Green's theorem

In vector calculus, Green’s theorem is the two-dimensional analogue to the divergence theorem and Stokes’ theorem. There are two main forms.

One is the flux version, the analogue to the divergence theorem. For $R\in {\mathbb{R}}^2$ and a bounding curve $S$, then:

$${\int }_S\vec{F}\cdot \hat{n}ds={\iint }_R\nabla \cdot \vec{F}dA$$

The other is the circulation form, which is essentially Stokes’ theorem without modifications.