An extension of Green's theorem with application

Abstract

The main result of this thesis is a generalization of Green's Theorem. Green' s Theorem states: If Omega is an open subset of R[logical and]2 containing a compact subset K with smooth boundary. Let P and Q be two real valued functions on Omega which are differentiable with continuous partial derivatives. Then the integral over the boundary of K of Pdx [plus] Qdy is equal to the double integral over K of a package of partial derivatives (namely the partial derivative of Q with respect to x minus the partial derivative of P with respect to y). In this thesis we prove that the conditions on P and Q can be weakened. In fact, we prove that the conclusion of Green's Theorem holds if P and Q are only differentiable on a neighborhood of K and the package of partial derivatives is continuous on K. After proving the main result we can conclude two further results, a generalization of the Divergence Theorem in R[logical and]2 and a generalization of Cauchy's Integral Formula.