FM5002-HW4-2.15.12

FM5002-HW4-2.15.12 - Financial Mathem atics 5002 Hom ework...

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Financial Mathematics 5002 : Homework 4 (0041) Due on 15 Februrary 2012 Scot Adams Solutions 0041 M 1 . Fixr G 0. Let D E L 0, r R x L 0, 2 Π R and E E lL x, y R x 2 4 P y 2 9 L r 2 r \ 0, r R x ² 0 ³R . Define f : D R E by f (s, t) = (2s Cos[t], 3 s Sin[t]). Then f is a smooth bijection. f ’ L x, t R E 2 Cos ± t ´ M 2 s Sin ± t ´ 3 Sin ± t ´ 3 s Cos ± t ´ . Area L E R E I I D det f ’ L s, t R D s D t E I 0 2 Π I 0 r det f ’ L s, t R D s D t. a. Finish this computation. b. Graph E. a. I 0 2 Π I 0 r det f ’ L s, t R D s D t E I 0 2 Π I 0 r 6 s Cos ± t ´ 2 P 6 s Sin ± t ´ 2 D s D t E I 0 2 Π I 0 r 6 s D s D t E I 0 2 Π 3 r 2 D t E 6 Π r 2 . b. Observe that E is just an ellipse centered at 0. The picture below is for r = 1. M 2 M 1 1 2 M 3 M 2 M 1 1 2 3 0041 − 2 . Let D = (0, 4) x (0, Π ) x (0, Π /3). Define Ψ : D R ± 3 by Ψ L r, Θ , Φ R E L r Cos ± Φ ´ Cos ± Θ ´ , r Cos ± Φ ´ Sin ± Θ ´ , 2 r Sin ± Φ ´R .
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Let E := Ψ (D). Then Ψ : D R E is bijective and smooth and has continuous extension to the closure of D. Compute the volume of E by computing I E 1 D x D y D z via the change of variables formula.
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FM5002-HW4-2.15.12 - Financial Mathem atics 5002 Hom ework...

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