# 旋度

## 定义

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

${\displaystyle \lim _{\Delta S\to 0}{\frac {1}{\left|\Delta S\right\vert }}\oint _{\Gamma }\mathbf {A} \cdot \mathrm {d} {\boldsymbol {l}}}$

${\displaystyle \mathbf {curl\,} \mathbf {A} (x)\cdot \mathbf {n} =\lim _{\Delta S_{\mathbf {n} }\to 0}{\frac {1}{\left|\Delta S_{\mathbf {n} }\right\vert }}\oint _{\Gamma _{\mathbf {n} }}\mathbf {A} \cdot \mathrm {d} {\boldsymbol {l}}}$

## 分量表示

### 直角坐标系

${\displaystyle \mathbf {A} (x,y,z)=A_{x}(x,y,z)\mathbf {i} +A_{y}(x,y,z)\mathbf {j} +A_{z}(x,y,z)\mathbf {k} }$

${\displaystyle {\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}},\quad {\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}},\quad {\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}}$

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

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

### 圆柱坐标系

${\displaystyle \mathbf {A} =A_{\rho }(\rho ,\varphi ,z){\boldsymbol {e}}_{\rho }+A_{\varphi }(\rho ,\varphi ,z){\boldsymbol {e}}_{\varphi }+A_{z}(\rho ,\varphi ,z){\boldsymbol {e}}_{z}}$

${\displaystyle \mathbf {curl\,} \mathbf {A} =\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\boldsymbol {e}}_{\rho }+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\boldsymbol {e}}_{\varphi }+{\frac {1}{\rho }}\left({\frac {\partial ({\rho }A_{\varphi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\boldsymbol {e}}_{z}}$

${\displaystyle {\boldsymbol {\nabla }}\times \mathbf {A} ={\begin{vmatrix}{\frac {1}{\rho }}\mathbf {e} _{\rho }&\mathbf {e} _{\varphi }&{\frac {1}{\rho }}\mathbf {e} _{z}\\{\frac {\partial }{\partial \rho }}&{\frac {\partial }{\partial \varphi }}&{\frac {\partial }{\partial z}}\\A_{\rho }&\rho A_{\varphi }&A_{z}\end{vmatrix}}}$

### 球坐标系

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

${\displaystyle \mathbf {curl\,} \mathbf {A} ={\frac {1}{r\sin \theta }}\left({\frac {\partial (A_{\varphi }\sin \theta )}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial \varphi }}\right){\boldsymbol {e}}_{r}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \varphi }}-{\frac {\partial (rA_{\varphi })}{\partial r}}\right){\boldsymbol {e}}_{\theta }+{\frac {1}{r}}\left({\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {\partial A_{r}}{\partial \theta }}\right){\boldsymbol {e}}_{\varphi }\,.}$

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

## 例子

${\displaystyle \mathbf {F} _{1}(x,y,z)=y{\boldsymbol {\hat {x}}}-x{\boldsymbol {\hat {y}}}.}$

${\displaystyle {\boldsymbol {\nabla }}\times \mathbf {F} _{1}=0{\boldsymbol {\hat {x}}}+0{\boldsymbol {\hat {y}}}+\left[{\frac {\partial }{\partial x}}(-x)-{\frac {\partial }{\partial y}}y\right]{\boldsymbol {\hat {z}}}=-2{\boldsymbol {\hat {z}}}}$[10]:70

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

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

## 性质

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

${\displaystyle \varphi }$是标量函数，F是向量场，则它们的乘积的旋度为[5]:9

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

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

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

F 的旋度场的旋度场则有公式[4]:14${\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times \mathbf {F} )={\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {F} )-\nabla ^{2}\mathbf {F} .}$

### 旋度的斯托克斯公式

${\displaystyle \iint \limits _{S}\left({\frac {\partial R}{\partial y}}-{\frac {\partial Q}{\partial z}}\right)\mathrm {d} y\,\mathrm {d} z+\left({\frac {\partial P}{\partial z}}-{\frac {\partial R}{\partial x}}\right)\mathrm {d} z\,\mathrm {d} x+\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\mathrm {d} x\,\mathrm {d} y=\oint \limits _{\Gamma }P\mathrm {d} x+Q\mathrm {d} y+R\mathrm {d} z}$

${\displaystyle \int _{S}({\boldsymbol {\nabla }}\times \mathbf {A} )\cdot \mathrm {d} \mathbf {S} =\oint _{\partial S}\mathbf {A} \cdot \mathrm {d} \mathbf {l} }$

## 历史

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

## 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${\displaystyle \mathbf {n} }$

## 参考来源

1. ^ David K. Cheng,Field and wave electromagnetics,Addison-Wesley publishing company,p49.
2. ^ Proceedings of the London Mathematical Society, March 9th, 1871 (PDF). [2019-06-25]. （原始内容存档 (PDF)于2021-02-19）.
3. ^ Collected works of James MacCullagh
4. 钟顺时. 《电磁场基础》. 清华大学出版社有限公司. 2006. ISBN 9787302126126.
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.
6. 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 （英语）.
7. ^ 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. [2012-08-18]. （原始内容存档于2021-08-12）.
8. ^ 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. [2012-08-18]. （原始内容存档于2021-01-21）.
9. ^ David K. Cheng. Field and Wave Electromagnetics. 2014: 第58頁. ISBN 9781292026565.
10. 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 （英语）.
11. 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 （英语）.