Green’s theorem

Green’s theorem provides a connection between path integrals over a well-connected region in the plane and the area of the region bounded in the plane. Given a closed path P bounding a region R with area A, and a vector-valued functionPlanetmathPlanetmath F=(f(x,y),g(x,y)) over the plane,


where an is the derivative of a with respect to the nth variable.


The closed path integral over a gradient of a function with continuousMathworldPlanetmath partial derivativesMathworldPlanetmath is always zero. Thus, gradients are conservative vector fields. The smooth functionMathworldPlanetmath is called the potential of the vector field.


The corollary states that


We can easily prove this using Green’s theorem.


But since this is a gradient…


Since h12=h21 for any function with continuous partials, the corollary is proven.

Title Green’s theorem
Canonical name GreensTheorem
Date of creation 2013-03-22 12:15:55
Last modified on 2013-03-22 12:15:55
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 10
Author mathcam (2727)
Entry type Theorem
Classification msc 26B20
Related topic GaussGreenTheorem
Related topic ClassicalStokesTheorem