Green's Theorem

Green's theorem is a vector identity which is equivalent to the Curl Theorem in the Plane. Over a region $D$ in the plane with boundary $\partial D$,

$$\int_{\partial D} f(x,y)\,dx+g(x,y)\,dy = \int\!\!\!\int _... ... g\over\partial x}-{\partial f\over\partial y}}\right)\,dx\,dy$$

$$\int_{\partial D} {\bf F}\cdot d{\bf s} = \int\!\!\!\int _D (\nabla \times{\bf F})\cdot{\bf k}\,dA.$$

If the region $D$ is on the left when traveling around $\partial D$, then Area of $D$ can be computed using

$$A = {\textstyle{1\over 2}}\int_{\partial D} x\,dy-y\,dx.$$

See also Curl Theorem, Divergence Theorem

References

Arfken, G. ``Gauss's Theorem.'' §1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985.