[ill. This exercise is used in Section 13.6. Let E c R 3 . Recall that the gradient of a C1 function f : E --+ R is defined by grad f := \7 f := (fx, fy, fz).

(a) Prove that if f is C2 at Xo, then curlgradf(xo) = O.

(b) If F: E --+ R3 is C1 on E and C2 at Xo E E, prove that divcurlF(xo) = O.

(c) Suppose that E satisfies the hypotheses of Gauss's Theorem and i : E --+ R is a C2 function that is harmonic on E (see Exercise 10d). If F = grad f on E, prove that flaE iF ·ndO' = ff LIIFI12 dV.