Green's Theorem is a result in real analysis. It is a special case of Stokes' Theorem.
Let be a bounded subset of with positively oriented boundary , and let and be functions with continuous partial derivatives mapping an open set containing into . Then
It suffices to show that the theorem holds when is a square, since can always be approximated arbitrarily well with a finite collection of squares.
Then let be a square with vertices , , , , with and . Then Now, by the Fundamental Theorem of Calculus, and Hence as desired.