Green's Theorem

Theorem 1: Green's Theorem:   Let D be a simple closed region in two dimensions, and let its boundary be C. For a vector field F(x,y) = (P(x,y), Q(x,y)), the line integral of F on the positive orientation of C is equal to the surface integral over the region D of the partial derivative of Q with repsect to x minus the partial derivative of P with respect to y

Theorem 2: Area of a Region:   If C is a simple closed curve that bounds a region to which Theorem 1 applies, then the area of the region D bounded by C is one-half the line integral along C of F(x,y)=(-y,x)

Theorem 3: Vector Form of Green's Theorem:   Let D be a subset of R-2 be a region to which Green's theorem applies, let C be its boundary (oriented counter-clockwise), and let F=(P,Q) be continuously differentiable vector field on D. Then, the line integral along C is equal to the surface integral of <curl(F),k> over the region D.

Green's Theorem prov.

[TMRC] For any story, in any group of people there will be at least one person who has not heard the story. A refinement of the theorem states that there will be exactly one person (if there were more than one, it wouldn't be as bad to re-tell the story). [The name of this theorem is a play on a fundamental theorem in calculus. --ESR]

--The Jargon File version 4.3.1, ed. ESR, autonoded by rescdsk.