Assuming that all the necessary derivatives exist and are continuous, show that if (x, y) satisfies the Laplace equation then for all closed curves C

Assuming that all the necessary derivatives exist and are continuous, show that if ƒ(x, y) satisfies the Laplace equation

then

for all closed curves C to which Green’s Theorem applies. (The converse is also true: If the line integral is always zero, then ƒ satisfies the Laplace equation.)

afaf ax + ay = 0,