The following formulation of Green's theorem is due to Spivak (Calculus on Manifolds, p. 134):
|Let be a compact two-dimensional manifold-with-boundary. Suppose that are differentiable. Then|
Green's theorem relates a closed line integral to a double integral of its curl. Green's theorem is actually a special case of Stokes' theorem, which, when dealing with a loop in the plane, simplifies as follows:
If the line integral is dotted with the normal, rather than tangent vector, Green's theorem takes the form
which is also known as the two-dimensional divergence theorem.
Extended Green's Theorem
If curl = 0 but there are points which are not differentiable, it is still possible to apply green's theorem by creating a secondary curve enclosing the first one, and appling the theorem to the region between the two curves. For example, if curve C1 encloses an undefined point, let C2 be a smaller region that encloses said point: