Quote:Original post by arithma
Green's Theorem:

double intergral ( curl vector(F) dA ) = closed integral ( vector(F) dot vector(ds) )

What I need is double integral ( x^2 + y^2 dA ), right?
Knowing that vector(F) = <F1, F2>.. and applying green's theorem to this problem we get: dF2/dy - dF1/x = x^2 + y^2. I just chose F1 = 0. So F2 = x^2*y + y^3/3.

My areas are polygonal, so I am just going to parameterize the boundaries into segments and from there it should go alright.

So my question is: Is my work correct?

It's almost correct. Curl(F) dot <0,0,1> gives you [(∂F2/∂x) - (∂F1/∂y)]. Letting that equal x2 + y2, and letting F1 = 0 (or in general any constant) gives you F2 = x3/3 + y2x. Note that this is only true if you integrate the contour with respect to the parallel component 'dr'. You can also integrate with respect to the perpendicular component nds = -perpdot(dr), and you'll end up with the divergence theorem (more info about its relationship to Green's theorem). This gives you [(∂F1/∂x) + (∂F2/∂y)], which when equal to x2 + y2 gives you F2 = x2y + y3/3 (with F1 being a constant again). I'm partial (no pun intended) toward the divergence theorem due to my studies in electrostatics, but how you evaluate the integrals is up to you.