Green's Theorem

Green's theorem gives the relationship between a line integral around a simple closed curveand a double integral over the plane regionbounded byIt is the two-dimensional special case of the more general Stokes' theorem, and is named after British mathematician George Green.

Letbe a positively oriented (counterclockwise), piecewise smooth, simple closed curve in the planeand let be the region bounded byIfandare functions ofdefined on an open region containingand have continuous partial derivatives there, then

For example letbe the triangular region illustrated below and let

We need to find the limits. If we integrate with respect tofirst then we must findas a function ofon the line BC:Our integral becomes

Expanding the integrand and simplifying gives