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(Solution) - Let E be a set in Rm For each u

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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

Let E be a set in Rm. For each u:
a) Show that if u is C2 on E, then ?u =  ? (u) on E.
b) Show that if E ? R3 satisfies the hypotheses of the Divergence Theorem, then
Let E be a set in Rm. For each u:
for all C2 functions u, v : E ? R.
c) Show that if E ? R3 satisfies the hypotheses of the Divergence Theorem, then
Let E be a set in Rm. For each u:
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. 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 ?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
Let E be a set in Rm. For each u:
for all two-dimensional regions E ? V which satisfy the hypotheses of Green's Theorem.

 







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This question was answered on: Jul 11, 2017

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