矢量分析公式

$$\begin{array}{l}a\cdot (b\times c)=b\cdot (c\times a)+c\cdot (a\times b)\\ a\times (b\times c)=(a\cdot c)b-(a\cdot b)c\\ a\times (b\times c)+(b\times c)\times a+c\times (a\times b)=0\\ (a\times b)\cdot (c\times d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)\\ (a\times b)\times (c\times d)=[(a\times b)\cdot d]c-[(a\times b)\cdot c]d\\ =[(c\times d)\cdot a]b-[(c\times d)\cdot b]a\\ ▽\times (▽\varphi )=0，▽\cdot (▽\times a)=0，▽\cdot ▽\varphi ={▽}^2\varphi =△\varphi \\ ▽\times (▽\times a)=▽(▽\cdot a)-△a,△a=▽\cdot (▽a)\\ ▽(\varphi \psi )=\varphi ▽\psi +\psi ▽\varphi \\ △(\varphi \psi )=\varphi △\psi +2(▽\varphi )\cdot (▽\psi )+\psi △\varphi \\ ▽\cdot (\varphi a)=\varphi ▽\cdot a+a\cdot ▽\varphi \\ ▽\times (\varphi a)=\varphi ▽\times a+(▽\varphi )\times a\\ ▽\cdot (a\times b)=b\cdot (▽\times a)-a\cdot (▽\times b)\\ ▽(a\cdot b)=a\times (▽\times b)+b\times (▽\times a)+(b\cdot ▽)a+(a\cdot ▽)b\\ ▽\times (a\times b)=a(▽\cdot b)-b(▽\cdot a)+(b\cdot ▽)a-(a\cdot ▽)b\\ l=-ihr\times ▽,▽=i\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}+k\frac{\partial }{\partial z}\\ ▽=e_r\frac{\partial }{\partial r}-\frac{i}{h{r}^2}r\times l\\ △=\frac{1}{r}\frac{{\partial }^2}{\partial r^2}r-\frac{l^2}{h^2{r}^2}\\ ▽\cdot r=3\\ ▽\times r=0\\ ▽\cdot e_r=\frac{2}{r}▽\times e_r=0\end{array}$$