Tutorial3_sol

Tutorial3_sol - MATH1211/10-11(1)/Tu3sol s Department of...

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MATH1211/10-11(1)/Tu3sol s Department of Mathematics, The University of Hong Kong MATH1211 Multivariable Calculus (2010–2011, First semester) Brief Solution to Tutorial 3 Note : 1. fg = e xy x sin2 y . D ( fg ) = £ (1 + xy ) e xy sin2 y xe xy ( x sin2 y + 2cos2 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 + 2cos2 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 , 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. 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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Tutorial3_sol - MATH1211/10-11(1)/Tu3sol s Department of...

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