各种坐标系下的散度、梯度、旋度公式

引言

本文介绍了散度、梯度和旋度在直角坐标系、柱坐标系和球坐标系三种常见坐标系下的表示。记录一下，具体可以利用梅拉系数进行推导。

谨记：
梯度：标量求梯度得到矢量。
散度：矢量求散度得到标量。
旋度：矢量求旋度得到矢量。

1.直角坐标系

标量表示
f=f(x,y,z)f=f(x,y,z)f=f(x,y,z)

矢量表示
f→=fxex→+fyey→+fzez→\mathbf{\overrightarrow{f}}=f_x\mathbf{\overrightarrow{e_x}}+f_y\mathbf{\overrightarrow{e_y}}+f_z\mathbf{\overrightarrow{e_z}}f=fxex+fyey+fzez

梯度：
▽f=∂f∂xex→+∂f∂yey→+∂f∂zez→\bigtriangledown {f}=\frac{\partial f}{\partial x}\mathbf{\overrightarrow{e_x}}+\frac{\partial f}{\partial y}\mathbf{\overrightarrow{e_y}}+\frac{\partial f}{\partial z}\mathbf{\overrightarrow{e_z}}▽f=∂x∂fex+∂y∂fey+∂z∂fez

散度：
▽⋅f→=∂fx∂x+∂fy∂y+∂fz∂z\bigtriangledown \cdot \mathbf{\overrightarrow{f}}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}▽⋅f=∂x∂fx+∂y∂fy+∂z∂fz

旋度：
▽×f→=[ex→ey→ez→∂∂x∂∂y∂∂zfxfyfz]=(∂fz∂y−∂fy∂z)ex→+(∂fx∂z−∂fz∂x)ey→+(∂fy∂x−∂fx∂y)ez→\begin{aligned} \bigtriangledown \times \mathbf{\overrightarrow{f}}&= \begin{bmatrix} \mathbf{\overrightarrow{e_x}} &\mathbf{\overrightarrow{e_y}} &\mathbf{\overrightarrow{e_z}} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\f_x & f_y & f_z\end{bmatrix}\\&=(\frac{\partial f_z}{\partial y}-\frac{\partial f_y}{\partial z})\mathbf{\overrightarrow{e_x}}+(\frac{\partial f_x}{\partial z}-\frac{\partial f_z}{\partial x})\mathbf{\overrightarrow{e_y}}+(\frac{\partial f_y}{\partial x}-\frac{\partial f_x}{\partial y})\mathbf{\overrightarrow{e_z}} \end{aligned} ▽×f=⎣⎡ex∂x∂fxey∂y∂fyez∂z∂fz⎦⎤=(∂y∂fz−∂z∂fy)ex+(∂z∂fx−∂x∂fz)ey+(∂x∂fy−∂y∂fx)ez

2.柱坐标系

标量表示
f=f(ρ,θ,z)f=f(\rho,\theta,z)f=f(ρ,θ,z)

矢量表示
f→=fρeρ→+fθeθ→+fzez→\mathbf{\overrightarrow{f}}=f_\rho\mathbf{\overrightarrow{e_\rho}}+f_\theta\mathbf{\overrightarrow{e_\theta}}+f_z\mathbf{\overrightarrow{e_z}}f=fρeρ+fθeθ+fzez

梯度：
▽f=∂f∂ρeρ→+1ρ∂f∂θeθ→+∂f∂zez→\bigtriangledown {f} =\frac{\partial f}{\partial \rho}\mathbf{\overrightarrow{e_\rho}}+\frac{1}{\rho}\frac{\partial f}{\partial \theta}\mathbf{\overrightarrow{e_\theta}}+\frac{\partial f}{\partial z}\mathbf{\overrightarrow{e_z}}▽f=∂ρ∂feρ+ρ1∂θ∂feθ+∂z∂fez

散度：
▽⋅f→=1ρ∂(ρfρ)∂ρ+1ρ∂fθ∂θ+∂fz∂z\bigtriangledown \cdot \mathbf{\overrightarrow{f}}=\frac{1}{\rho}\frac{\partial (\rho f_\rho)}{\partial \rho}+\frac{1}{\rho}\frac{\partial f_\theta}{\partial \theta}+\frac{\partial f_z}{\partial z}▽⋅f=ρ1∂ρ∂(ρfρ)+ρ1∂θ∂fθ+∂z∂fz

旋度：
▽×f→=1ρ[eρ→ρeθ→ez→∂∂ρ∂∂θ∂∂zfρρfθfz]=1ρ[(∂fz∂θ−∂(ρfθ)∂z)eρ→+(∂fρ∂z−∂fz∂ρ)ρeθ→+(∂(ρfθ)∂ρ−∂fρ∂θ)ez→]\begin{aligned} \bigtriangledown \times \mathbf{\overrightarrow{f}}&=\frac{1}{\rho} \begin{bmatrix} \mathbf{\overrightarrow{e_\rho}} &\rho\mathbf{\overrightarrow{e_\theta}} &\mathbf{\overrightarrow{e_z}} \\ \frac{\partial}{\partial \rho} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial z} \\f_\rho & \rho f_\theta & f_z\end{bmatrix}\\&=\frac{1}{\rho}\left[(\frac{\partial f_z}{\partial \theta}-\frac{\partial (\rho f_\theta)}{\partial z})\mathbf{\overrightarrow{e_\rho}}+(\frac{\partial f_\rho}{\partial z}-\frac{\partial f_z}{\partial \rho})\rho\mathbf{\overrightarrow{e_\theta}}+(\frac{\partial (\rho f_\theta)}{\partial \rho}-\frac{\partial f_\rho}{\partial \theta})\mathbf{\overrightarrow{e_z}}\right] \end{aligned} ▽×f=ρ1⎣⎡eρ∂ρ∂fρρeθ∂θ∂ρfθez∂z∂fz⎦⎤=ρ1[(∂θ∂fz−∂z∂(ρfθ))eρ+(∂z∂fρ−∂ρ∂fz)ρeθ+(∂ρ∂(ρfθ)−∂θ∂fρ)ez]

3.球坐标系

(请注意这里的θ\thetaθ和柱坐标系中的θ\thetaθ的定义不同，详细见图)

标量表示
f=f(ρ,θ,ϕ)f=f(\rho,\theta,\phi)f=f(ρ,θ,ϕ)

矢量表示
f→=fρeρ→+fθeθ→+fϕeϕ→\mathbf{\overrightarrow{f}}=f_\rho\mathbf{\overrightarrow{e_\rho}}+f_\theta\mathbf{\overrightarrow{e_\theta}}+f_\phi\mathbf{\overrightarrow{e_\phi}}f=fρeρ+fθeθ+fϕeϕ

梯度：
▽f=∂f∂ρeρ→+1ρ∂f∂θeθ→+1ρsinθ∂f∂ϕeϕ→\bigtriangledown {f} =\frac{\partial f}{\partial \rho}\mathbf{\overrightarrow{e_\rho}}+\frac{1}{\rho}\frac{\partial f}{\partial \theta}\mathbf{\overrightarrow{e_\theta}}+\frac{1}{\rho sin\theta}\frac{\partial f}{\partial \phi}\mathbf{\overrightarrow{e_\phi}}▽f=∂ρ∂feρ+ρ1∂θ∂feθ+ρsinθ1∂ϕ∂feϕ

散度：
▽⋅f→=1ρ2∂(ρ2fρ)∂ρ+1ρsinθ∂(sinθfθ)∂θ+1ρsinθ∂fϕ∂ϕ\bigtriangledown \cdot \mathbf{\overrightarrow{f}}=\frac{1}{\rho^2}\frac{\partial (\rho^2 f_\rho)}{\partial \rho}+\frac{1}{\rho sin\theta}\frac{\partial (sin\theta f_\theta)}{\partial \theta}+\frac{1}{\rho sin\theta}\frac{\partial f_\phi}{\partial \phi}▽⋅f=ρ21∂ρ∂(ρ2fρ)+ρsinθ1∂θ∂(sinθfθ)+ρsinθ1∂ϕ∂fϕ

旋度：
▽×f→=1ρ2sinθ[eρ→ρeθ→ρsinθeϕ→∂∂ρ∂∂θ∂∂ϕfρρfθρsinθfϕ]=1ρ2sinθ[(∂(ρsinθfϕ)∂θ−∂(ρfθ)∂ϕ)eρ→+(∂fρ∂ϕ−∂(ρsinθfϕ)∂ρ)ρeθ→+(∂(ρfθ)∂ρ−∂fρ∂θ)ρsinθeϕ→]\begin{aligned} \bigtriangledown \times \mathbf{\overrightarrow{f}}&=\frac{1}{\rho^2sin\theta} \begin{bmatrix} \mathbf{\overrightarrow{e_\rho}} &\rho\mathbf{\overrightarrow{e_\theta}} &\rho sin\theta\mathbf{\overrightarrow{e_\phi}} \\ \frac{\partial}{\partial \rho} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial \phi} \\f_\rho & \rho f_\theta & \rho sin\theta f_\phi\end{bmatrix}\\&=\frac{1}{\rho^2sin\theta}\left[(\frac{\partial (\rho sin\theta f_\phi)}{\partial \theta}-\frac{\partial (\rho f_\theta)}{\partial \phi})\mathbf{\overrightarrow{e_\rho}}+(\frac{\partial f_\rho}{\partial \phi}-\frac{\partial (\rho sin\theta f_\phi)}{\partial \rho})\rho\mathbf{\overrightarrow{e_\theta}}+(\frac{\partial (\rho f_\theta)}{\partial \rho}-\frac{\partial f_\rho}{\partial \theta})\rho sin\theta \mathbf{\overrightarrow{e_\phi}}\right] \end{aligned} ▽×f=ρ2sinθ1⎣⎡eρ∂ρ∂fρρeθ∂θ∂ρfθρsinθeϕ∂ϕ∂ρsinθfϕ⎦⎤=ρ2sinθ1[(∂θ∂(ρsinθfϕ)−∂ϕ∂(ρfθ))eρ+(∂ϕ∂fρ−∂ρ∂(ρsinθfϕ))ρeθ+(∂ρ∂(ρfθ)−∂θ∂fρ)ρsinθeϕ]

由于笔者之前一直想不清楚柱坐标系和球坐标系下三度的变换，于是查阅资料，其中发现[1]是极好的，推荐一看！！！详细推导见下面的参考资料[1]，利用梅拉系数很容易地可以进行推导，利用拉梅系数还可以很直接地推导了斯托克斯公式和高斯公式，非常地简单易懂！！感谢知乎@弧长长长长长。另外[2]写得也比较详细，可以用作其他的参考。

[1] 浅谈：拉梅系数那些事儿
[2] 柱面及球面坐标系中散度、旋度的应用

附：

1.常用的梅拉系数

直角坐标系：
Hx=1,Hy=1,Hz=1H_x=1,H_y=1,H_z=1Hx=1,Hy=1,Hz=1

柱坐标系：
Hr=1,Hθ=r,Hz=1H_r=1,H_\theta=r,H_z=1Hr=1,Hθ=r,Hz=1

球坐标系：
Hr=1,Hθ=r,Hϕ=rsinθH_r=1,H_\theta=r,H_\phi=rsin\thetaHr=1,Hθ=r,Hϕ=rsinθ

2.通用公式

梯度：

▽f=1H1∂f∂q1e1→+1H2∂f∂q2e2→+1H3∂f∂q3e3→\bigtriangledown f=\frac{1}{H_{1}} \frac{\partial f}{\partial q_{1}} \mathbf{\overrightarrow{e_1}}+\frac{1}{H_{2}} \frac{\partial f}{\partial q_{2}} \mathbf{\overrightarrow{e_2}}+\frac{1}{H_{3}} \frac{\partial f}{\partial q_{3}} \mathbf{\overrightarrow{e_3}}▽f=H11∂q1∂fe1+H21∂q2∂fe2+H31∂q3∂fe3

散度：
div⁡r=lim⁡V→0∮SridSV=1H1H2H3(∂(r1H2H3)∂q1+∂(r2H1H3)∂q2+∂(r3H2H1)∂q3)\operatorname{div} \boldsymbol{r}=\lim _{V \rightarrow 0} \frac{\oint_{S} r_{i} d S}{V}=\frac{1}{H_{1} H_{2} H_{3}}\left(\frac{\partial\left(r_{1} H_{2} H_{3}\right)}{\partial q_{1}}+\frac{\partial\left(r_{2} H_{1} H_{3}\right)}{\partial q_{2}}+\frac{\partial\left(r_{3} H_{2} H_{1}\right)}{\partial q_{3}}\right)divr=V→0limV∮SridS=H1H2H31(∂q1∂(r1H2H3)+∂q2∂(r2H1H3)+∂q3∂(r3H2H1))

旋度：

{rot⁡q1r=1H2H3[∂(r3H3)∂q2−∂(r2H2)∂q3]rot⁡q2r=1H1H3[∂(r1H1)∂q3−∂(r3H3)∂q1]rot⁡q3r=1H2H1[∂(r2H2)∂q1−∂(r1H1)∂q2]\left\{\begin{array}{l} \operatorname{rot}_{q_{1}} \boldsymbol{r}=\frac{1}{H_{2} H_{3}}\left[\frac{\partial\left(r_{3} H_{3}\right)}{\partial q_{2}}-\frac{\partial\left(r_{2} H_{2}\right)}{\partial q_{3}}\right] \\ \operatorname{rot}_{q_{2}} \boldsymbol{r}=\frac{1}{H_{1} H_{3}}\left[\frac{\partial\left(r_{1} H_{1}\right)}{\partial q_{3}}-\frac{\partial\left(r_{3} H_{3}\right)}{\partial q_{1}}\right] \\ \operatorname{rot}_{q_{3}} \boldsymbol{r}=\frac{1}{H_{2} H_{1}}\left[\frac{\partial\left(r_{2} H_{2}\right)}{\partial q_{1}}-\frac{\partial\left(r_{1} H_{1}\right)}{\partial q_{2}}\right] \end{array}\right.⎩⎪⎪⎪⎨⎪⎪⎪⎧rotq1r=H2H31[∂q2∂(r3H3)−∂q3∂(r2H2)]rotq2r=H1H31[∂q3∂(r1H1)−∂q1∂(r3H3)]rotq3r=H2H11[∂q1∂(r2H2)−∂q2∂(r1H1)]
或
rot⁡r=1H1H2H3∣H1e1H2e2H3e3∂∂q1∂∂q2∂∂q3H1r1H2r2H3r3∣\operatorname{rot} \boldsymbol{r}=\frac{1}{H_{1} H_{2} H_{3}}\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} r_{1} & H_{2} r_{2} & H_{3} r_{3} \end{array}\right|rotr=H1H2H31∣∣∣∣∣∣H1e1∂q1∂H1r1H2e2∂q2∂H2r2H3e3∂q3∂H3r3∣∣∣∣∣∣