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

Theorem
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.