# 旋度

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

## 定义

$\operatorname{Circ}_{\mathbf{A}}( \Gamma ) =\oint_{\Gamma}\mathbf{A}\cdot\mathrm{d} \boldsymbol{l}$

$\lim_{\Delta S \to 0 } \frac{1}{\left | \Delta S \right \vert} \oint_{\Gamma}\mathbf{A}\cdot\mathrm{d} \boldsymbol{l}$

$\mathbf{curl\,} \mathbf{A}(x)\cdot \mathbf{n} = \lim_{ \Delta S_{\mathbf{n}} \to 0 } \frac{1}{\left | \Delta S_{\mathbf{n}} \right \vert} \oint_{\Gamma_{\mathbf{n}}}\mathbf{A}\cdot\mathrm{d} \boldsymbol{l}$

## 分量表示

### 直角坐标系

$\mathbf{A}(x,y,z)=P(x,y,z)\mathbf{i}+Q(x,y,z)\mathbf{j}+R(x,y,z)\mathbf{k}$

$\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \quad \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \quad \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$

$\mathbf{curl\,} \ \mathbf{A}=\boldsymbol\nabla\times\mathbf{A}=\left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{i}+\left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathbf{j}+\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k}$

$\mathbf{curl\,}\mathbf{A}= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac {\partial}{\partial z} \\ P & Q & R \end{vmatrix}$

### 圆柱坐标系

$\mathbf A = A_r(r, \theta, z) \boldsymbol{e}_{r} + A_z(r, \theta, z) \boldsymbol{e}_{z} + A_{\theta}(r, \theta, z)\boldsymbol{e}_{\theta},$

$\mathbf{curl\,} \mathbf A = \left(\frac1r \frac{\partial A_z}{\partial \theta} - \frac{\partial A_{\theta}}{\partial z} \right) \boldsymbol{e}_{r} + \left( \frac{\partial A_r}{\partial z} - \frac{\partial A_{z}}{\partial r} \right)\boldsymbol{e}_{\theta} + \frac1r \left( \frac{\partial (rA_{\theta})}{\partial r} - \frac{\partial A_{r}}{\partial \theta} \right)\boldsymbol{e}_{z} \, .$

$\boldsymbol\nabla\times\mathbf{A}= \begin{vmatrix} \frac{1}{r}\mathbf{e}_{r} & \mathbf{e}_{\theta} & \frac{1}{r}\mathbf{e}_z \\ \frac{\partial}{\partial r} &\frac{\partial}{\partial \theta} & \frac {\partial}{\partial z} \\ A_r & r A_{\theta} & A_z \end{vmatrix}$

### 球坐标系

$\mathbf A = A_r (r , \theta , \varphi ) \boldsymbol{e}_{r} + A_{\theta} (r , \theta , \varphi ) \boldsymbol{e}_{\theta} + A_{ \varphi } (r , \theta , \varphi ) \boldsymbol{e}_{\varphi } ,$

$\mathbf{curl\,} \mathbf A = \frac1{r \sin\theta} \left( \frac{\partial (A_{\varphi}\sin\theta )}{\partial \theta} - \frac{\partial A_{\theta}}{\partial \varphi} \right) \boldsymbol{e}_{r} + \frac1r \left( \frac1{ \sin\theta}\frac{\partial A_r}{\partial \varphi } - \frac{\partial (r A_{ \varphi })}{\partial r} \right)\boldsymbol{e}_{\theta} + \frac1r \left( \frac{\partial (rA_{\theta})}{\partial r} - \frac{\partial A_{r}}{\partial \theta} \right)\boldsymbol{e}_{ \varphi } \, .$

$\boldsymbol\nabla\times\mathbf{A}= \frac{1}{r^2 \sin\theta}\begin{vmatrix} \mathbf{e}_{r} & r\mathbf{e}_{\theta} & r\sin\theta \mathbf{e}_{\varphi} \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac {\partial}{\partial \varphi} \\ A_r & r A_{\theta} & r\sin\theta A_{\varphi} \end{vmatrix}$

## 例子

$\mathbf{F}_1(x,y,z)=y\boldsymbol{\hat{x}}-x\boldsymbol{\hat{y}}.$

$\boldsymbol{\nabla} \times \mathbf{F}_1=0\boldsymbol{\hat{x}}+0\boldsymbol{\hat{y}}+ \left[{\frac{\partial}{\partial x}}(-x) -{\frac{\partial}{\partial y}} y\right]\boldsymbol{\hat{z}}=-2\boldsymbol{\hat{z}}$[6]:70

$\mathbf{F}_2(x,y,z)=-x^2\boldsymbol{\hat{y}}.$

$\boldsymbol{\nabla} \times \mathbf{F}_2 =0\boldsymbol{\hat{x}}+0\boldsymbol{\hat{y}}+ {\frac{\partial}{\partial x}}(-x^2) \boldsymbol{\hat{z}}=-2x\boldsymbol{\hat{z}}.$

## 性质

$\mathbf{curl\,} ( a\mathbf{F} + b\mathbf{G} ) = a\;\mathbf{curl\,} ( \mathbf{F} ) + b\;\mathbf{curl\,} ( \mathbf{G} )$

$\varphi$是标量函数，F是向量场，则它们的乘积的旋度为[2]:9

$\mathbf{curl\,} (\varphi \mathbf{F}) = \mathbf{grad\,}(\varphi) \times \mathbf{F} + \varphi \;\mathbf{curl\,}(\mathbf{F}),$

$\boldsymbol\nabla\times (\varphi \mathbf{F}) = (\boldsymbol\nabla\varphi) \times \mathbf{F} + \varphi \;\boldsymbol\nabla \times \mathbf{F}.$

$\boldsymbol\nabla\times (\mathbf{F}\times\mathbf{G}) = (\mathbf{G}\cdot\boldsymbol\nabla )\mathbf{F} \;-\; (\boldsymbol\nabla\cdot \mathbf{F})\mathbf{G}-(\mathbf{F}\cdot\boldsymbol\nabla)\mathbf{G} +(\boldsymbol\nabla\cdot \mathbf{G} ) \mathbf{F}$

F 的旋度场的旋度场则有公式[1]:14$\boldsymbol\nabla \times (\boldsymbol\nabla \times \mathbf{F} ) = \boldsymbol\nabla(\boldsymbol\nabla\cdot \mathbf{F}) - \nabla^2 \mathbf{F} .$

### 旋度的斯托克斯公式

$\iint\limits_{S}\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\mathrm dy\,\mathrm dz+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\mathrm dz\,\mathrm dx+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\mathrm dx\,\mathrm dy=\oint\limits_{\Gamma}P\mathrm dx+Q\mathrm dy+R\mathrm dz$

$\int_{S} (\boldsymbol\nabla \times \mathbf{A}) \cdot \mathrm d\mathbf{S} = \oint_{\partial S} \mathbf{A} \cdot \mathrm{d} \mathbf{l}$

## 历史

$\boldsymbol\nabla \sigma = (\boldsymbol{i}\frac{\mathrm{d}}{\mathrm{d}x}+ \boldsymbol{j} \frac{\mathrm{d}}{\mathrm{d}y} + \boldsymbol{k} \frac{\mathrm{d}}{\mathrm{d}z})( B\boldsymbol{i}+C\boldsymbol{j}+D\boldsymbol{k})$
$= -\left(\frac{\mathrm{d}B}{\mathrm{d}x}+ \frac{\mathrm{d}C}{\mathrm{d}y}+\frac{\mathrm{d}D}{\mathrm{d}z} \right) +\left( \left(\frac{\mathrm{d}D}{\mathrm{d}y} - \frac{\mathrm{d} C}{\mathrm{d} z}\right)\boldsymbol{i}+\left(\frac{\mathrm{d}B}{\mathrm{d}z} - \frac{\mathrm{d}D}{\mathrm{d} x}\right)\boldsymbol{j}+\left(\frac{\mathrm{d}C}{\mathrm{d}x} - \frac{\mathrm{d}B}{\mathrm{d}y}\right)\boldsymbol{k}\right)$

## 注释

1. ^ 一般选取过这一点的平面上，包含这一点的有界的一部分作为面元。为了其后定义方便起见，一般还会假定这个部分的边界是一个简单闭合有向曲线
2. ^ 指面元所在平面的法向量方向
3. ^ 指面元所在平面的法向量方向是单位向量$\mathbf{n}$

## 参考来源

1. ^ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 钟顺时. 《电磁场基础》. 清华大学出版社有限公司. 2006. ISBN 9787302126126.
2. ^ 2.0 2.1 2.2 2.3 2.4 2.5 王蔷, 李国定, 龚克. 《电磁场理论基础》. 清华大学出版社有限公司. 2001. ISBN 9787302042518.
3. ^ 3.0 3.1 3.2 Roel Snieder. A Guided Tour of Mathematical Methods: For the Physical Sciences. Cambridge University Press, 2, 插图版, 修订版. 2004. ISBN 9780521834926 （英文）.
4. ^ 梯度、散度、旋度和调和量在柱面坐标系中的表达式. 浙江大学远程教育学院. [08 18, 2012].
5. ^ 梯度、散度、旋度和调和量在球坐标系中的表达式. 浙江大学远程教育学院. [08 18, 2012].
6. ^ 6.0 6.1 6.2 K.T. Tang. Mathematical Methods for Engineers and Scientists 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms. Springer，插图版. 2006. ISBN 9783540302681 （英文）.
7. ^ 7.0 7.1 7.2 Michael J. Crowe. A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. Dover books on advanced mathematics, 2nd Edition. 1994. ISBN 9780486679105 （英文）.