梯度、散度、旋度总结

散度定理（Gauss定理）：穿过整个体积表面$\partial V$（闭曲面）的通量等于其体积微元散度之和，即

$${\oint }_{\partial V}\overrightarrow{F}\cdot \overrightarrow{n}dS={\oint }_V\mathrm{div}\overrightarrow{F}dV$$

三维Gauss定理:

$${\iint }_{\partial V}{f}_{1}dydz+f_{2}dzdx+f_{3}dxdy={\iiint }_V(\frac{\partial f_1}{\partial x}+\frac{\partial f_2}{\partial y}+\frac{\partial f_3}{\partial z})dxdydz$$

旋度定理：沿区域边界$\partial S$（闭曲线）的环量等于其区域面积微元旋度之和，即

$${\oint }_{\partial S^{+}}\overrightarrow{F}\cdot d\overrightarrow{L}={\oint }_S\mathrm{rot}\overrightarrow{F}\cdot \overrightarrow{n}dS$$

二维Green定理：

$${\int }_{\partial S^{+}}{f}_{1}dx+f_{2}dy={\iint }_S(\frac{\partial f_2}{\partial x}-\frac{\partial f_1}{\partial y})dxdy$$

三维Stokes定理：

$${\int }_{\partial S^{+}}{f}_{1}dx+f_{2}dy+f_{3}dz={\iint }_S(\frac{\partial f_3}{\partial y}-\frac{\partial f_2}{\partial z})dydz+(\frac{\partial f_1}{\partial z}-\frac{\partial f_3}{\partial x})dzdx+(\frac{\partial f_2}{\partial x}-\frac{\partial f_1}{\partial y})dxdy$$