10. Let E be a set in Rm. For each u : E --+ R that has second-order partial derivatives on E, Laplace's equation is defined by m {)2U

~u:= L () 2· x·

j=l J

(a) Show that if u is C2 on E, then ~u = 'V. ('Vu) on E.

(b) [GREEN'S FIRST IDENTITY]. Show that if E c R3 satisfies the hypotheses of Gauss's Theorem, then f f Ie (u~v + 'Vu· 'Vv) dV = f hE u'Vv ·nda for all C2 functions u, v : E --+ R.

(c) [GREEN'S SECOND IDENTITY]. Show that if E c R3 satisfies the hypotheses of Gauss's Theorem, then f fie (u~v - v~u) dV = f hE (u'Vv - v'Vu) ·nda 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 ~u(x) = 0 for all x E E. Suppose that E is a nonempty open region in R3 that satisfies the hypotheses of Gauss's Theorem. If u is harmonic on E, u is continuous on E, and u = 0 on ()E, prove that u = 0 on E.

(e) Suppose that V is open and nonempty in R 2 , U is C2 on V, and u is continuous on V. Prove that u is harmonic on V if and only if

{ (ux dy - uy dx) = 0 iaE for all two-dimensional regions E c V that satisfy the hypotheses of Green's Theorem.