tutorial4_sol

tutorial4_sol - DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF...

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DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF HONG KONG MATH1211 Multivariable Calculus 2011-12 Second Semester: Solution to Tutorial 4 1. fg = e xy x sin2 y . D ( fg ) = [ (1 + xy ) e xy sin2 y xe xy ( x sin2 y + 2 cos 2 y ) ] . On the other hand, g Df + f Dg = x sin2 y [ ye xy xe xy ] + e xy [ sin2 y 2 x cos2 y ] = [ (1 + xy ) e xy sin2 y xe xy ( x sin2 y + 2cos 2 y ) ] . Also, f ( x, y ) g ( x, y ) = e xy x sin2 y , D ( f g ) = ± xye xy sin2 y e xy sin2 y x 2 sin 2 2 y x 2 e xy sin2 y 2 xe xy cos2 y x 2 sin 2 2 y ² , and g Df f Dg g 2 = x sin2 y [ ye xy xe xy ] e xy [ sin2 y 2 x cos2 y ] x 2 sin 2 2 y = ± xye xy sin2 y e xy sin2 y x 2 sin 2 2 y x 2 e xy sin2 y 2 xe xy cos2 y x 2 sin 2 2 y ² . 2. F x = 3 x 2 y 2 2 yz 4 + ze xz , F y = 2 x 3 y 2 xz 4 , F z = 8 xyz 3 + xe xz . (a) F xx = ∂F x ∂x = 6 xy 2 + z 2 e xz , F yy = ∂F y ∂y = 2 x 3 , F zz = ∂F z ∂z = 24 xyz 2 + x 2 e xz . (b)
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This note was uploaded on 03/13/2012 for the course MATH 1211 taught by Professor Wang during the Spring '11 term at HKU.

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tutorial4_sol - DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF...

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