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9. (a) We call Laplacian the operator 22 + dy? 22 A Let u(x, y, z) be a twice differentiable function. Show that Au = div(Vu). (b) Let BC R³ be the closed unit ball a2 +y? +2² < 1 and S2 the unit sphere x²+ y? +z² = 1. Suppose that the function u(x, Y, 2) satisfies Au = 0 in B and Vu · n n S² . (1) on (i) Compute Vu ·n dS . S2

9. (a) We call Laplacian the operator 22 + dy? 22 A Let u(x, y, z) be a twice differentiable function. Show that Au = div(Vu). (b) Let BC R³ be the closed unit ball a2 +y? +2² < 1 and S2 the unit sphere x²+ y? +z² = 1. Suppose that the function u(x, Y, 2) satisfies Au = 0 in B and Vu · n n S² . (1) on (i) Compute Vu ·n dS . S2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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9. (a) We call Laplacian the operator A dx2 dy? Let u(x, y, z) be a twice differentiable function. Show that Δυ - div(Vu). (b) Let BCR³ be the closed unit ball x? +y? + 22 <1 and S² the unit sphere x? +y? + 22 = 1. Suppose that the function u(x, y, z) satisfies Δυ 0 in B and Vu · n 1 on S2. (1) (i) Compute Vu · n dS . S2 (ii) Can there exist a function u satisfying the problem (1)?
Transcribed Image Text:9. (a) We call Laplacian the operator A dx2 dy? Let u(x, y, z) be a twice differentiable function. Show that Δυ - div(Vu). (b) Let BCR³ be the closed unit ball x? +y? + 22 <1 and S² the unit sphere x? +y? + 22 = 1. Suppose that the function u(x, y, z) satisfies Δυ 0 in B and Vu · n 1 on S2. (1) (i) Compute Vu · n dS . S2 (ii) Can there exist a function u satisfying the problem (1)?
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