在圆柱和球坐标系中的del

下面是常用于正交Template:En-link中的一些向量微积分公式。

注释

• 本文对球坐标使用标准符号ISO 80000-2，它取代了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Template:Nowrap可以用于替代数学函数Template:Nowrap。这是由于它的定义域和像的缘故，经典arctan函数的像为Template:Nowrap，而atan2定义的像为Template:Nowrap。

坐标转换

在直角、圆柱和球坐标间的变换

		从
		直角	圆柱	球
到	直角	$\begin{array}{cc}x& =x\\ y& =y\\ z& =z\end{array}$	$\begin{array}{cc}x& =\rho \cos \phi \\ y& =\rho \sin \phi \\ z& =z\end{array}$	$\begin{array}{cc}x& =r\sin \theta \cos \phi \\ y& =r\sin \theta \sin \phi \\ z& =r\cos \theta \end{array}$
	圆柱	$\begin{array}{cc}\rho & =\sqrt{x^2+y^2}\\ \phi & =\arctan \left(\frac{y}{x}\right)\\ z& =z\end{array}$	$\begin{array}{cc}\rho & =\rho \\ \phi & =\phi \\ z& =z\end{array}$	$\begin{array}{cc}\rho & =r\sin \theta \\ \phi & =\phi \\ z& =r\cos \theta \end{array}$
	球	$\begin{array}{cc}r& =\sqrt{x^2+y^2+z^2}\\ \theta & =\arctan \left(\frac{\sqrt{x^2+y^2}}{z}\right)\\ \phi & =\arctan \left(\frac{y}{x}\right)\end{array}$	$\begin{array}{cc}r& =\sqrt{{\rho }^2+z^2}\\ \theta & =\arctan \left(\frac{\rho }{z}\right)\\ \phi & =\phi \end{array}$	$\begin{array}{cc}r& =r\\ \theta & =\theta \\ \phi & =\phi \end{array}$

单位向量转换

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。

	直角	圆柱	球
直角	Template:N/a	$\begin{array}{cc}\widehat{𝐱}& =\cos \phi \hat{\mathit{\rho }}-\sin \phi \hat{\mathit{\phi }}\\ \widehat{𝐲}& =\sin \phi \hat{\mathit{\rho }}+\cos \phi \hat{\mathit{\phi }}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	$\begin{array}{cc}\widehat{𝐱}& =\sin \theta \cos \phi \widehat{𝐫}+\cos \theta \cos \phi \hat{\mathit{\theta }}-\sin \phi \hat{\mathit{\phi }}\\ \widehat{𝐲}& =\sin \theta \sin \phi \widehat{𝐫}+\cos \theta \sin \phi \hat{\mathit{\theta }}+\cos \phi \hat{\mathit{\phi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝐫}-\sin \theta \hat{\mathit{\theta }}\end{array}$
圆柱	$\begin{array}{cc}\hat{\mathit{\rho }}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \hat{\mathit{\phi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	Template:N/a	$\begin{array}{cc}\hat{\mathit{\rho }}& =\sin \theta \widehat{𝐫}+\cos \theta \hat{\mathit{\theta }}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝐫}-\sin \theta \hat{\mathit{\theta }}\end{array}$
球	$\begin{array}{cc}\widehat{𝐫}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}+z\widehat{𝐳}}{\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{(x\widehat{𝐱}+y\widehat{𝐲})z-(x^2+y^2)\widehat{𝐳}}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}\\ \hat{\mathit{\phi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\end{array}$	$\begin{array}{cc}\widehat{𝐫}& =\frac{\rho \hat{\mathit{\rho }}+z\widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{z\hat{\mathit{\rho }}-\rho \widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\end{array}$	Template:N/a

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	Template:N/a	$\begin{array}{cc}\widehat{𝐱}& =\frac{x\hat{\mathit{\rho }}-y\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}}\\ \widehat{𝐲}& =\frac{y\hat{\mathit{\rho }}+x\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	$\begin{array}{cc}\widehat{𝐱}& =\frac{x(\sqrt{x^2+y^2}\widehat{𝐫}+z\hat{\mathit{\theta }})-y\sqrt{x^2+y^2+z^2}\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \widehat{𝐲}& =\frac{y(\sqrt{x^2+y^2}\widehat{𝐫}+z\hat{\mathit{\theta }})+x\sqrt{x^2+y^2+z^2}\hat{\mathit{\phi }}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \widehat{𝐳}& =\frac{z\widehat{𝐫}-\sqrt{x^2+y^2}\hat{\mathit{\theta }}}{\sqrt{x^2+y^2+z^2}}\end{array}$
圆柱	$\begin{array}{cc}\hat{\mathit{\rho }}& =\cos \phi \widehat{𝐱}+\sin \phi \widehat{𝐲}\\ \hat{\mathit{\phi }}& =-\sin \phi \widehat{𝐱}+\cos \phi \widehat{𝐲}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$	Template:N/a	$\begin{array}{cc}\hat{\mathit{\rho }}& =\frac{\rho \widehat{𝐫}+z\hat{\mathit{\theta }}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\\ \widehat{𝐳}& =\frac{z\widehat{𝐫}-\rho \hat{\mathit{\theta }}}{\sqrt{{\rho }^2+z^2}}\end{array}$
球	$\begin{array}{cc}\widehat{𝐫}& =\sin \theta (\cos \phi \widehat{𝐱}+\sin \phi \widehat{𝐲})+\cos \theta \widehat{𝐳}\\ \hat{\mathit{\theta }}& =\cos \theta (\cos \phi \widehat{𝐱}+\sin \phi \widehat{𝐲})-\sin \theta \widehat{𝐳}\\ \hat{\mathit{\phi }}& =-\sin \phi \widehat{𝐱}+\cos \phi \widehat{𝐲}\end{array}$	$\begin{array}{cc}\widehat{𝐫}& =\sin \theta \hat{\mathit{\rho }}+\cos \theta \widehat{𝐳}\\ \hat{\mathit{\theta }}& =\cos \theta \hat{\mathit{\rho }}-\sin \theta \widehat{𝐳}\\ \hat{\mathit{\phi }}& =\hat{\mathit{\phi }}\end{array}$	Template:N/a

Del公式

在直角、圆柱和球坐标下的del算子的表格

运算	直角坐标 Template:Math	圆柱坐标 Template:Math	球坐标 Template:Math，这里的θ是极角而Template:Math是方位角Template:Ref
向量场 Template:Math	$A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$	$A_{\rho }\hat{\mathit{\rho }}+A_{\phi }\hat{\mathit{\phi }}+A_z\widehat{𝐳}$	$A_r\widehat{𝐫}+A_{\theta }\hat{\mathit{\theta }}+A_{\phi }\hat{\mathit{\phi }}$
梯度 Template:Math	$\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}+\frac{\partial f}{\partial z}\widehat{𝐳}$	$\frac{\partial f}{\partial r}\widehat{𝐫}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \phi }\hat{\mathit{\phi }}$
散度 Template:Math	$\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$	$\frac{1}{\rho }\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\phi }}{\partial \phi }+\frac{\partial A_z}{\partial z}$	$\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\phi }}{\partial \phi }$
旋度 Template:Math	$\begin{array}{cc}(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})& \widehat{𝐱}\\ +(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})& \widehat{𝐲}\\ +(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})& \widehat{𝐳}\end{array}$	$\begin{array}{cc}(\frac{1}{\rho }\frac{\partial A_z}{\partial \phi }-\frac{\partial A_{\phi }}{\partial z})& \hat{\mathit{\rho }}\\ +(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })& \hat{\mathit{\phi }}\\ +\frac{1}{\rho }(\frac{\partial \left(\rho A_{\phi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \phi })& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\phi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \phi })& \widehat{𝐫}\\ +\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \phi }-\frac{\partial }{\partial r}\left(r{A}_{\phi }\right))& \hat{\mathit{\theta }}\\ +\frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\theta }\right)-\frac{\partial A_r}{\partial \theta })& \hat{\mathit{\phi }}\end{array}$
拉普拉斯算子 Template:Math	$\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\phi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$	$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\phi }^2}$
向量拉普拉斯算子 Template:Math	${\mathrm{\nabla }}^2{A}_x\widehat{𝐱}+{\mathrm{\nabla }}^2{A}_y\widehat{𝐲}+{\mathrm{\nabla }}^2{A}_z\widehat{𝐳}$	Template:Collapsible section	Template:Collapsible section
物质导数Template:Ref Template:Math	$𝐀\cdot \mathrm{\nabla }{B}_x\widehat{𝐱}+𝐀\cdot \mathrm{\nabla }{B}_y\widehat{𝐲}+𝐀\cdot \mathrm{\nabla }{B}_z\widehat{𝐳}$	$\begin{array}{cc}(A_{\rho }\frac{\partial B_{\rho }}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_{\rho }}{\partial \phi }+A_z\frac{\partial B_{\rho }}{\partial z}-\frac{A_{\phi }{B}_{\phi }}{\rho })& \hat{\mathit{\rho }}\\ +(A_{\rho }\frac{\partial B_{\phi }}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_{\phi }}{\partial \phi }+A_z\frac{\partial B_{\phi }}{\partial z}+\frac{A_{\phi }{B}_{\rho }}{\rho })& \hat{\mathit{\phi }}\\ +(A_{\rho }\frac{\partial B_z}{\partial \rho }+\frac{A_{\phi }}{\rho }\frac{\partial B_z}{\partial \phi }+A_z\frac{\partial B_z}{\partial z})& \widehat{𝐳}\end{array}$	Template:Collapsible section
张量散度 Template:Math	Template:Collapsible section	Template:Collapsible section	Template:Collapsible section
微分位移 Template:Math	$dx\widehat{𝐱}+dy\widehat{𝐲}+dz\widehat{𝐳}$	$d\rho \hat{\mathit{\rho }}+\rho d\phi \hat{\mathit{\phi }}+dz\widehat{𝐳}$	$dr\widehat{𝐫}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\phi \hat{\mathit{\phi }}$
微分正规面积 Template:Math	$\begin{array}{cc}dydz& \widehat{𝐱}\\ +dxdz& \widehat{𝐲}\\ +dxdy& \widehat{𝐳}\end{array}$	$\begin{array}{cc}\rho d\phi dz& \hat{\mathit{\rho }}\\ +d\rho dz& \hat{\mathit{\phi }}\\ +\rho d\rho d\phi & \widehat{𝐳}\end{array}$	$\begin{array}{cc}{r}^2\sin \theta d\theta d\phi & \widehat{𝐫}\\ +r\sin \theta drd\phi & \hat{\mathit{\theta }}\\ +rdrd\theta & \hat{\mathit{\phi }}\end{array}$
微分体积 Template:Math	$dxdydz$	$\rho d\rho d\phi dz$	$r^2\sin \theta drd\theta d\phi$

Template:Note本页对极角采用$\theta$对方位角采用$\phi$，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$\theta$对极角采用$\phi$，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$\theta$和$\phi$。

非平凡的演算规则

1. $\mathrm{div}\mathrm{grad}f\equiv \mathrm{\nabla }\cdot \mathrm{\nabla }f\equiv {\mathrm{\nabla }}^{2}f$
2. $\mathrm{curl}\mathrm{grad}f\equiv \mathrm{\nabla }\times \mathrm{\nabla }f=\mathrm{𝟎}$
3. $\mathrm{div}\mathrm{curl}𝐀\equiv \mathrm{\nabla }\cdot (\mathrm{\nabla }\times 𝐀)=0$
4. $\mathrm{curl}\mathrm{curl}𝐀\equiv \mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐀)=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐀)-{\mathrm{\nabla }}^2𝐀$（del的拉格朗日公式）
5. ${\mathrm{\nabla }}^2(fg)=f{\mathrm{\nabla }}^{2}g+2\mathrm{\nabla }f\cdot \mathrm{\nabla }g+g{\mathrm{\nabla }}^{2}f$

直角坐标系推导

$\begin{array}{cc}\mathrm{div}𝐀=\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }𝐀\cdot d𝐒}{\underset{V}{\iiint }dV}& =\frac{A_x(x+dx)dydz-A_x(x)dydz+A_y(y+dy)dxdz-A_y(y)dxdz+A_z(z+dz)dxdy-A_z(z)dxdy}{dxdydz}\\ & =\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_x=\underset{S^{\perp \widehat{𝐱}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_z(y+dy)dz-A_z(y)dz+A_y(z)dy-A_y(z+dz)dy}{dydz}\\ & =\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}\end{array}$

$(\mathrm{curl}𝐀{)}_y$

和

$(\mathrm{curl}𝐀{)}_z$

的表达式可以同理得出。

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。

圆柱坐标系推导

$\begin{array}{cc}\mathrm{div}𝐀& =\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }𝐀\cdot d𝐒}{\underset{V}{\iiint }dV}\\ & =\frac{A_{\rho }(\rho +d\rho )(\rho +d\rho )d\varphi dz-A_{\rho }(\rho )\rho d\varphi dz+A_{\varphi }(\varphi +d\varphi )d\rho dz-A_{\varphi }(\varphi )d\rho dz+A_z(z+dz)d\rho (\rho +d\rho /2)d\varphi -A_z(z)d\rho (\rho +d\rho /2)d\varphi }{\rho d\varphi d\rho dz}\\ & =\frac{1}{\rho }\frac{\partial (\rho A_{\rho })}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_{\rho }& =\underset{S^{\perp \hat{\mathit{\rho }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_{\varphi }(z)(\rho +d\rho )d\varphi -A_{\varphi }(z+dz)(\rho +d\rho )d\varphi +A_z(\varphi +d\varphi )dz-A_z(\varphi )dz}{(\rho +d\rho )d\varphi dz}\\ & =-\frac{\partial A_{\varphi }}{\partial z}+\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_{\varphi }& =\underset{S^{\perp \hat{\mathit{\varphi }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_z(\rho )dz-A_z(\rho +d\rho )dz+A_{\rho }(z+dz)d\rho -A_{\rho }(z)d\rho }{d\rho dz}\\ & =-\frac{\partial A_z}{\partial \rho }+\frac{\partial A_{\rho }}{\partial z}\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_z& =\underset{S^{\perp \widehat{𝒛}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}\\ & =\frac{A_{\rho }(\varphi )d\rho -A_{\rho }(\varphi +d\varphi )d\rho +A_{\varphi }(\rho +d\rho )(\rho +d\rho )d\varphi -A_{\varphi }(\rho )\rho d\varphi }{\rho d\rho d\varphi }\\ & =-\frac{1}{\rho }\frac{\partial A_{\rho }}{\partial \varphi }+\frac{1}{\rho }\frac{\partial (\rho A_{\varphi })}{\partial \rho }\end{array}$

$\begin{array}{cc}\mathrm{curl}𝐀& =(\mathrm{curl}𝐀{)}_{\rho }\hat{\mathit{\rho }}+(\mathrm{curl}𝐀{)}_{\varphi }\hat{\mathit{\varphi }}+(\mathrm{curl}𝐀{)}_z\widehat{𝒛}\\ & =(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})\hat{\mathit{\rho }}+(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })\hat{\mathit{\varphi }}+\frac{1}{\rho }(\frac{\partial (\rho A_{\varphi })}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })\widehat{𝒛}\end{array}$

球坐标系推导

$\begin{array}{cc}\mathrm{div}𝐀& =\underset{V\to 0}{\lim }\frac{\underset{\partial V}{\iint }𝐀\cdot d𝐒}{\underset{V}{\iiint }dV}\\ & =\frac{A_r(r+dr)(r+dr)d\theta (r+dr)\sin \theta d\varphi -A_r(r)rd\theta r\sin \theta d\varphi +A_{\theta }(\theta +d\theta )\sin (\theta +d\theta )rdrd\varphi -A_{\theta }(\theta )\sin (\theta )rdrd\varphi +A_{\varphi }(\varphi +d\varphi )(r+dr/2)drd\theta -A_{\varphi }(\varphi )(r+dr/2)drd\theta }{drrd\theta r\sin \theta d\varphi }\\ & =\frac{1}{r^2}\frac{\partial (r^2{A}_r)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_r=\underset{S^{\perp \widehat{𝒓}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_{\theta }(\varphi )rd\theta +A_{\varphi }(\theta +d\theta )r\sin (\theta +d\theta )d\varphi -A_{\theta }(\varphi +d\varphi )rd\theta -A_{\varphi }(\theta )r\sin (\theta )d\varphi }{rd\theta r\sin \theta d\varphi }\\ & =\frac{1}{r\sin \theta }\frac{\partial (A_{\varphi }\sin \theta )}{\partial \theta }-\frac{1}{r\sin \theta }\frac{\partial A_{\theta }}{\partial \varphi }\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_{\theta }=\underset{S^{\perp \hat{\mathit{\theta }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_{\varphi }(r)r\sin \theta d\varphi +A_r(\varphi +d\varphi )dr-A_{\varphi }(r+dr)(r+dr)\sin \theta d\varphi -A_r(\varphi )dr}{drr\sin \theta d\varphi }\\ & =\frac{1}{r\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{1}{r}\frac{\partial (r{A}_{\varphi })}{\partial r}\end{array}$

$\begin{array}{cc}(\mathrm{curl}𝐀{)}_{\varphi }=\underset{S^{\perp \hat{\mathit{\varphi }}}\to 0}{\lim }\frac{\underset{\partial S}{\int }𝐀\cdot d\mathit{\ell }}{\underset{S}{\iint }dS}& =\frac{A_r(\theta )dr+A_{\theta }(r+dr)(r+dr)d\theta -A_r(\theta +d\theta )dr-A_{\theta }(r)rd\theta }{(r+dr/2)drd\theta }\\ & =\frac{1}{r}\frac{\partial (r{A}_{\theta })}{\partial r}-\frac{1}{r}\frac{\partial A_r}{\partial \theta }\end{array}$

$\mathrm{curl}𝐀=(\mathrm{curl}𝐀{)}_r\widehat{𝒓}+(\mathrm{curl}𝐀{)}_{\theta }\hat{\mathit{\theta }}+(\mathrm{curl}𝐀{)}_{\varphi }\hat{\mathit{\varphi }}=\frac{1}{r\sin \theta }(\frac{\partial (A_{\varphi }\sin \theta )}{\partial \theta }-\frac{\partial A_{\theta }}{\partial \varphi })\widehat{𝒓}+\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial (r{A}_{\varphi })}{\partial r})\hat{\mathit{\theta }}+\frac{1}{r}(\frac{\partial (r{A}_{\theta })}{\partial r}-\frac{\partial A_r}{\partial \theta })\hat{\mathit{\varphi }}$

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$\overrightarrow{𝒓}$在$\widehat{𝒖}$方向上的改变。因此：

$\frac{\partial \overrightarrow{𝒓}}{\partial u}=\frac{\partial s}{\partial u}\widehat{𝒖}$

这里的s是弧长参数。

对于两组坐标系$u_i$和$v_j$，依据链式法则：

$d\overrightarrow{𝒓}=\sum _i\frac{\partial \overrightarrow{𝒓}}{\partial u_i}d{u}_i=\sum _i\frac{\partial s}{\partial u_i}\widehat{{𝒖}_{𝒊}}d{u}_i=\sum _j\frac{\partial s}{\partial v_j}\widehat{{𝒗}_{𝒋}}d{v}_j=\sum _j\frac{\partial s}{\partial v_j}\widehat{{𝒗}_{𝒋}}\sum _i\frac{\partial v_j}{\partial u_i}d{u}_i=\sum _i\sum _j\frac{\partial s}{\partial v_j}\frac{\partial v_j}{\partial u_i}\widehat{{𝒗}_{𝒋}}d{u}_i$

现在，使除了一个之外的所有$d{u}_i=0$并在两边除以对应的坐标参数的微分，得到：

$\frac{\partial s}{\partial u_i}\widehat{{𝒖}_{𝒊}}=\sum _j\frac{\partial s}{\partial v_j}\frac{\partial v_j}{\partial u_i}\widehat{{𝒗}_{𝒋}}$

参见

• 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

外部链接

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