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真空中麦克斯韦方程组
$$ \begin{gathered}\nabla\times \boldsymbol E=-\frac{\partial \boldsymbol B}{\partial t} \\\nabla\times \boldsymbol B=\mu_0\boldsymbol{J}+\mu_0\boldsymbol{\varepsilon}_0\frac{\partial{\boldsymbol E}}{\partial t} \\\nabla\cdot\boldsymbol{E}=\frac\rho{\varepsilon_0} \\\nabla\cdot\boldsymbol{B}=0 \end{gathered} $$
介质中麦克斯韦方程组
$\rho=\rho_f+\rho_p$自由电荷，极化电荷$\rho_p=-\nabla\cdot \boldsymbol{\mathcal P}$；$J=J_d+J_M+J_p.$传导电流，磁化电流$J_M=\nabla\times M$，极化电流$J_P=-\frac{\partial \mathcal P}{\partial t}$
$$ \begin{aligned}&\nabla\times\boldsymbol{E}= -\frac{\partial \boldsymbol B}{\partial t} \\&\nabla\times\boldsymbol{H}=\boldsymbol{J}_d+\frac{\partial\boldsymbol{D}}{\partial t} \\&\nabla\cdot\boldsymbol{D}=\rho_f \\&\nabla\cdot\boldsymbol{B}=0\end{aligned} $$
$$ \begin{aligned}&\oint_{L}\boldsymbol{E}\cdot\mathrm{d}\boldsymbol{l}=-\frac{\mathrm{d}}{\mathrm{d}\boldsymbol{t}}\int_{s}\boldsymbol{B}\cdot\mathrm{d}\boldsymbol{S} \\&\oint_{L}\boldsymbol{H}\cdot\mathrm{d}\boldsymbol{l}=I_f+\frac{\mathrm{d}}{\mathrm{d}t}\underset{\mathrm{s}}{\operatorname*{\int}}\boldsymbol{D}\cdot\mathrm{d}\boldsymbol{S} \\&\oint_{s}\boldsymbol{D}\cdot\mathrm{d}\boldsymbol{S}=Q_{t} \\&\oint_s\boldsymbol{B}\cdot\mathrm{d}\boldsymbol{S}=0\end{aligned} $$
$$ \begin{aligned} \oint_{\partial \Sigma}\bm E\cdot d\bm l&=-\iint_\Sigma \frac{\partial \bm B}{\partial t}\cdot d\bm \sigma\\
\oint_{\partial \Sigma}\bm H\cdot d\bm l&=\iint_\Sigma \left(\bm J+\frac{\partial \bm D}{\partial t}\right)\cdot d\bm \sigma\\
\oiint_\Sigma \bm D\cdot d\bm \sigma&＝\iiint_\Omega \rho dV\\
\oiint_\Sigma \bm B\cdot d\bm \sigma &= 0 \end{aligned} $$
斯托克斯公式 → 格林公式 - 积分定理
封闭曲线的环量，等于旋度的（封闭曲线为边界的曲面）通量
$$ \oint_l\bm A\cdot d\bm l=\iint_\Sigma\left(\nabla\times \bm A\right)\cdot d\bm \sigma $$
高斯公式 → 格林公式 - 积分定理
封闭曲面的通量，等于散度的（封闭曲面所围体积的）体积分
$$ \oiint_\Sigma \bm A\cdot d\bm \sigma=\iiint_\Omega \nabla \cdot \bm A dV $$
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maxwell equations 中的偏导数（电磁效应导致，固定面元的积分的求导）
transverse wave 横波
there is an infinite number of sunch to and fro directin, all lying in a plane normal to the direction of propagation, in which the vibrations may occur.
longitudinal wave 纵波
there is only one to and fro direction 一个来回震荡的方向 in which the vibrations 震荡 can occur for a given direction of propagation 传播
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Gauge invariance in electromagnetism规范不变性
平面谐振波 PHM Plane Harmonic Motion
真空中
$$ \begin{aligned}
\boldsymbol H=\frac1{\mu_0}\boldsymbol B,\quad \boldsymbol D=\varepsilon_0\boldsymbol E
\end{aligned} $$
玻印亭矢量 $\bm S$；
能流密度 $w$ 可以视作逐点电场能和磁场能的和，但更一般的定义是变分法（？）
$$ \begin{aligned}
\boldsymbol{S}&= \frac1{{\mu}_0}\boldsymbol{E}\times\boldsymbol{B} \\
\boldsymbol{w}&= \frac12\left({\varepsilon}_0\boldsymbol E^2 +\frac1{\mu_0}\boldsymbol B^2\right)
\end{aligned} $$
$$ \delta \omega=\boldsymbol E\cdot\delta \boldsymbol D+\boldsymbol H\cdot\delta \boldsymbol B $$
线性介质