Would someone please help me understand Green's Theorem and Stoke's Theorem?

Question

I just want a decent conceptual explanation of the theorems, I've read through some books and received a technical explanation, but my professor doesn't explain things so well imho, so I'm just looking for more of a "dumbed down" explanation. I think I'm starting to get the idea, but I don't quite fully grasp it yet. I'm hoping that your explanations might help me get there. Thanks! :)

Answer 1

Green's theorem basically just tells us the relationship between the integral in a (simple) region, and the integral around the boundary of the region. The two are (obviously) not equal, but it's much easier to integrate one area than it is to do several integrals around the different sides of some region.

Stokes's theorem is quite a bit more general, although what you see in, say, the standard third semester of calculus isn't (usually) the full-blown theorem. It does the same (relating integral in a region to integral around the region). The most general form of Stokes's theorem is probably way too abstract to be "dumbed down" at all:
http://en.wikipedia.org/wiki/Stokes%27_theorem#General_formulation

I'm afraid it's impossible to explain exactly what each term in the theorems means without being at least a little bit technical. But the general meaning is integral in a region vs. integral around its boundary, and how the two relate.