Daniel Castillo's Favorite Theorem
My favorite theorem is Green's Theorem. Here it is as it is stated in Jerrold E. Marsden & Anthony J. Tromba's Vector Calculus:
Let D be a simple region and let C be its boundary. Suppose P: D -> R and Q: D -> R are of class C1. Then
$ \int_C+ \P * dx + Q * dy = \int_ \int_D \ ((part(Q) / part(x) - part(P) / part(y))) * dx * dy. $
