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
→
=
f
x
e
x
→
+
f
y
e
y
→
+
f
z
e
z
→
\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
∂
x
e
x
→
+
∂
f
∂
y
e
y
→
+
∂
f
∂
z
e
z
→
\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
→
=
∂
f
x
∂
x
+
∂
f
y
∂
y
+
∂
f
z
∂
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
→
=
[
e
x
→
e
y
→
e
z
→
∂
∂
x
∂
∂
y
∂
∂
z
f
x
f
y
f
z
]
=
(
∂
f
z
∂
y
−
∂
f
y
∂
z
)
e
x
→
+
(
∂
f
x
∂
z
−
∂
f
z
∂
x
)
e
y
→
+
(
∂
f
y
∂
x
−
∂
f
x
∂
y
)
e
z
→
\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
→
=
f
ρ
e
ρ
→
+
f
θ
e
θ
→
+
f
z
e
z
→
\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
∂
ρ
e
ρ
→
+
1
ρ
∂
f
∂
θ
e
θ
→
+
∂
f
∂
z
e
z
→
\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
→
=
1
ρ
∂
(
ρ
f
ρ
)
∂
ρ
+
1
ρ
∂
f
θ
∂
θ
+
∂
f
z
∂
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
→
=
1
ρ
[
e
ρ
→
ρ
e
θ
→
e
z
→
∂
∂
ρ
∂
∂
θ
∂
∂
z
f
ρ
ρ
f
θ
f
z
]
=
1
ρ
[
(
∂
f
z
∂
θ
−
∂
(
ρ
f
θ
)
∂
z
)
e
ρ
→
+
(
∂
f
ρ
∂
z
−
∂
f
z
∂
ρ
)
ρ
e
θ
→
+
(
∂
(
ρ
f
θ
)
∂
ρ
−
∂
f
ρ
∂
θ
)
e
z
→
]
\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
\theta
θ和柱坐标系中的
θ
\theta
θ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
→
=
f
ρ
e
ρ
→
+
f
θ
e
θ
→
+
f
ϕ
e
ϕ
→
\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
∂
ρ
e
ρ
→
+
1
ρ
∂
f
∂
θ
e
θ
→
+
1
ρ
s
i
n
θ
∂
f
∂
ϕ
e
ϕ
→
\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
→
=
1
ρ
2
∂
(
ρ
2
f
ρ
)
∂
ρ
+
1
ρ
s
i
n
θ
∂
(
s
i
n
θ
f
θ
)
∂
θ
+
1
ρ
s
i
n
θ
∂
f
ϕ
∂
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
→
=
1
ρ
2
s
i
n
θ
[
e
ρ
→
ρ
e
θ
→
ρ
s
i
n
θ
e
ϕ
→
∂
∂
ρ
∂
∂
θ
∂
∂
ϕ
f
ρ
ρ
f
θ
ρ
s
i
n
θ
f
ϕ
]
=
1
ρ
2
s
i
n
θ
[
(
∂
(
ρ
s
i
n
θ
f
ϕ
)
∂
θ
−
∂
(
ρ
f
θ
)
∂
ϕ
)
e
ρ
→
+
(
∂
f
ρ
∂
ϕ
−
∂
(
ρ
s
i
n
θ
f
ϕ
)
∂
ρ
)
ρ
e
θ
→
+
(
∂
(
ρ
f
θ
)
∂
ρ
−
∂
f
ρ
∂
θ
)
ρ
s
i
n
θ
e
ϕ
→
]
\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：
H
x
=
1
,
H
y
=
1
,
H
z
=
1
H_x=1,H_y=1,H_z=1
Hx=1,Hy=1,Hz=1
柱坐标系：
H
r
=
1
,
H
θ
=
r
,
H
z
=
1
H_r=1,H_\theta=r,H_z=1
Hr=1,Hθ=r,Hz=1
球坐标系：
H
r
=
1
,
H
θ
=
r
,
H
ϕ
=
r
s
i
n
θ
H_r=1,H_\theta=r,H_\phi=rsin\theta
Hr=1,Hθ=r,Hϕ=rsinθ
2.通用公式
梯度：
▽
f
=
1
H
1
∂
f
∂
q
1
e
1
→
+
1
H
2
∂
f
∂
q
2
e
2
→
+
1
H
3
∂
f
∂
q
3
e
3
→
\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
∮
S
r
i
d
S
V
=
1
H
1
H
2
H
3
(
∂
(
r
1
H
2
H
3
)
∂
q
1
+
∂
(
r
2
H
1
H
3
)
∂
q
2
+
∂
(
r
3
H
2
H
1
)
∂
q
3
)
\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
q
1
r
=
1
H
2
H
3
[
∂
(
r
3
H
3
)
∂
q
2
−
∂
(
r
2
H
2
)
∂
q
3
]
rot
q
2
r
=
1
H
1
H
3
[
∂
(
r
1
H
1
)
∂
q
3
−
∂
(
r
3
H
3
)
∂
q
1
]
rot
q
3
r
=
1
H
2
H
1
[
∂
(
r
2
H
2
)
∂
q
1
−
∂
(
r
1
H
1
)
∂
q
2
]
\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
=
1
H
1
H
2
H
3
∣
H
1
e
1
H
2
e
2
H
3
e
3
∂
∂
q
1
∂
∂
q
2
∂
∂
q
3
H
1
r
1
H
2
r
2
H
3
r
3
∣
\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∣∣∣∣∣∣