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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


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

© 1996-9 Eric W. Weisstein