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,

PF𝑑x=R[g1(x,y)-f2(x,y)]𝑑A

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

Corollary:

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.

Proof:

The corollary states that

Ph𝑑x=0

We can easily prove this using Green’s theorem.

Ph𝑑x=R[g1(x,y)-f2(x,y)]𝑑A

But since this is a gradient…

R[g1(x,y)-f2(x,y)]𝑑A=R[h21(x,y)-h12(x,y)]𝑑A

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