Theorem 1: Green's Theorem: Let D be a simple closed region in two dimensions, and let its boundary be C. For a vector field F(x,y) = (P(x,y), Q(x,y)), the line integral of F on the positive orientation of C is equal to the surface integral over the region D of the partial derivative of Q with repsect to x minus the partial derivative of P with respect to y
Theorem 2: Area of a Region: If C is a simple closed curve that bounds a region to which Theorem 1 applies, then the area of the region D bounded by C is one-half the line integral along C of F(x,y)=(-y,x)
Theorem 3: Vector Form of Green's Theorem: Let D be a subset of R-2 be a region to which Green's theorem applies, let C be its boundary (oriented counter-clockwise), and let F=(P,Q) be continuously differentiable vector field on D. Then, the line integral along C is equal to the surface integral of <curl(F),k> over the region D.