial B_\varphi}{\partial \rho} + \frac{A_\varphi}{\rho}\frac{\partial B_\varphi}{\partial \varphi} + A_z\frac{\partial B_\varphi}{\partial z} + \frac{A_\varphi B_\rho}{\rho}\right) &\hat{\boldsymbol \varphi}\\ + \left(A_\rho \frac{\partial B_z}{\partial \rho}+\frac{A_\varphi}{\rho}\frac{\partial B_z}{\partial \varphi}+A_z\frac{\partial B_z}{\partial z}\right) &\hat{\mathbf z} \end{align}$｜$\begin{align} \left( A_r \frac{\partial B_r}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_r}{\partial \varphi} - \frac{A_\theta B_\theta + A_\varphi B_\varphi}{r} \right) &\hat{\mathbf r} \\ + \left( A_r \frac{\partial B_\theta}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\theta}{\partial \varphi} + \frac{A_\theta B_r}{r} - \frac{A_\varphi B_\varphi\cot\theta}{r} \right) &\hat{\boldsymbol \theta} \\ + \left( A_r \frac{\partial B_\varphi}{\partial r} + \frac{A_\theta}{r} \frac{\partial B_\varphi}{\partial \theta} + \frac{A_\varphi}{r\sin\theta} \frac{\partial B_\varphi}{\partial \varphi} + \frac{A_\varphi B_r}{r} + \frac{A_\varphi B_\theta \cot\theta}{r} \right) &\hat{\boldsymbol \varphi} \end{align}$

張量散度 ∇ ⋅ T｜$\begin{align} \left(\frac{\partial T_{xx}}{\partial x}+\frac{\partial T_{yx}}{\partial y}+\frac{\partial T_{zx}}{\partial z}\right)&\hat{\mathbf x} \\ +\left(\frac{\partial T_{xy}}{\partial x}+\frac{\partial T_{yy}}{\partial y}+\frac{\partial T_{zy}}{\partial z}\right)&\hat{\mathbf y} \\ +\left(\frac{\partial T_{xz}}{\partial x}+\frac{\partial T_{yz}}{\partial y}+\frac{\partial T_{zz}}{\partial z}\right)&\hat{\mathbf z} \end{align}$｜$\begin{align} \left[\frac{\partial T_{\rho\rho}}{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi\rho}}{\partial\varphi}+\frac{\partial T_{z\rho}}{\partial z}+\frac1\rho(T_{\rho\rho}-T_{\varphi\varphi})\right]&\hat{\boldsymbol\rho} \\ +\left[\frac{\partial T_{\rho\varphi}}{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi\varphi}}{\partial\varphi}+\frac{\partial T_{z\varphi}}{\partial z}+\frac1\rho(T_{\rho\varphi}+T_{\varphi\rho})\right]&\hat{\boldsymbol\varphi} \\ +\left[\frac{\partial T_{\rho z}}{\partial\rho}+\frac1\rho\frac{\partial T_{\varphi z}}{\partial\varphi}+\frac{\partial T_{zz}}{\partial z}+\frac{T_{\rho z}}\rho\right]&\hat{\mathbf z} \end{align}$｜$\begin{align} \left[\frac{\partial T_{rr}}{\partial r}+2\frac{T_{rr}}r+\frac1r\frac{\partial T_{\theta r}}{\partial\theta}+\frac{\cot\theta}rT_{\theta r}+\frac1{r\sin\theta}\frac{\partial T_{\varphi r}}{\partial\varphi}-\frac1r(T_{\theta\theta}+T_{\varphi\varphi})\right]&\hat{\mathbf r} \\ +\left[\frac{\partial T_{r\theta}}{\partial r}+2\frac{T_{r\theta}}r+\frac1r\frac{\partial T_{\theta\theta}}{\partial\theta}+\frac{\cot\theta}rT_{\theta\theta}+\frac1{r\sin\theta}\frac{\partial T_{\varphi\theta}}{\partial\varphi}+\frac{T_{\theta r}}r-\frac{\cot\theta}rT_{\varphi\varphi}\right]&\hat{\boldsymbol\theta} \\ +\left[\frac{\partial T_{r\varphi}}{\partial r}+2\frac{T_{r\varphi}}r+\frac1r\frac{\partial T_{\theta\varphi}}{\partial\theta}+\frac1{r\sin\theta}\frac{\partial T_{\varphi\varphi}}{\partial\varphi}+\frac {T_{\varphi r}}{r}+\frac{\cot\theta}{r} (T_{\theta\varphi}+T_{\varphi\theta})\right]&\hat{\boldsymbol\varphi} \end{align}$

微分位移 dℓ｜$dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}$｜$d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z}$｜$dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi}$

微分正規面積 dS｜$\begin{align} dy \, dz &\, \hat{\mathbf x} \\ {} + dx \, dz &\, \hat{\mathbf y} \\ {} + dx \, dy &\, \hat{\mathbf z} \end{align}$｜$\begin{align} \rho \, d\varphi \, dz &\, \hat{\boldsymbol \rho} \\ {} + d\rho \, dz &\, \hat{\boldsymbol \varphi} \\ {} + \rho \, d\rho \, d\varphi &\, \hat{\mathbf z} \end{align}$｜$\begin{align} r^2 \sin\theta \, d\theta \, d\varphi &\, \hat{\mathbf r} \\ {} + r \sin\theta \, dr \, d\varphi &\, \hat{\boldsymbol \theta} \\ {} + r \, dr \, d\theta &\, \hat{\boldsymbol \varphi} \end{align}$

微分體積 dV｜$dx \, dy \, dz$｜$\rho \, d\rho \, d\varphi \, dz$｜$r^2 \sin\theta \, dr \, d\theta \, d\varphi$

Alpha本頁對極角採用$\theta$對方位角採用$\varphi$，這是在物理學中常用的符號。某些來源在這些公式中對方位角採用$\theta$對極角採用$\varphi$，這是常用數學符號，如果需要這種數學公式，可對換上表公式中的$\theta$和$\varphi$。

### 非平凡的演算規則

- $$\operatorname{div} \, \operatorname{grad} f \equiv \nabla \cdot \nabla f \equiv \nabla^2 f$$
- $$\operatorname{curl} \, \operatorname{grad} f \equiv \nabla \times \nabla f = \mathbf 0$$
- $$\operatorname{div} \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot (\nabla \times \mathbf{A}) = 0$$
- $\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$（del的拉格朗日公式）
- $$\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f$$

## 直角坐標系推導

$$\begin{align}\operatorname{div} \mathbf A = \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S}}{\iiint_V dV} &= \frac{A_x(x+dx)dydz - A_x(x)dydz + A_y(y+dy)dxdz - A_y(y)dxdz + A_z(z+dz)dxdy - A_z(z)dxdy}{dxdydz} \\ &= \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}\end{align}$$

$$\begin{align}(\operatorname{curl} \mathbf A)_x = \lim_{S^{\perp \mathbf{\hat x}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} &= \frac{A_z(y+dy)dz - A_z(y)dz + A_y(z)dy - A_y(z+dz)dy }{dydz} \\ &= \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\end{align}$$

$(\operatorname{curl} \mathbf A)_y$和$(\operatorname{curl} \mathbf A)_z$的表達式可以同理得出。

註：第一式中的$A_x(x+dx)$是$A_x$在$x+dx$時的量值，並非$A_x$值乘上$x+dx$。以下圓柱座標、球座標的推導中亦然。

## 圓柱坐標系推導

$$\begin{align} \operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S}}{\iiint_V dV} \\ &= \frac{A_\rho(\rho+d\rho)(\rho+d\rho)d\phi dz - A_\rho(\rho)\rho d\phi dz + A_\phi(\phi+d\phi)d\rho dz - A_\phi(\phi)d\rho dz + A_z(z+dz)d\rho (\rho +d\rho/2)d\phi - A_z(z)d\rho (\rho +d\rho/2) d\phi}{\rho d\phi d\rho dz} \\ &= \frac 1 \rho \frac{\partial (\rho A_\rho)}{\partial \rho} + \frac 1 \rho \frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{\partial z} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_\rho &= \lim_{S^{\perp \boldsymbol{\hat \rho}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} \\ &= \frac{A_\phi (z)(\rho+d\rho)d\phi - A_\phi(z+dz)(\rho+d\rho)d\phi + A_z(\phi + d\phi)dz - A_z(\phi)dz}{(\rho+d\rho)d\phi dz} \\ &= -\frac{\partial A_\phi}{\partial z} + \frac{1}{\rho} \frac{\partial A_z}{\partial \phi} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_\phi &= \lim_{S^{\perp \boldsymbol{\hat \phi}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} \\ &= \frac{A_z (\rho)dz - A_z(\rho + d\rho)dz + A_\rho(z+dz)d\rho - A_\rho(z)d\rho}{d\rho dz} \\ &= -\frac{\partial A_z}{\partial \rho} + \frac{\partial A_\rho}{\partial z} \end{align}$$

$$\begin{align} (\operatorname{curl} \mathbf A)_z &= \lim_{S^{\perp \boldsymbol{\hat z}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} \\ &= \frac{A_\rho(\phi)d\rho - A_\rho(\phi + d\phi)d\rho + A_\phi(\rho + d\rho)(\rho + d\rho)d\phi - A_\phi(\rho)\rho d\phi}{\rho d\rho d\phi} \\ &= -\frac{1}{\rho}\frac{\partial A_\rho}{\partial \phi} + \frac{1}{\rho} \frac{\partial (\rho A_\phi)}{\partial \rho} \end{align}$$

$$\begin{align} \operatorname{curl} \mathbf A &= (\operatorname{curl} \mathbf A)_\rho \hat{\boldsymbol \rho} + (\operatorname{curl} \mathbf A)_\phi \hat{\boldsymbol \phi} + (\operatorname{curl} \mathbf A)_z \hat{\boldsymbol z} \\ &= \left(\frac{1}{\rho} \frac{\partial A_z}{\partial \phi} -\frac{\partial A_\phi}{\partial z} \right) \hat{\boldsymbol \rho} + \left(\frac{\partial A_\rho}{\partial z}-\frac{\partial A_z}{\partial \rho} \right) \hat{\boldsymbol \phi} + \frac{1}{\rho}\left(\frac{\partial (\rho A_\phi)}{\partial \rho} - \frac{\partial A_\rho}{\partial \phi} \right) \hat{\boldsymbol z} \end{align}$$

## 球坐標系推導

$$\begin{align}\operatorname{div} \mathbf A &= \lim_{V\to 0} \frac{\iint_{\partial V} \mathbf A \cdot d\mathbf{S}}{\iiint_V dV} \\ &= \frac{A_r(r+dr)(r+dr)d\theta\, (r+dr)\sin\theta d\phi - A_r(r)rd\theta\, r\sin\theta d\phi + A_\theta(\theta+d\theta)\sin(\theta + d\theta)\,r dr d\phi - A_\theta(\theta)\sin(\theta)\,r dr d\phi + A_\phi(\phi + d\phi) (r + dr/2)dr d\theta - A_\phi(\phi)(r + dr/2)dr d\theta}{dr\,rd\theta\,r\sin\theta d\phi} \\ &= \frac{1}{r^2}\frac{\partial (r^2A_r)}{\partial r} + \frac{1}{r \sin\theta} \frac{\partial(A_\theta\sin\theta)}{\partial \theta} + \frac{1}{r \sin\theta} \frac{\partial A_\phi}{\partial \phi} \end{align}$$

$$\begin{align}(\operatorname{curl} \mathbf A)_r = \lim_{S^{\perp \boldsymbol{\hat r}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} &= \frac{A_\theta(\phi)\,r d\theta + A_\phi(\theta + d\theta)\,r \sin(\theta + d\theta) d\phi <pre> - A_\theta(\phi + d\phi)\,r d\theta - A_\phi(\theta)\,r\sin(\theta) d\phi}{r d\theta\,r\sin\theta d\phi} \\ </pre> &= \frac{1}{r\sin\theta}\frac{\partial(A_\phi \sin\theta)}{\partial \theta} <pre> - \frac{1}{r\sin\theta} \frac{\partial A_\theta}{\partial \phi}\end{align}$$

$$\begin{align}(\operatorname{curl} \mathbf A)_\theta = \lim_{S^{\perp \boldsymbol{\hat \theta}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} &= \frac{A_\phi(r)\,r \sin\theta d\phi + A_r(\phi + d\phi)dr <pre> - A_\phi(r+dr)(r+dr)\sin\theta d\phi - A_r(\phi)dr}{dr \, r \sin \theta d\phi} \\ </pre> &= \frac{1}{r\sin\theta}\frac{\partial A_r}{\partial \phi} <pre> - \frac{1}{r} \frac{\partial (rA_\phi)}{\partial r}\end{align}$$

$$\begin{align}(\operatorname{curl} \mathbf A)_\phi = \lim_{S^{\perp \boldsymbol{\hat \phi}}\to 0} \frac{\int_{\partial S} \mathbf A \cdot d\mathbf{\ell}}{\iint_{S} dS} &= \frac{A_r(\theta)dr + A_\theta(r+dr)(r+dr)d\theta <pre> - A_r(\theta+d\theta)dr - A_\theta(r)\, r d\theta}{(r+dr/2) dr d\theta} \\ </pre> &= \frac{1}{r}\frac{\partial(rA_\theta)}{\partial r} <pre> - \frac{1}{r} \frac{\partial A_r}{\partial \theta}\end{align}$$

$$\operatorname{curl} \mathbf A = (\operatorname{curl} \mathbf A)_r \, \hat{\boldsymbol r} + (\operatorname{curl} \mathbf A)_\theta \, \hat{\boldsymbol \theta} + (\operatorname{curl} \mathbf A)_\phi \, \hat{\boldsymbol \phi} = \frac{1}{r\sin\theta} \left(\frac{\partial(A_\phi \sin\theta)}{\partial \theta}-\frac{\partial A_\theta}{\partial \phi} \right) \hat{\boldsymbol r} +\frac{1}{r} \left(\frac{1}{\sin\theta}\frac{\partial A_r}{\partial \phi} - \frac{\partial (rA_\phi)}{\partial r} \right) \hat{\boldsymbol \theta} + \frac{1}{r}\left(\frac{\partial(rA_\theta)}{\partial r} - \frac{\partial A_r}{\partial \theta} \right) \hat{\boldsymbol \phi}$$

## 單位向量轉換公式

坐標參數u的單位向量以如下方式定義，u的小的正值改變導致位置向量$\boldsymbol\vec{r}$在$\boldsymbol\hat{u}$方向上的改變。因此：

$${\partial\boldsymbol\vec{r} \over \partial u}={\partial{s} \over \partial u}{\boldsymbol\hat{u}}$$

這裡的s是弧長參數。

對於兩組坐標系$u_i$和$v_j$，依據鏈式法則：

$$d\boldsymbol\vec{r}=\sum_{i}{\partial{\boldsymbol\vec{r}}\over\partial u_i}du_i=\sum_{i}{\partial{s}\over\partial u_i}\boldsymbol\hat{u_i}du_i=\sum_{j}{\partial{s}\over\partial v_j}\boldsymbol\hat{v_j}dv_j=\sum_{j}{\partial{s}\over\partial v_j}\boldsymbol\hat{v_j}\sum_{i}{\partial{v_j}\over\partial u_i}du_i=\sum_{i}\sum_{j}{\partial{s}\over\partial v_j}{\partial{v_j}\over\partial u_i}\boldsymbol\hat{v_j}du_i$$

現在，使除了一個之外的所有$du_i=0$並在兩邊除以對應的坐標參數的微分，得到：

$${\partial{s}\over\partial u_i}\boldsymbol\hat{u_i}=\sum_{j}{\partial{s}\over\partial v_j}{\partial{v_j}\over\partial u_i}\boldsymbol\hat{v_j}$$

## 外部連結

- Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.
- Curl, Divergence, and Gradient in Cylindrical and Spherical Coordinate Systems.