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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

Informal description

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.[1] For example, if curve C1 encloses an undefined point, let C2 be a smaller region that encloses said point:

See also


  1. Session 71: Extended Green's Theorem - MIT Open Courseware, Multivariable Calculus (Fall 2010)