Green's Theorem

[red team]

I don't understand Green's theorem or how or why it works. Can anybody help me on this?
Please include a detailed explanation on Stoke's theorem also.

Thank You

[ComradeRed]

Are you familiar with oriented piecewise-smooth surfaces? Ones bounded by a simple, closed, piecewise-smooth boundary curve with positive orientation?

Basically, think of the oriented piecewise-smooth surface as a hill, that as you walk around it your head is still up and the hill is on your left. (So we won't be dealing with a Mobius surface or anything that's wacky like that :)).

Well, if we integrate over such a surface a function F·dr, it's equal to a double integral over the surface for curlF·dS.

Or in English the line integral around the boundary curve of the surface of the tangential component of F is equal to the surface integral of the normal component of the curl of F. :)

Green's theorem is a vector form of Stoke's theorem (or at least, that's how I think of it, since I only deal with it in terms of Path integration in quantum field theory). If we have a surface S that is flat and lies in the xy-plane with upward orientation, the unit normal is k, the surface integral becomes a double integral, and stoke's theorem is of the form:

Integrate over the boundary curve ("C") F·dr = double integral over S (curl F)·kdA. This is green's theorem for vectors :)

[red team]

Got it.

Just looked at the proof in the text book. Really convoluted, but pretty ingenious how he came up with it.

[ComradeRed]

There are probably shorter methods to do the same proof to derive Green's theorem nowadays ;) but people are lazy, so I may be wrong (too lazy to find an alternate form of Green's theorem proof :P).

If you don't understand some of it, may I suggest writing it down? I mean the work, start from where you understand, then write down what follows exactly verbatim! Skip nothing!

If you still don't understand, go back, step by step, and say "What happened here? Why?". This is what I do when I don't understand some math ;) Check out Polya's List, it's a gold mine for math.