proof of Green’s theorem

Consider the region R boundedPlanetmathPlanetmathPlanetmath by the closed curve P in a simply connected space. P can be given by a vector valued function F(x,y)=(f(x,y),g(x,y)). The region R can then be described by


The double integrals above can be evaluated separately. Let’s look at


Evaluating the above double integral, we get


According to the fundamental theorem of line integrals, the above equation is actually equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the evaluation of the line integral of the function F1(x,y)=(0,g(x,y)) over a path P=P1+P2, where P1=(A(y),y) and P2=(B(y),y).


Thus we have


By a similar argumentMathworldPlanetmathPlanetmath, we can show that


where F2=(f(x,y),0). Putting all of the above together, we can see that


which is Green’s theorem.

Title proof of Green’s theorem
Canonical name ProofOfGreensTheorem
Date of creation 2013-03-22 12:28:47
Last modified on 2013-03-22 12:28:47
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Proof
Classification msc 26B12