三度其三——矢量场的旋度

环量
取有向闭合曲线，方向符合右手螺旋
$Q=\oint_{L}\vec A\cdot\vec l$
一个空间有无穷多个曲面穿套而成，用有向曲线把一个接一个的曲面填充，研究任意一个环路情况足以描述这个图，这只是一种整体的概念，大小几何形状不变，空间方位
河流有流速场，沿着流线对速度进行积分，环量不为0，这里有漩涡。漩涡围着转，相当于一个轴，当积分路径与轴垂直的平面在一个平面，积分值最大，轴与法线方向重合。
矢量场中我们关心通量的环路 $\vec n$ 取什么方向，环量最大，与数量场方向导数最大有相同概念。

旋度
限定环路的面积
$q=\lim_{\Delta l\to0}\frac{\oint_{l}\vec A\cdot\vec l}{\Delta l}$
这个环量对应平均的密度
$q=(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})cos\alpha+(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})cos\beta+(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})cos\gamma$
$cos\alpha,cos\beta,cos\gamma$ 是 $\vec n$ 方向的单位矢量。
$\vec n ^{\circ}=cos\alpha\vec e_x+cos\beta\vec e_y+cos\gamma\vec e_z$
取
$\vec V=(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\vec e_x+(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\vec e_y+(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\vec e_z\\ q=\vec V\cdot\vec n ^{\circ}$
$\vec n$ 方向的环量密度，即 $\vec V$ 在 $\vec n$ 方向的投影， $\vec V$ 取决于 $\vec A$ 在 $x,y$ ,方向的偏导数， $\vec A$ 一旦确定则 $\vec V$ 也确定。

$rot \vec A=(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\vec e_x+(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\vec e_y+(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\vec e_z$
$rot \vec A = \bigtriangledown \times \vec A = \left | \begin{matrix} \vec e_x &\vec e_y &\vec e_z\\ \frac{\partial}{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ A_x & A_y & A_z \end{matrix} \right |$

重要公式
$\triangledown \cdot(\vec A \times \vec B)= \vec B \cdot \triangledown \times\vec A-\vec A\cdot\triangledown\times\vec B$
证明
$\triangledown=\frac{\partial}{\partial x}\vec e_x+\frac{\partial }{\partial y}\vec e_y+\frac{\partial }{\partial z}\vec e_z$
$\vec A \times \vec B=\left | \begin{matrix} \vec e_x &\vec e_y &\vec e_z\\ A_x & A_y & A_z\\ B_x & B_y & B_z \end{matrix} \right |\\ =(A_yB_z-A_zB_y)\vec e_x+(A_zB_x-A_xB_z)\vec e_y+(A_xB_y-A_yB_x)\vec e_z$
$\triangledown \cdot(\vec A \times \vec B)=(\frac{\partial}{\partial x}\vec e_x+\frac{\partial }{\partial y}\vec e_y+\frac{\partial }{\partial z}\vec e_z) \cdot\bigg((A_yB_z-A_zB_y)\vec e_x+(A_zB_x-A_xB_z)\vec e_y+(A_xB_y-A_yB_x)\vec e_z\bigg)\\ =\frac{\partial{(A_yB_z-A_zB_y)}}{\partial x}+\frac{\partial{(A_zB_x-A_xB_z)}}{\partial y}+\frac{\partial{(A_xB_y-A_yB_x)}}{\partial z}\\ =\left | \begin{matrix} \frac{\partial}{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ A_x & A_y & A_z\\ B_x & B_y & B_z \end{matrix} \right |\\$

$\vec B \cdot \triangledown \times\vec A-\vec A\cdot\triangledown\times\vec B=\vec B \cdot \left | \begin{matrix} \vec e_x &\vec e_y &\vec e_z\\ \frac{\partial}{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ A_x & A_y & A_z \end{matrix} \right | - \vec A \cdot \left | \begin{matrix} \vec e_x &\vec e_y &\vec e_z\\ \frac{\partial}{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ B_x & B_y & B_z \end{matrix}\right | \\ = (B_x\vec e_x+B_y\vec e_y+B_z\vec e_z)\cdot\bigg( (\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\vec e_x+(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\vec e_y+(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\vec e_z \bigg)\\ -(A_x\vec e_x+A_y\vec e_y+A_z\vec e_z)\cdot\bigg( (\frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z})\vec e_x+(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x})\vec e_y+(\frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y})\vec e_z \bigg)\\ =B_x( \frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})+ B_y(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})+ B_z(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\\ -\bigg(A_x( \frac{\partial B_z}{\partial y}-\frac{\partial B_y}{\partial z})+ A_y(\frac{\partial B_x}{\partial z}-\frac{\partial B_z}{\partial x})+ A_z(\frac{\partial B_y}{\partial x}-\frac{\partial B_x}{\partial y})\bigg)\\ \because B_z\frac{\partial A_y}{\partial x}-B_y\frac{\partial A_z}{\partial x}+ A_y\frac{\partial B_z}{\partial x}- A_z\frac{\partial B_y}{\partial x}= \bigg(B_z\frac{\partial A_y}{\partial x}+ A_y\frac{\partial B_z}{\partial x}\bigg)-\bigg(B_y\frac{\partial A_z}{\partial x}+A_z\frac{\partial B_y}{\partial x} \bigg)\\ =\frac{\partial( A_yB_z)}{\partial x}-\frac{\partial( A_zB_y)}{\partial x}=\frac{\partial( A_yB_z-A_zB_y)}{\partial x}$ B⋅▽×A−A⋅▽×B=B⋅∣∣∣∣∣∣ex∂x∂Axey∂y∂Ayez∂z∂Az∣∣∣∣∣∣−A⋅∣∣∣∣∣∣ex∂x∂Bxey∂y∂Byez∂z∂Bz∣∣∣∣∣∣=(Bxex+Byey+Bzez)⋅((∂y∂Az−∂z∂Ay)ex+(∂z∂Ax−∂x∂Az)ey+(∂x∂Ay−∂y∂Ax)ez)−(Axex+Ayey+Azez)⋅((∂y∂Bz−∂z∂By)ex+(∂z∂Bx−∂x∂Bz)ey+(∂x∂By−∂y∂Bx)ez)=Bx(∂y∂Az−∂z∂Ay)+By(∂z∂Ax−∂x∂Az)+Bz(∂x∂Ay−∂y∂Ax)−(Ax(∂y∂Bz−∂z∂By)+Ay(∂z∂Bx−∂x∂Bz)+Az(∂x∂By−∂y∂Bx))∵Bz∂x∂Ay−By∂x∂Az+Ay∂x∂Bz−Az∂x∂By=(Bz∂x∂Ay+Ay∂x∂Bz)−(By∂x∂Az+Az∂x∂By)=∂x∂(AyBz)−∂x∂(AzBy)=∂x∂(AyBz−AzBy)

同理其他三项可得，证毕
哈密尔顿算子首先考虑矢量性再考虑微分性
$\triangledown\times(\mu\vec A)=\mu\triangledown\times\vec A+\triangledown\mu\times\vec A$
$\mu$ 是标量函数，取了梯度变为矢量
$\triangledown\times(\triangledown\mu)\equiv 0$
$\triangledown\mu=\frac{\partial \mu}{\partial x}\vec e_x+\frac{\partial \mu}{\partial y}\vec e_y+\frac{\partial\mu}{\partial x}\vec e_z\\ \triangledown\times(\triangledown\mu)=\left | \begin{matrix} \vec e_x &\vec e_y &\vec e_z\\ \frac{\partial}{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ \frac{\partial\mu}{\partial x} & \frac{\partial\mu}{\partial y} & \frac{\partial\mu}{\partial z} \end{matrix} \right |\\ =\bigg(\frac{\partial^2\mu }{\partial z\partial y}-\frac{\partial^2\mu }{\partial y\partial z}\bigg)\vec e_x+\cdots=0$
取旋度仍为矢量
$\triangledown\cdot(\triangledown\times\vec A)\equiv 0$
$\triangledown\cdot(\triangledown\times\vec A)= \left | \begin{matrix} \frac{\partial}{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ \frac{\partial}{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ A_x & A_y & A_z \end{matrix} \right | =0$
两个恒等式在静电场分析和恒定磁场分析里很重要
拉普拉斯算子
$\triangledown\cdot(\triangledown\mu)=\triangledown^2\mu$
$\triangledown\cdot(\triangledown\mu)=(\frac{\partial}{\partial x}\vec e_x+\frac{\partial }{\partial y}\vec e_y+\frac{\partial }{\partial z}\vec e_z)(\frac{\partial \mu}{\partial x}\vec e_x+\frac{\partial \mu}{\partial y}\vec e_y+\frac{\partial\mu}{\partial x}\vec e_z)\\ =\frac{\partial^2\mu }{\partial x^2}+\frac{\partial^2\mu }{\partial y^2}+\frac{\partial^2\mu }{\partial z^2}$
旋了又旋，矢量的拉普拉斯算子
$\triangledown\times(\triangledown\times\vec A)=\triangledown\cdot(\triangledown\cdot\vec A)-\triangledown^2\vec A$
$\triangledown\times(\triangledown\times\vec A)=\triangledown\times\bigg((\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\vec e_x+(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\vec e_y+(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\vec e_z\bigg)\\ = \left | \begin{matrix} \vec e_x &\vec e_y &\vec e_z\\ \frac{\partial}{\partial x}&\frac{\partial }{\partial y}&\frac{\partial }{\partial z}\\ (\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}) &(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}) &(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}) \end{matrix} \right | \\ =(\frac{\partial A_y}{\partial x\partial y}-\frac{\partial A_x}{\partial^2 y}-\frac{\partial A_x}{\partial^2 z}+\frac{\partial A_z}{\partial x\partial z})\vec e_x\\ +(\frac{\partial A_z}{\partial y\partial z}-\frac{\partial A_y}{\partial^2 z}-\frac{\partial A_y}{\partial^2 x}+\frac{\partial A_x}{\partial y\partial x})\vec e_y\\ +(\frac{\partial A_x}{\partial z\partial x}-\frac{\partial A_z}{\partial^2 x}-\frac{\partial A_z}{\partial^2 y}+\frac{\partial A_y}{\partial z\partial y})\vec e_z\\ =(\frac{\partial A_y}{\partial x\partial y}+\frac{\partial A_z}{\partial x\partial z})\vec e_x+(\frac{\partial A_z}{\partial y\partial z}+\frac{\partial A_x}{\partial y\partial x})\vec e_y+(\frac{\partial A_x}{\partial z\partial x}+\frac{\partial A_y}{\partial z\partial y})\vec e_z\\ -\bigg((\frac{\partial A_x}{\partial^2 y}+\frac{\partial A_x}{\partial^2 z})\vec e_x+(\frac{\partial A_y}{\partial^2 z}+\frac{\partial A_y}{\partial^2 x})\vec e_y+(\frac{\partial A_z}{\partial^2 x}+\frac{\partial A_z}{\partial^2 y})\vec e_z \bigg)$ ▽×(▽×A)=▽×((∂y∂Az−∂z∂Ay)ex+(∂z∂Ax−∂x∂Az)ey+(∂x∂Ay−∂y∂Ax)ez)=∣∣∣∣∣∣∣ex∂x∂(∂y∂Az−∂z∂Ay)ey∂y∂(∂z∂Ax−∂x∂Az)ez∂z∂(∂x∂Ay−∂y∂Ax)∣∣∣∣∣∣∣=(∂x∂y∂Ay−∂2y∂Ax−∂2z∂Ax+∂x∂z∂Az)ex+(∂y∂z∂Az−∂2z∂Ay−∂2x∂Ay+∂y∂x∂Ax)ey+(∂z∂x∂Ax−∂2x∂Az−∂2y∂Az+∂z∂y∂Ay)ez=(∂x∂y∂Ay+∂x∂z∂Az)ex+(∂y∂z∂Az+∂y∂x∂Ax)ey+(∂z∂x∂Ax+∂z∂y∂Ay)ez−((∂2y∂Ax+∂2z∂Ax)ex+(∂2z∂Ay+∂2x∂Ay)ey+(∂2x∂Az+∂2y∂Az)ez)
$\triangledown\cdot(\triangledown\cdot\vec A)=\triangledown\cdot\bigg(\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\bigg)\\ =\bigg(\frac{\partial}{\partial x}\vec e_x+\frac{\partial }{\partial y}\vec e_y+\frac{\partial }{\partial z}\vec e_z\bigg)\cdot\bigg(\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}\bigg)\\ =\frac{\partial A_x}{\partial^2 x}\vec e_x+\bigg(\frac{\partial A_y}{\partial y\partial x}+\frac{\partial A_z}{\partial z\partial x}\bigg)\vec e_x+\cdots$
$\triangledown^2\vec A=\triangledown^2A_x\vec e_x+\triangledown^2A_y\vec e_y+\triangledown^2A_z\vec e_z\\ =\bigg(\frac{\partial^2A_x }{\partial x^2}+\frac{\partial^2A_x }{\partial y^2}+\frac{\partial^2A_x }{\partial z^2}\bigg)\vec e_x\\ +\bigg(\frac{\partial^2A_y }{\partial x^2}+\frac{\partial^2A_y }{\partial y^2}+\frac{\partial^2A_y }{\partial z^2}\bigg)\vec e_y\\ +\bigg(\frac{\partial^2A_z }{\partial x^2}+\frac{\partial^2A_z }{\partial y^2}+\frac{\partial^2A_z }{\partial z^2}\bigg)\vec e_z$
$\triangledown\cdot(\triangledown\cdot\vec A)-\triangledown^2\vec A\\ =\frac{\partial A_x}{\partial^2 x}\vec e_x+\bigg(\frac{\partial A_y}{\partial y\partial x}+\frac{\partial A_z}{\partial z\partial x}\bigg)\vec e_x-\bigg(\frac{\partial^2A_x }{\partial x^2}+\frac{\partial^2A_x }{\partial y^2}+\frac{\partial^2A_x }{\partial z^2}\bigg)\vec e_x+\cdots\\ =\bigg(\frac{\partial A_y}{\partial y\partial x}+\frac{\partial A_z}{\partial z\partial x}\bigg)\vec e_x-\bigg(\frac{\partial^2A_x }{\partial y^2}+\frac{\partial^2A_x }{\partial z^2}\bigg)\vec e_x+\cdots$
其它三项均可证得，证毕

三度其三——矢量场的旋度

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