Let E R3 Recall that the gradient of a C1 function f E R is defined by a) Prove that if f is C2 at x0, then curl grad f(x0) 0 b) If F E R3 is C1 on E and C2 at x0 E, prove that div curl F(x0) 0 c) Suppose that E satisfies the hypotheses of the Divergence Theorem and that f E R is a C2 function whic...

a By hypothesis F f x f y Thus by Greens Theorem and Theorem ...

Let E R3. Recall that the gradient of a C1 function f: E R is

Question:

Let E Š‚ R3. Recall that the gradient of a C1 function f: E †’ R is defined by

a) Prove that if f is C2 at x0, then curl grad f(x0) = 0.
b) If F: E R3 is C1 on E and C2 at x0 ˆˆ E, prove that div curl F(x0) = 0.
c) Suppose that E satisfies the hypotheses of the Divergence Theorem and that f: E †’ R is a C2 function which is harmonic on E (see Exercise 13.5.l0d). If F = grad f on E, prove that