nLab Green theorem

Green's Theorem (also the Green Theorem, but that is easily misunderstood in English) is the classical version of the Stokes Theorem for surfaces in the plane (which just amounts to regions in $\mathbb{R}^2$). It is in a way the most basic form of the Stokes Theorem beyond the Fundamental Theorem of Calculus, containing all of the analytic subtleties. The classical Kelvin–Stokes Theorem (for surfaces in $\mathbb{R}^3$) is a direct corollary (as indeed is the Stokes Theorem for surfaces in any ambient manifold); the other forms are proved in an analogous fashion. Other corollaries include the Cauchy integral theorem and the equality of mixed partial derivatives.