Integral Topic Notes - Path Independent Conditions

一、The path-independent condition of the plane curve integral

定理（Four equivalent conditions for the curve integral of the second kind of plane curve to be independent of the path）

若 $D$ is a plane simply connected region,且 $P$,$Q$ 在 $D$ 上连续且具有连续的一阶偏导数,Then the following four conditions are equivalent：

(1) 对 $D$ any closed curve $\Gamma$ The curve integral of the second kind is $0$,即：${\oint }_{\Gamma }Pdx+Qdy=0$ .

(2) 对任给 ${\Gamma }_{ACB},{\Gamma }_{ADB}\subset D$,则：${\oint }_{{\Gamma }_{ACB}}Pdx+Qdy={\oint }_{{\Gamma }_{ADB}}Pdx+Qdy$ .

​ 即在 $D$ Integral over a closed curve in Africa,It's only about the starting point and the ending point,与 $D$ The path in is irrelevant.

(3) 存在 $D$ 上的一个二元函数 $\mu (x,y)$,使 $du=Pdx+Qdy$,即 $\frac{\partial u}{\partial x}=P$ , $\frac{\partial u}{\partial y}=Q$ .

(4)$\forall (x,y)\in D$,都有 $\frac{\partial Q}{\partial x}\equiv \frac{\partial P}{\partial y}$ .

二、空间曲线积分与路径无关的条件

Four equivalent conditions for the curve integral of the second kind of space to be independent of paths

设 $\Omega$ is the first-order partial derivative of a line simply connected in the space,Then the following four conditions are equivalent：

(1) 对 $\Omega$ any closed curve in $L$ The curve integral of the second kind on is $0$,即 ${\oint }_{L}Pdx+Qdy+Rdz=0$

(2) 对 $\Omega$ any two non-closed curves in ${\Gamma }_{ACB},{\Gamma }_{ADB}$,

​ 有 ${\int }_{{\Gamma }_{ACB}}Pdx+Qdy+Rdz={\int }_{{\Gamma }_{ADB}}Pdx+Qdy+Rdz$ ,

​ 即 $\Omega$ The curve integral of the second kind on the middle non-closed curve is only related to the starting point and the ending point,与 $\Omega$ The path in is irrelevant.

(3) 存在 $\Omega$ a ternary function on $u(x,y,z)$ ,使 $du=Pdx+Qdy+Rdz$,

​ 即 $\frac{\partial u}{\partial x}=P,\frac{\partial u}{\partial y}=Q,\frac{\partial u}{\partial z}=R$,称 $u$ 是 $Pdx+Qdy+Rdz$ 的一个原函数.

(4) $\frac{\partial Q}{\partial x}\equiv \frac{\partial P}{\partial y},\frac{\partial R}{\partial y}\equiv \frac{\partial Q}{\partial z},\frac{\partial P}{\partial z}\equiv \frac{\partial R}{\partial x}$,即 $\mathrm{rot}\vec{A}\equiv 0,(x,y,z)\in \Omega$ .