名称

曲线积分

曲面积分

思想方法

分割(化整为零)→近似(以不变代变)→求和(积零为整)→取极限(精确值)

性质

线性性质、可加性、比较大小、反向变号

记号

${\displaystyle \underset{L}{\int }f\left(x,y\right)\text{d}S}$ (第一类)

${\displaystyle {\int }_{L}P\left(x,y\right)\text{d}x+Q\left(x,y\right)\text{d}y}$

(第二类)

${\displaystyle \underset{\Sigma }{\iint }f\left(x,y,z\right)\text{d}S}$ (第一类) ${\displaystyle \underset{\Sigma }{\iint }P\left(x,y,z\right)\text{d}y\text{d}z}+{\displaystyle \underset{\Sigma }{\iint }Q\left(x,y,z\right)\text{d}z\text{d}x}+{\displaystyle \underset{\Sigma }{\iint }R\left(x,y,z\right)\text{d}x\text{d}y}$ (第二类)

计算思路

明确曲线方程 ↓

化为定积分

↓

计算定积分

明确曲面方程

↓

化为二重积分

↓

计算二重积分

关系

格林公式

$\begin{array}{l}{\displaystyle \underset{D}{\iint }\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\text{d}x\text{d}y}\\ ={\displaystyle {\oint }_{L}P\left(x,y\right)\text{d}x+Q\left(x,y\right)\text{d}y}\end{array}$

高斯公式

$\begin{array}{l}{\displaystyle \underset{\Omega }{\iiint }\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right)\text{d}v}\\ ={\displaystyle \underset{\Sigma }{∯}P\left(x,y,z\right)\text{d}y\text{d}z+Q\left(x,y,z\right)\text{d}z\text{d}x+R\left(x,y,z\right)\text{d}x\text{d}y}\end{array}$

斯托克斯公式(曲线积分与曲面积分关系)

$\begin{array}{l}{\displaystyle \underset{\Sigma }{\iint }\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\text{d}y\text{d}z}+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\text{d}z\text{d}x+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\text{d}x\text{d}y\\ ={\displaystyle {\oint }_{\Gamma }P\left(x,y,z\right)\text{d}x+Q\left(x,y,z\right)\text{d}y+R\left(x,y,z\right)\text{d}z}\end{array}$