The following formulation of Green's theorem is due to Spivak (Calculus on Manifolds, p. 134):

 Let M\subset\R^2 be a compact two-dimensional manifold-with-boundary. Suppose that \alpha,\beta:M\to\R are differentiable. Then \int_{\part M}\alpha\,dx+\beta\,dy=\int_M (D_1\beta-D_2\alpha)dx\and dy=\iint_M\left(\frac{\part\beta}{\part x}-\frac{\part\alpha}{\part y}\right)dx\,dy

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:

\oint_C\vec{F}\cdot d\vec{r}=\iint_S\nabla\times\vec{F}\cdot dS=\iint_S\left(\frac{\part\beta}{\part x}-\frac{\part\alpha}{\part y}\right)dA

If the line integral is dotted with the normal, rather than tangent vector, Green's theorem takes the form

\oint_C\vec{F}\cdot\vec{\text{n}}\,ds=\iint_S\nabla\cdot\vec{F}\,dA=\iint_S\left(\frac{\part\alpha}{\part x}+\frac{\part\beta}{\part y}\right)dA

which is also known as the two-dimensional divergence theorem.

Extended Green's Theorem

If curl \vec{F} = 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:

\oint_{C_1} \vec{F} \cdot dr - \oint_{C_2} \vec{F} \cdot dr = \iint curl \vec{F} \cdot k dA = 0

See also


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