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高等数学拾遗 矢量分析

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立体角

d\Omega = \frac{dA}{r^2}


各坐标系下的梯度，散度，旋度表达式




  • 直角坐标系



梯度

 		\bigtriangledown f=\frac{\partial f}{\partial x}\vec{x}+\frac{\partial f}{\partial y}\vec{y}+\frac{\partial f}{\partial z}\vec{z}


散度

 		\bigtriangledown \cdot f=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}


旋度

 		\bigtriangledown \times f=\begin{bmatrix} \vec{e_{x}} & \vec{e_{y}} & \vec{e_{z}} \\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ f_{x} &f_{y} & f_{z} \end{bmatrix}



  • 柱坐标系




梯度

 		\bigtriangledown f=\frac{\partial f}{\partial \rho }\vec{e_{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \theta }\vec{e_{\theta }}+\frac{\partial f}{\partial z }\vec{e_{z}}


散度

 		\bigtriangledown \cdot 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}


旋度

 		\bigtriangledown \times f=\frac{1}{\rho }\begin{vmatrix} \vec{e_{p}} &\rho \vec{e_{\theta }} & \vec{e_{z}}\\ \frac{\partial }{\partial \rho }&\frac{\partial }{\partial \theta } &\frac{\partial }{\partial z} \\ f_{\rho }&\rho f_{\theta } & f_{z} \end{vmatrix}



  • 球坐标系




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梯度

\bigtriangledown f=\frac{\partial f}{\partial \rho }\vec{e_{\rho }}+\frac{1}{\rho sin\varphi }\frac{\partial f}{\partial \theta }\vec{e_{\theta }}+\frac{1}{\rho }\frac{\partial f}{\partial \varphi }\vec{e_{\varphi }}



散度


 		\bigtriangledown \cdot f=\frac{1}{\rho ^{2}}\frac{\partial (\rho ^{2}f_{\rho })}{\partial \rho }+\frac{1}{\rho sin\varphi }\frac{\partial f_{\theta }}{\partial \theta }+\frac{1}{\rho sin\varphi }\frac{\partial (sin\varphi f_{\varphi })}{\partial \varphi }



旋度


 		\bigtriangledown \times f=\frac{1}{\rho ^{2}sin\varphi }\begin{vmatrix} \vec{e_{\rho }} & \rho \vec{e_{\varphi }} &\rho sin\varphi \vec{e_{\theta }} \\ \frac{\partial }{\partial \rho }& \frac{\partial }{\partial \varphi }&\frac{\partial }{\partial \theta } \\ f_{\rho } &\rho f_{\varphi } & psin\varphi f_{\theta } \end{vmatrix}

矢量运算

矢量积

\vec{A}\times \vec{B}=a_{n}ABsin\theta

不满足交换律，满足分配率

三重积

标量三重积
\mathbf{C \cdot (A\times B))}=ABCsin\theta cos\varphi

θ表示矢量A与B之间的夹角

φ表示矢量C与A×B之间的夹角

标量三重积满足
\mathbf{A\cdot (B\times C)=B\cdot (C\times A)=C\cdot \left ( A\times B \right )}

矢量三重积
A\times (B\times C)=(A\cdot C)B-(A\cdot B)C

矢量函数和微分

设F(u)是单变量u的矢量函数，它对u的导数定义为

\frac{dF}{du}=\lim_{\Delta u->0}\frac{\Delta F}{\Delta u}=\frac{F(u+\Delta u)-F(u)}{\Delta u}

设f和F分别是变量u的标量函数和矢量函数，二者积的导数为

\frac{d(fF)}{du}=f\frac{dF}{du}+F\frac{df}{du}

正交坐标系

圆柱坐标系

线元，面积元与体积元

d\mathbf{l=a_\rho}d\rho+\mathbf{a_\phi}\rho d\phi+\mathbf{a_z}dz

\begin{matrix} dS_\rho=a_\rho \rho d\phi dz\\dS_\phi = a_\phi d\rho dz \\ dS_z=a_z \rho d\rho d\phi \end{matrix}

dV=\rho d\rho d\phi dz

与正坐标系的转换

\left [ \begin{matrix} A_x\\A_y \\ A_z \end{matrix} \right ]=\left [ \begin{matrix} cos\varphi & -sin\varphi & 0\\sin\varphi &cos\varphi &0 \\ 0& 0& 1 \end{matrix} \right ]\left [ \begin{matrix} A_\rho \\A_\varphi \\A_z \end{matrix} \right ]

\left [ \begin{matrix} A_\rho\\A_\phi \\ A_z \end{matrix} \right ]=\left [ \begin{matrix} cos\phi & sin\phi & 0\\ -sin\phi &cos\phi &0 \\ 0& 0 &1 \end{matrix} \right ]\left [ \begin{matrix} A_x\\A_y \\ A_z \end{matrix} \right ]

球坐标系

线元，面积元与体积元

\begin{matrix} d\mathbf{l}=\mathbf{a_r}dr+\mathbf{a_\theta}rd\theta +\mathbf{a_\phi}rsin\theta d\phi\\\left\{\begin{matrix} dS_r=a_r r^2sin\theta d\theta d\phi\\ dS_\phi = a_\phi rdrd\theta \\dS_\theta = a_\theta r sin\theta drd\phi \end{matrix}\right. \\ dV=r^2sin\theta dr d\theta d\phi \end{matrix}

与正坐标系的转换

\left [ \begin{matrix} A_x\\A_y \\ A_z \end{matrix} \right ]=[\begin{matrix} sin\theta cos\phi& cos\theta cos\phi &-sin\phi \\ sin\theta sin\phi &cos\theta sin\phi & cos\phi\\ cos\phi &-sin\phi &0 \end{matrix}]\left [ \begin{matrix} A_r\\A_\theta \\ A_\rho \end{matrix} \right ]

\begin{bmatrix} A_r\\A_\theta \\ A_\phi \end{bmatrix}=\begin{bmatrix} sin\theta cos\phi &sin\theta sin\phi &cos\theta \\ cos\theta cos\phi& cos\theta sin\phi & -sin\theta\\ -sin\phi&cos\phi &0 \end{bmatrix}\begin{bmatrix} A_x\\ A_y \\ A_z \end{bmatrix}

标量场的梯度

梯度运算基本公式

与导数相同

梯度在柱坐标系和球坐标系中的表达式

\bigtriangledown u=a_\rho \frac{\partial u}{\partial \rho}+a_\phi \frac{1}{\rho}\frac{\partial u}{\partial \phi}+a_z\frac{\partial u}{\partial z}

\bigtriangledown u=a_r\frac{\partial u}{\partial r}+a_\theta \frac{1}{r}\frac{\partial u }{\partial \theta}+a_\phi \frac{1}{rsin\theta}\frac{\partial u}{\partial \phi}

重要性质

现有
R=\begin{bmatrix} (x-x^{'})^2+(y-y^{'})^2+(z-z^{'})^2 \end{bmatrix}
,有
\bigtriangledown (\frac{1}{R})=-\bigtriangledown ^{'}(\frac{1}{R})

其中，
\bigtriangledown ^{'}
表示对
x^{'},y^{'},z^{'}
微分

矢量场的散度

散度在柱坐标系和球坐标系中的表达式

\bigtriangledown \cdot F=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y}+\frac{\partial F}{\partial z}

\bigtriangledown \cdot F = \frac{1}{\rho}\frac{\partial (\rho F_\rho)}{\partial \rho}+\frac{1}{\rho}\frac{\partial F}{\partial \phi}+\frac{\partial F}{\partial z}

\bigtriangledown \cdot F=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 F_r)+\frac{1}{rsin\theta}\frac{\partial}{\partial \theta}(sin\theta F_\theta)+\frac{1}{r sin\theta}\frac{\partial F_\phi}{\partial \phi}

散度运算基本公式

\begin{bmatrix} \bigtriangledown \cdot \mathbf{C}=0\\ \bigtriangledown \cdot \mathbf{C}f=\mathbf{C} \cdot \bigtriangledown f\\ \bigtriangledown \cdot (kF)=k\bigtriangledown \cdot F\\ \bigtriangledown \cdot (fF)=f\bigtriangledown \cdot F+F\bigtriangledown \cdot f\\ \bigtriangledown \cdot (F\pm G)=\bigtriangledown \cdot F \pm \bigtriangledown \cdot G \end{bmatrix}

高斯散度定理

\oint _s F\cdot dS=\int_V \bigtriangledown \cdot F dV

矢量场的旋度

旋度计算基本公式

\left\{\begin{matrix} \bigtriangledown \times C=0\\ \bigtriangledown \times (kF)=k \bigtriangledown \times F\\ \bigtriangledown \times(fC)=\bigtriangledown f\times C\\ \bigtriangledown \times(fF)=f\bigtriangledown \times F+\bigtriangledown f\times F\\ \bigtriangledown \times (F \pm G)= \bigtriangledown \times F \pm \bigtriangledown \times G \\ \bigtriangledown \cdot (F \times G)=G \cdot (\bigtriangledown \times F)-F \cdot (\bigtriangledown \times G) \end{matrix}\right.

C为常矢量；k为常数；f为标量函数

旋度在柱坐标系和球坐标系中的表达式

rotF=\begin{vmatrix} a_x & a_y &a_z \\ \frac{\partial}{\partial x} &\frac{\partial}{\partial y} &\frac{\partial}{\partial z} \\ F_x & F_y &F_z \end{vmatrix}

rotF=\frac{1}{\rho}\begin{vmatrix} a_\rho & \rho a_\phi & a_z\\ \frac{\partial}{\partial \rho} & \frac{\partial }{\partial \phi} & \frac{\partial}{\partial z}\\ F_\rho&\rho F_\phi &F_z \end{vmatrix}

rotF=\frac{1}{r^2 sin\theta}\begin{vmatrix} a_r & ra_\theta & rsin\theta a_\phi\\ \frac{\partial}{\partial \rho} &\frac{\partial}{\partial \theta} &\frac{\partial}{\partial \phi} \\ F_r& rF_\theta & rsin\theta F_\phi \end{vmatrix}

斯托克斯定理

\oint _C F\cdot dl= \int _S \bigtriangledown \times F \cdot dS

场函数的二阶微分运算

零恒等式

数量场的梯度场为无旋场

向量场的旋度场为无源场

拉普拉斯运算

标性拉普拉斯运算

\bigtriangledown ^2u=\bigtriangledown \cdot (\bigtriangledown u)

\left\{\begin{matrix} \bigtriangledown ^2 u=(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}+\frac{\partial ^2}{\partial z^2})u\\ \bigtriangledown ^2 u=\frac{1}{\rho}\frac{\partial}{\partial \rho}(\rho \frac{\partial u}{\partial \rho})+\frac{1}{\rho ^2}\frac{\partial ^2 u}{\partial \phi ^2}+\frac{\partial ^2 u}{\partial z^2}\\ \bigtriangledown ^2 u=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial u}{\partial r})+\frac{1}{r ^2 sin\theta}\frac{\partial }{\partial \theta}(sin \theta \frac{\partial u}{\partial \theta})+\frac{1}{r^2 sin^2 \theta}\frac{\partial ^2 u}{\partial \phi ^2} \end{matrix}\right.

矢性拉普拉斯运算

\bigtriangledown ^2F=\bigtriangledown(\bigtriangledown \cdot F)-\bigtriangledown \times(\bigtriangledown \times F)

亥姆霍兹定理

在空间的有限区域内的任意一个矢量场，若已知它的散度，旋度与边界条件，那么该矢量场就被唯一地确定，并可表示为一个无旋场和一个无源场之和

例题


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