Since it is just a sort of discussion,I will just give the formula and condition without proving them or leaving examples.

General:

• Line integral(Work and in the plane)

　　　　$displaystyle int_{C}vec{F}cdot mathrm{d}vec{r} = int_{C}Mmathrm{d}x+Nmathrm{d}y$,in which $vec{F} = <M,N>$

　　　　　　Method: Express $x$ and $y$ in a single variable (OR means parameterization).

• Gradient fields & path-independence

Condition:

　　$curl(vec{F}) = 0$ and $vec{F}$ is defined in a simple-connected region,

　　　　　　　　in which $displaystyle curl(vec{F}) = N_{x} - M_{y}$ if $vec{F} = <M,N>$ AND $displaystyle curl(vec{F}) = nablatimesvec{F}$(namely$displaystyle begin{vmatrix}hat{i} & hat{j} & hat{k} frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} P & Q & Rend{vmatrix}) $,if $vec{F} = <P,Q,R>$

　　　　then $vec{F} = nabla f$,or $vec{F}$ is the partial derivative vector of some vector field.

The method of finding the potential:

　　Method 1. Do line integral.?Integral along the x-axis and y-axis and z-axis,if they exist. (Using path-independence)

　　Method 2. Integral one component of $vec{F}$ and then differential it over another variable and compare. (...)

• Flux in plane & space

in the plane:

　　$hat{n} = hat{T}$ rotated 90 degrees clockwise(standard) $=<mathrm{d}y,-mathrm{d}x>$

　　$displaystyle int_{C}vec{F}cdothat{n}mathrm{d}s = int_{C}Pmathrm{d}y-Qmathrm{d}x$,in which $vec{F} = <P,Q>$

in the space(or specifically,surface):

　　Case 1. $displaystyle iint_{S}vec{F}cdothat{n}mathrm{d}S = iint_{S}vec{F}cdot(<-f_{x},-f_{y},1>mathrm{d}xmathrm{d}y)$,if we use $z = f(x,y)$ to describe the surface.

　　Case 2. $displaystyle iint_{S}vec{F}cdothat{n}mathrm{d}S=iint_{S}vec{F}cdot(pmfrac{vec{N}}{vec{N}cdothat{k}}mathrm{d}xmathrm{d}y)$,if we are given the normal vector of the surface,or specifically,$g(x,y,z) = 0$

Addition(general case of the second): let‘s say $x = x(u,v)$ and $y = y(u,v)$ and $z = z(u,v)$ describe a surface,then we can get the $displaystyle hat{n}mathrm{d}S$ by changing $u$ and $v$ a little bit. Specifically,we begin at $displaystyle (x(u,v),y(u,z(u,v))$. By changing $u$ a little bit($Delta u$),then we arrive at $displaystyle (x(u+Delta u,y(u+Delta u,z(u+Delta u,v))$.

Using linear approximation,we get $displaystyle (x(u,v) + x_{u}Delta u,v) + y_{u}Delta u,v) + z_{u}Delta u)$,so the difference is $displaystyle (x_{u}Delta u,y_{u}Delta u,z_{u}Delta u) = Delta u(x_{u},y_{u},z_{u})$,let‘s set it to be $vec{r_{1}}$.

In the same way,we can get $displaystyle vec{r_{2}} = Delta v(x_{v},y_{v},z_{v})$ by changing $v$ a little bit. Thus we take the limits (meaning replace $Delta$ with $mathrm{d}$) and we can derive the corresponding $displaystyle hat{n}mathrm{d}S$ from it.

So $displaystyle hat{n}mathrm{d}S = vec{r_{1}}timesvec{r_{2}} = <x_{u},z_{u}>times<x_{v},z_{v}>mathrm{d}umathrm{d}v$.

(of course we can use position vector $displaystyle vec{r} = vec{r}(u,v) = <x,z>$ to simplify the problem,namely $displaystyle frac{partial vec{r}}{partial u}mathrm{d}u$ is the $vec{r_{1}}$)

Association:

Work(line integral):

• 2-D(Green‘s Theorem):

　　$displaystyle oint_{C}vec{F}cdotmathrm{d}vec{r} = iint_{R}curl(vec{F})mathrm{d}A$

• 3-D(Stoke‘s Theorem):

　　$displaystyle oint_{C}vec{F}cdotmathrm{d}vec{r} = iint_{S}curl(vec{F})hat{n}mathrm{d}S$,in which $S$ means any surface bounded by this curve and $curl(vec{F})=nablatimesvec{F}$.

Flux:

• 2-D(Green‘s Theorem):

　　$displaystyle oint_{C}vec{F}cdothat{n}mathrm{d}s = iint_{R}div(vec{F})mathrm{d}A$,Q>$ and $div(vec{F}) = P_{x} + Q_{y}$.

• 3-D(Divergence Theorem):

　　$displaystyleoiint_{S}vec{F}cdothat{n}mathrm{d}S = iiint_{R}div(vec{F})mathrm{d}V$,R>$ and $div(vec{F}) = P_{x} + Q_{y} + R_{z}$,in which the direction of $hat{n}$ is determined by the right-hand rule.