Let E be a set in Rm. For each u: E †’ R which has second-order partial derivatives on E, Laplace's equation is defined by

a) Show that if u is C2 on E, then Δu = ˆ™ (u) on E.
b) Show that if E Š‚ R3 satisfies the hypotheses of the Divergence Theorem, then

for all C2 functions u, v : E †’ R.
c) Show that if E Š‚ R3 satisfies the hypotheses of the Divergence Theorem, then

for all C2 functions u, v: E †’ R.
d) A function u: E †’ R is said to be harmonic on E if and only if u is C2 on E and Δu(x) = 0 for all x ˆˆ E. Suppose that E is a nonempty open region in R3 which satisfies the hypotheses of the Divergence Theorem. If u is harmonic on E, u is continuous on , and u = 0 on Ï‘E, prove that u = 0 on .
e) Suppose that V is open and nonempty in R2, that u is C2 on V, and that u is continuous on . Prove that u is harmonic on V if and only if

for all two-dimensional regions E Š‚ V which satisfy the hypotheses of Green's Theorem.

(Uz dy-uydx) = 0