W5V2 - Green\'s Theorem PDF

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Green’s Theorem We learned the FTC for line integrals:

Z

∇f · d~r = f (~r (b)) − f (~r (a)) C

This means the integral of a derivative of f over a curve is determined by the values of f at the endpoints (boundary points) of the curve.

The equivalent statement about double integrals is called Green’s Theorem: If D a simple region in R2 , and C = ∂D is the boundary of D (oriented counter-clockwise) and P (x, y) andZQ(x, y) have continuous on D, then Z Z  partial derivatives  ∂Q ∂P P dx + Qdy = dA − ∂y ∂x D C

Assume that D is both type I and type II in R2 . Here is the proof of Green’s Theorem:...