Solve the problem.Assuming all the necessary derivatives exist, show that if = 0 for all closed curves C to which Green's Theorem applies, then f satisfies the Laplace equation + = 0 for all regions bounded by closed curves C to which Green's Theorem applies.

Question

What will be an ideal response?

Answer 1

Let C be a closed curve for which Green's Theorem applies. Then If the Laplace equation does not hold for f on R, then there is a simple closed curve C' in the interior of R which bounds a simply connected region R' on which  +   ? 0 for all (x, y) in R'. Moreover, without loss of generality, we can assume that  +  > 0 . Hence  As Green's Theorem applies to C', we also have that
0 ==  which is a contradiction.