Green's Theorem Problem

a) Let C be a simple closed curve which is the boundary of a simple domain D. Use Green's theorem to show the area of D = 1/2 (integral) F (dot) ds where F = -yi +xj

b) Use (a) to compute the area of a disk with radius r

c) Use (a) to compute the area of an ellipse with semi-axes a and b

I was under the impression that dQ/dx - dP/dy had to equal one in order to give that equation. If somone could generally clear things up that would be so appreciated!

Read http://tutorial.math.lamar.edu/Classes/C… for a proof and some good explanations. Part b) of your question is worked out at the very end of the page. In pat c) you should get the answer Pi times ab with similar calculations to part b)

if your vector field is (-y,x) and you want to use green theorem obiusly it would be like (S as integral)
S(2)dA ---> 2SdA.
but i dont understand what are you trying to say with that D = 1/2 integral F.ds, because the notation ds is used for integral of surface and green is not an integral of surface.
But if you want to use green theorem in this case for a disk with raius r it would be like 2(Area of disk):
2(PI)(r^2), and with an elipse you would have to use cartesians coordinates and it would be for order dydx it would be like:
2(Area of elipse): 2((pi)ab) being a the radius of axis-x and b the radius of axis-y because we know that the area of a elipse is (PI)AB.
I hope it helps.