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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 sin 2 y . D ( fg ) = [ (1 + xy ) e xy sin 2 y xe xy ( x sin 2 y + 2 cos 2 y ) ] . On the other hand, g Df + f Dg = x sin 2 y [ ye xy xe xy ] + e xy [ sin 2 y 2 x cos 2 y ] = [ (1 + xy ) e xy sin 2 y xe xy ( x sin 2 y + 2 cos 2 y ) ] . Also, f ( x, y ) g ( x, y ) = e xy x sin 2 y , D ( f g ) = [ xye xy sin 2 y e xy sin 2 y x 2 sin 2 2 y x 2 e xy sin 2 y 2 xe xy cos 2 y x 2 sin 2 2 y ] , and g Df f Dg g 2 = x sin 2 y [ ye xy xe xy ] e xy [ sin 2 y 2 x cos 2 y ] x 2 sin 2 2 y = [ xye xy sin 2 y e xy sin 2 y x 2 sin 2 2 y x 2 e xy sin 2 y 2 xe xy cos 2 y x 2 sin 2 2 y ] . 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) F xy = ∂F x ∂y = 6 x 2 y 2 z 4 , F yx = ∂F y ∂x = 6 x 2 y 2 z 4 , F xz = ∂F x ∂z = 8 yz 3 + e xz + xze xz , F zx = ∂F z ∂x = 8 yz 3 + e xz + xz xz , F yz = ∂F y ∂z = 8 xz 3 , F zy = ∂F z
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