Let E be a set in Rm. For each u: E R which has second-order partial derivatives
Question:
a) Show that if u is C2 on E, then Îu = (u) on E.
b) Show that if E R3 satisfies the hypotheses of the Divergence Theorem, then
for all C2 functions u, v : E R.
c) Show that if E 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 Îu(x) = 0 for all x 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 ÏE, 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 V which satisfy the hypotheses of Green's Theorem.
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