查看“︁在圆柱和球坐标系中的del”︁的源代码

下面是常用于[[正交坐标系|正交]]{{en-link|曲线坐标系|Curvilinear coordinates}}中的一些[[向量微积分]]公式。

== 注释 ==

* 本文对[[球坐标系|球坐标]]使用标准符号[[ISO 80000-2]]，它取代了[[ISO 31-11#坐标系|ISO 31-11]]，（部分其他来源可能有着颠倒θ和φ的定义）：
**极角表示为θ：它是在z轴与连接原点和目标点的径向向量之间的角度。
**[[方位角]]表示为φ：它是在x轴与径向向量在xy面上的投影之间的角度。
* 函数{{nowrap|[[atan2]](y, x)}}可以用于替代数学函数{{nowrap|[[arctan]](y/x)}}。这是由于它的[[定义域]]和[[像 (数学)|像]]的缘故，经典arctan函数的像为{{nowrap|(−π/2, +π/2)}}，而atan2定义的像为{{nowrap|(−π, π]}}。
<!--(The expressions for the Del in spherical coordinates may need to be corrected)-->

== 坐标转换 ==

{| class="wikitable"
|+ 在直角、圆柱和球坐标间的变换<ref name="griffiths">{{Cite book|title=Introduction to Electrodynamics|last=Griffiths|first=David J.|publisher=Pearson|year=2012|isbn=978-0-321-85656-2}}</ref>
! colspan="2" rowspan="2" |
! colspan="3" | 从
|-
! 直角
! 圆柱
! 球
|-
! rowspan="3" | 到
! 直角
| <math>\begin{align}
  x &= x \\
  y &= y \\
  z &= z
\end{align}</math>
| <math>\begin{align}
  x &= \rho \cos\varphi \\
  y &= \rho \sin\varphi \\
  z &= z
\end{align}</math>
| <math>\begin{align}
  x &= r \sin\theta \cos\varphi \\
  y &= r \sin\theta \sin\varphi \\
  z &= r \cos\theta
\end{align}</math>
|-
! 圆柱
| <math>\begin{align}
  \rho &= \sqrt{x^2 + y^2} \\
  \varphi &= \arctan\left(\frac{y}{x}\right) \\
  z &= z
\end{align}</math>
| <math>\begin{align}
     \rho &= \rho \\
  \varphi &= \varphi \\
        z &= z
\end{align}</math>
| <math>\begin{align}
     \rho &= r \sin\theta \\
  \varphi &= \varphi \\
        z &= r\cos\theta
\end{align}</math>
|-
! 球
| <math>\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}</math>
| <math>\begin{align}
        r &= \sqrt{\rho^2 + z^2} \\
   \theta &= \arctan{\left(\frac{\rho}{z}\right)} \\
  \varphi &= \varphi
\end{align}</math>
| <math>\begin{align}
        r &= r \\
   \theta &= \theta \\
  \varphi &= \varphi
\end{align}</math>
|}

== 单位向量转换 ==

{| class="wikitable"
|+ 在直角、圆柱和球坐标系间的单位向量转换，从目的坐标的角度。<ref name="griffiths"/>
|-
!
! 直角
! 圆柱
! 球
|-
! 直角
| {{n/a}}
| <math>\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}</math>
| <math>\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}</math>
|-
! 圆柱
| <math>\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}</math>
| {{n/a}}
| <math>\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}</math>
|-
! 球
| <math>\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}</math>
| <math>\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}</math>
| {{n/a}}
|}

{| class="wikitable"
|+ 在直角、圆柱和球坐标系间的单位向量转换，从源坐标的角度。
|-
!
! 直角
! 圆柱
! 球
|-
! 直角
| {{n/a}}
| <math>\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}</math>
| <math>\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}</math>
|-
! 圆柱
| <math>\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}</math>
| {{n/a}}
| <math>\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}</math>
|-
! 球
| <math>\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}</math>
| <math>\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}</math>
| {{n/a}}
|}

== Del公式 ==

{| class="wikitable" style="background: white"
|+ 在直角、圆柱和球坐标下的[[del算子]]的表格
<!-- Header -->
|-
! style="background: white" ! width=5%| 运算
! style="background: white" ! width=13%| [[直角坐标]] {{math|(''x'', ''y'', ''z'')}}
! style="background: white" ! width=41%| [[圆柱坐标]] {{math|(''ρ'', ''φ'', ''z'')}}
! style="background: white" ! width=41%| [[球坐标]] {{math|(''r'', ''θ'', ''φ'')}}，这里的θ是极角而{{math|''φ''}}是方位角{{ref|Alpha|α}} 

<!-- Definition of A -->
|- align="center"
! style="background: white" | [[向量場|向量场]] <span style="font-weight: normal">{{math|'''A'''}}</span>
| <math>A_x      \hat{\mathbf x}         + A_y      \hat{\mathbf y}         + A_z    \hat{\mathbf z}</math>
| <math>A_\rho   \hat{\boldsymbol \rho}   + A_\varphi   \hat{\boldsymbol \varphi}   + A_z    \hat{\mathbf z}</math>
| <math>A_r      \hat{\mathbf r}     + A_\theta \hat{\boldsymbol \theta} + A_\varphi \hat{\boldsymbol \varphi}</math>

<!-- grad f -->
|- align="center"
! style="background: white" | [[梯度]] <span style="font-weight: normal">{{math|∇''f''}}</span><ref name="griffiths"/>
| <math>{\partial f \over \partial x}\hat{\mathbf x} + {\partial f \over \partial y}\hat{\mathbf y}
+ {\partial f \over \partial z}\hat{\mathbf z}</math>
| <math>{\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}</math>
| <math>{\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}</math>

<!-- div A -->
|- align="center"
! style="background: white" | [[散度]] <span style="font-weight: normal">{{math|∇ ⋅ '''A'''}}</span><ref name="griffiths"/>
| <math>{\partial A_x \over \partial x} + {\partial A_y \over \partial y} + {\partial A_z \over \partial z}</math>
| <math>{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}</math>
| <math>{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}</math>

<!-- curl A -->
|- align="center"
! style="background: white" | [[旋度]] <span style="font-weight: normal">{{math|∇ × '''A'''}}</span><ref name="griffiths"/>
| <math>\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}</math>
| <math>\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}</math>
| <math>\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}</math>

<!-- Laplacian f -->
|- align="center"
! style="background: white" | [[拉普拉斯算子]] <span style="font-weight: normal">{{math|∇<sup>2</sup>''f'' ≡ ∆''f''}}</span><ref name="griffiths"/>
| <math>{\partial^2 f \over \partial x^2} + {\partial^2 f \over \partial y^2} + {\partial^2 f \over \partial z^2}</math>
| <math>{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}</math>
| <math>{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}
</math>

<!-- vector Laplacian A -->
|- align="center"
! style="background: white" | [[拉普拉斯算子|向量拉普拉斯算子]] <span style="font-weight: normal">{{math|∇<sup>2</sup>'''A''' ≡ ∆'''A'''}}</span>
| <math>\nabla^2 A_x \hat{\mathbf x} + \nabla^2 A_y \hat{\mathbf y} + \nabla^2 A_z \hat{\mathbf z} </math>
| {{Collapsible section |content =
<math>\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}</math>
}}
| align="center" | {{Collapsible section |content =
<math>\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}</math>
}}

<!-- Material derivative (A dot del)B -->
|- align="center"
! style="background: white" | [[物质导数]]{{ref|Alpha|α}}<ref name="Mathworld">{{cite mathworld|urlname=ConvectiveOperator|title=Convective Operator |date= |accessdate=23 March 2011 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303221612/http://mathworld.wolfram.com/ConvectiveOperator.html |dead-url=no }}</ref> <span style="font-weight: normal">{{math|('''A''' ⋅ ∇)'''B'''}}</span>
<!--         Cartesian -->
| <math>\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}}</math>
<!--         Cylindrical \frac{\partial B_}{\partial } -->
|<math>\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}</math>
<!--         Sp -->
| align="center" | {{Collapsible section |content =
<math>\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}</math>
}}

<!-- Tensor divergence del dot T -->
|- align="center"
! style="background: white" | [[张量散度]] <span style="font-weight: normal">{{math|∇ ⋅ '''T'''}}</span>
<!-- Cartesian -->
| {{Collapsible section|content =
<math>\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}</math>
}}
<!-- cylindrical -->
| {{Collapsible section|content = 
<math>\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}</math>
}}
<!-- spherical -->
| {{Collapsible section|content = 
<math>\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}</math>
}}

<!-- Differential displacement -->
|- align="center"
! style="background: white" | 微分位移 <span style="font-weight: normal">{{math|''d'''ℓ'''''}}</span><ref name="griffiths"/>
| <math>dx \, \hat{\mathbf x} + dy \, \hat{\mathbf y} + dz \, \hat{\mathbf z}</math>
| <math>d\rho \, \hat{\boldsymbol \rho} + \rho \, d\varphi \, \hat{\boldsymbol \varphi} + dz \, \hat{\mathbf z}</math>
| <math>dr \, \hat{\mathbf r} + r \, d\theta \, \hat{\boldsymbol \theta} + r \, \sin\theta \, d\varphi \, \hat{\boldsymbol \varphi}</math>

<!-- Differential normal area -->
|- align="center"
! style="background: white" | 微分正规面积 <span style="font-weight: normal">{{math|''d'''''S'''}}</span>
| <math>\begin{align}
  dy \, dz &\, \hat{\mathbf x} \\
{} + dx \, dz &\, \hat{\mathbf y} \\
{} + dx \, dy &\, \hat{\mathbf z}
\end{align}</math>
| <math>\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}</math>
| <math>\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}</math>

<!-- Differential volume -->
|- align="center"
! style="background: white" | 微分体积 <span style="font-weight: normal">{{math|''dV''}}</span><ref name="griffiths"/>
| <math>dx \, dy \, dz</math>
| <math>\rho \, d\rho \, d\varphi \, dz</math>
| <math>r^2 \sin\theta \, dr \, d\theta \, d\varphi</math>

|}

:{{note|Alpha|α}}本页对极角采用<math>\theta</math>对方位角采用<math>\varphi</math>，这是在物理学中常用的符号。某些来源在这些公式中对方位角采用<math>\theta</math>对极角采用<math>\varphi</math>，这是常用数学符号，如果需要这种数学公式，可对换上表公式中的<math>\theta</math>和<math>\varphi</math>。

=== 非平凡的演算规则 ===

#  <math>\operatorname{div}  \, \operatorname{grad} f          \equiv \nabla \cdot  \nabla f \equiv \nabla^2 f</math>
#  <math>\operatorname{curl} \, \operatorname{grad} f          \equiv \nabla \times \nabla f = \mathbf 0</math>
#  <math>\operatorname{div}  \, \operatorname{curl} \mathbf{A} \equiv \nabla \cdot  (\nabla \times \mathbf{A}) = 0</math>
#  <math>\operatorname{curl} \, \operatorname{curl} \mathbf{A} \equiv \nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}</math>（del的[[三重积#向量三重积|拉格朗日公式]]）
#  <math>\nabla^2 (f g) = f \nabla^2 g + 2 \nabla f \cdot \nabla g + g \nabla^2 f</math>

== 直角坐标系推导 ==

[[File:Nabla cartesian.svg]]

<math>\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}</math>


<math>\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}</math>

<math>(\operatorname{curl} \mathbf A)_y</math>和<math>(\operatorname{curl} \mathbf A)_z</math>的表达式可以同理得出。 <p>
註：第一式中的<math>A_x(x+dx)</math>是<math>A_x</math>在<math>x+dx</math>時的量值，並非<math>A_x</math>值乘上<math>x+dx</math>。以下圓柱座標、球座標的推導中亦然。

== 圆柱坐标系推导 ==

[[File:Nabla cylindrical2.svg]]

:<math>\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}</math>

:<math>\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}</math>

:<math>\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}</math>

:<math>\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}</math>

:<math>\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}</math>

== 球坐标系推导 ==

[[File:Nabla spherical2.svg]]

<math>\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}</math>

<math>\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
  - A_\theta(\phi + d\phi)\,r d\theta - A_\phi(\theta)\,r\sin(\theta) d\phi}{r d\theta\,r\sin\theta d\phi} \\
&= \frac{1}{r\sin\theta}\frac{\partial(A_\phi \sin\theta)}{\partial \theta}
  - \frac{1}{r\sin\theta} \frac{\partial A_\theta}{\partial \phi}\end{align}</math>

<math>\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
  - A_\phi(r+dr)(r+dr)\sin\theta d\phi - A_r(\phi)dr}{dr \, r \sin \theta d\phi} \\
&= \frac{1}{r\sin\theta}\frac{\partial A_r}{\partial \phi}
  - \frac{1}{r} \frac{\partial (rA_\phi)}{\partial r}\end{align}</math>

<math>\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
  - A_r(\theta+d\theta)dr - A_\theta(r)\, r d\theta}{(r+dr/2) dr d\theta} \\
&= \frac{1}{r}\frac{\partial(rA_\theta)}{\partial r}
  - \frac{1}{r} \frac{\partial A_r}{\partial \theta}\end{align}</math>

<math>\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}</math>

== 单位向量转换公式 ==
坐标参数''u''的单位向量以如下方式定义，''u''的小的正值改变导致位置向量<math>\boldsymbol\vec{r}</math>在<math>\boldsymbol\hat{u}</math>方向上的改变。因此：

:<math>{\partial\boldsymbol\vec{r} \over \partial u}={\partial{s} \over \partial u}{\boldsymbol\hat{u}}</math> 

这里的''s''是[[弧长]]参数。

对于两组坐标系<math>u_i</math>和<math>v_j</math>，依据[[链式法则]]： 

:<math>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</math>

现在，使除了一个之外的所有<math>du_i=0</math>并在两边除以对应的坐标参数的微分，得到：
:<math>{\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}</math>

== 参见 ==
* [[Nabla算子|Del算子]]
* [[正交坐标系]]
* [[曲线坐标系]]
* [[在圆柱和球坐标中的向量场]]

== 引用 ==
{{Reflist}}

== 外部链接 ==
* [http://www.csulb.edu/~woollett/ Maxima Computer Algebra system scripts]{{Wayback|url=http://www.csulb.edu/~woollett/ |date=20140906133257 }} to generate some of these operators in cylindrical and spherical coordinates.
* [https://web.archive.org/web/20200301063345/https://ece.illinois.edu/webooks/nnrao2009/12%20Rao%20Fundamentals%202009%20appB.pdf Curl, Divergence, and Gradient in Cylindrical and Spherical Coordinate Systems].

[[Category:向量分析]]
[[Category:坐标系]]