quiz2sol - MATH 203 FALL 2009 TAKE HOME QUIZ 2 SOLUTION...

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Unformatted text preview: MATH 203, FALL 2009, TAKE HOME QUIZ # 2: SOLUTION This is a closed book quiz, take 20 minutes. Due Monday 10/19/09. Question 1: Let f ( x, y ) be a map from R 2 → R , and x = g ( s, t ), y = h ( s, t ) for some functions g, h : R 2 → R . Assume f, g, h have derivatives of all orders. Let k ( s, t ) = f ( g ( s, t ) , h ( s, t )). Using the chain rule, calculate ∂ 2 k ∂s∂t . Solution: ∂ ∂s∂t k ( s, t ) = ∂ ∂s∂t f ( g ( s, t ) , h ( s, t )) = ∂ ∂s ( ∂f ∂x ∂g ∂t + ∂f ∂y ∂h ∂t ) (by the chain rule) = ∂ ∂s ( ∂f ∂x ) ∂g ∂t + ∂f ∂x ∂ ∂s ( ∂g ∂t ) + ∂ ∂s ( ∂f ∂y ) ∂h ∂t + ∂f ∂y ∂ ∂s ( ∂h ∂t ) (product rule) =( ∂ 2 f ∂x 2 ∂g ∂s + ∂ 2 f ∂y∂x ∂h ∂s ) ∂g ∂t + ∂f ∂x ∂ 2 g ∂s∂t + ( ∂ 2 f ∂x∂y ∂g ∂s + ∂ 2 f ∂y 2 ∂h ∂s ) ∂h ∂t + ∂f ∂y ∂ 2 h ∂s∂t (chain rule) = f xx g s g t + f xy h s g t + f x g ts + f yx g s h t + f yy h s h t + f y h ts (different notation)...
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This note was uploaded on 11/05/2011 for the course MAT 203 at Princeton.

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quiz2sol - MATH 203 FALL 2009 TAKE HOME QUIZ 2 SOLUTION...

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