1211a3 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 First Semester: Assignment 3 Due date: 2010 October 15 (Friday), 5:00pm. 1. The partial differential equation 2 f ∂x 2 1 + 2 f ∂x 2 2 + ··· + 2 f ∂x 2 n = 0 , where f = f ( x 1 ,x 2 ,...,x n ) : X R n R , is known as the Laplace’s equation . Any function f of class C 2 that satisfies Laplace’s equation is called a harmonic function . (a) Show that f ( x,y,z ) = e 3 x +4 y sin(5 z ) is harmonic in all of R 3 . (b) Suppose f ( x,y ) is harmonic in the xy -plane. What condition should the constants a,b,c satisfy to ensure that the function g ( x,y,z ) = f ( ax + by,cz ) is harmonic in R 3 . 2. ( § 2.5 no.22) Let z = f ( x,y ), where f has continuous partial derivatives. If we make the standard polar/rectangular substitution x = r cos θ, y = r sin θ , show that ± ∂z ∂x ² 2 + ± ∂z ∂y ² 2 = ± ∂z ∂r ² 2 + 1 r 2 ± ∂z ∂θ ² 2 .
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This note was uploaded on 05/04/2011 for the course MATH 1211 taught by Professor Wang during the Spring '11 term at HKU.

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1211a3 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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