基础定义及公式请参见百科。

电磁学数学基础

右旋叉乘恒等式:

• $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{c}\cdot \overrightarrow{a}\cdot \overrightarrow{b}-\overrightarrow{a}\cdot \overrightarrow{b}\cdot \overrightarrow{c}$ (难点)

算子算式:

• $\mathrm{\nabla }f=d_i\cdot {\overrightarrow{e}}_i\cdot f$

• $\mathrm{\nabla }\cdot \overrightarrow{f}=d_i\cdot f_i$

• $\mathrm{\nabla }\times \overrightarrow{f}=\epsilon \cdot d_i\cdot f_j\cdot {\overrightarrow{e}}_k$

分裂公式:

• $\partial (\overrightarrow{a}\cdot \overrightarrow{b})=\overrightarrow{a}\cdot \partial \overrightarrow{b}+\partial \overrightarrow{a}\cdot \overrightarrow{b}$

• $\partial (\overrightarrow{a}\times \overrightarrow{b})=\overrightarrow{a}\times \partial \overrightarrow{b}+\partial \overrightarrow{a}\times \overrightarrow{b}$

• $\mathrm{\nabla }(f\cdot g)=f\cdot \mathrm{\nabla }g+\mathrm{\nabla }f\cdot g$

• $\mathrm{\nabla }(\overrightarrow{a}\cdot \overrightarrow{b})=(\overrightarrow{a}\cdot \mathrm{\nabla })\cdot \overrightarrow{b}+(\overrightarrow{b}\cdot \mathrm{\nabla })\cdot \overrightarrow{a}+\overrightarrow{a}\times (\mathrm{\nabla }\times \overrightarrow{b})+\overrightarrow{b}\times (\mathrm{\nabla }\times \overrightarrow{a})$

函数分解通式:

• $\mathrm{\nabla }\ast \overrightarrow{F}(\phi (t))=\mathrm{\nabla }\phi (t)\ast {\displaystyle \frac{\partial \overrightarrow{F}}{\partial \phi }}$ (难点)

双重算子(二阶微分)公式:

• $∆f=(d_i{)}^{2}f$

• $\mathrm{\nabla }\times (\mathrm{\nabla }\cdot \overrightarrow{f})=0$

• $\mathrm{\nabla }\times (\mathrm{\nabla }f)=0$

• $\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \overrightarrow{f})=0$

• $(\mathrm{\Delta }\times {)}^2\overrightarrow{f}=\mathrm{\nabla }\times (\mathrm{\nabla }\times \overrightarrow{f})=\mathrm{\nabla }(\mathrm{\nabla }\cdot \overrightarrow{f})-\mathrm{\Delta }\overrightarrow{f}$

其他公式:

• $\mathrm{\nabla }\cdot (f\cdot \overrightarrow{F})=f\cdot (\mathrm{\nabla }\cdot \overrightarrow{F})+\overrightarrow{F}\cdot \mathrm{\nabla }f$

• $\mathrm{\nabla }\times (f\cdot \overrightarrow{F})=\mathrm{\nabla }f\times \overrightarrow{F}+f\cdot (\mathrm{\nabla }\times \overrightarrow{F})$

  • ${\int }_{S}f\cdot (\mathrm{\nabla }\times \overrightarrow{F})\cdot d\overrightarrow{a}={\int }_S\overrightarrow{F}\times (\mathrm{\nabla }f)\cdot d\overrightarrow{a}+{\oint }_l(f\cdot \overrightarrow{F})\cdot d\overrightarrow{l}$

• $\mathrm{\nabla }\cdot (\overrightarrow{a}\times \overrightarrow{b})=\overrightarrow{b}\cdot (\mathrm{\nabla }\times \overrightarrow{a})-\overrightarrow{a}\cdot (\mathrm{\nabla }\times \overrightarrow{b})$

  • ${\int }_V\overrightarrow{b}\cdot (\mathrm{\nabla }\times \overrightarrow{a})\cdot d\tau ={\int }_V\overrightarrow{a}\cdot (\mathrm{\nabla }\times \overrightarrow{b})\cdot d\tau +{\oint }_S(\overrightarrow{a}\times \overrightarrow{b})\cdot d\overrightarrow{a}$

• $\mathrm{\nabla }\times (\overrightarrow{a}\times \overrightarrow{b})=\overrightarrow{a}\cdot (\mathrm{\nabla }\cdot \overrightarrow{b})-\overrightarrow{b}\cdot (\mathrm{\nabla }\cdot \overrightarrow{a})+(\overrightarrow{b}\cdot \mathrm{\nabla })\cdot \overrightarrow{a}-(\overrightarrow{a}\cdot \mathrm{\nabla })\cdot \overrightarrow{b}$

狄拉克公式:

• ${\int }^{\mathrm{\forall }}{\delta }^{(n)}(t-t_0)\cdot f(t)\cdot dt=(-1{)}^n\cdot f^{(n)}(t_0)$

• $\delta (z=x^2-1)=\frac{1}{2}\cdot \delta (x+1)+\frac{1}{2}\cdot \delta (x-1)$

• $\delta (z=k(x-x_0))={\displaystyle \frac{1}{|k|}}\delta (x-x_0)$

冲击偶:

• $x\cdot {\delta }^{\prime }(x)=-\delta (x)$

• $f(t)\cdot {\delta }^{\prime }(t-t_0)=f(t_0){\delta }^{\prime }(t)-f^{\prime }(t_0)\delta (t)$

• 尺度变换 ${\delta }^{\prime }(at)={\displaystyle \frac{1}{a\cdot \left|a\right|}}\cdot {\delta }^{\prime }(t)$

抽样函数:

• $Sa(x)={\displaystyle \frac{sin(x)}{x}}$