a6 solu - MATH 251-3 Fall 2007 Simon Fraser University...

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Unformatted text preview: MATH 251-3, Fall 2007 Simon Fraser University Assignment 6: Solutions Additional Question Due: 4:30pm, Monday 29 October 2007 1. Define the function of two variables u ( x, y ) by u = ln p x 2 + y 2- 4 e- πx cos πy + x 3- 3 xy 2 + 2 x for ( x, y ) 6 = (0 , 0). (a) Compute the first partial derivatives u x = ∂u/∂x and u y = ∂u/∂y . Solution: We compute u x by differentiating with respect to x , keeping y constant. Using ∂ ∂x ln p x 2 + y 2 = 1 p x 2 + y 2 ∂ ∂x ( x 2 + y 2 ) 1 / 2 = 1 p x 2 + y 2 1 2 ( x 2 + y 2 )- 1 / 2 2 x = x x 2 + y 2 , we find ∂u ∂x = x x 2 + y 2 + 4 πe- πx cos πy + 3 x 2- 3 y 2 + 2 . (1) Similarly, ∂u ∂y = y x 2 + y 2 + 4 πe- πx sin πy- 6 xy . (2) (b) Compute u xy = ∂ 2 u ∂y∂x (differentiate first with respect to x , then with respect to y ) and u yx , and show that they are equal; that is, verify Clairaut’s theorem for this function....
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a6 solu - MATH 251-3 Fall 2007 Simon Fraser University...

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