FM5002-HW4-2.15.12

# Let d 0 4 x 0 x 0 3 2 define d 3 by r

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Unformatted text preview: ) x (0, Π/ 3). 2 Define Ψ : D 3 by Ψ r , Θ, Φ r Cos Φ Cos Θ , r Cos Φ Sin Θ , 2 r Sin Φ . 2Π 6s s t b. Observe that E is just an ellipse cen tered at 0. The picture below is for r = 1. 2 r st 2 0 0 3 r2 t 6 Π r 2. 2 FM5002−HW4−2.15.12.nb Let E := Ψ (D). Then Ψ : D E is bijective an d smooth an d has con tin uous extension to the closure of D. Compute the volume of E by computing 1 x y z via the change of variables formula. E We compute that J r , Θ, Φ Cos Θ Cos Φ det Cos Φ Sin Θ 2 Sin Φ r Cos Φ Sin Θ r Cos Θ Cos Φ 0 r Cos Θ Sin Φ r Sin Θ Sin Φ 2 r Cos Φ 2 r 2 Cos Θ 2 Cos Φ 3 2 r 2 Cos Φ 3 Sin Θ 2 2 r 2 Cos Θ 2 Cos Φ Sin Φ 2 2 r 2 Cos Φ Sin Θ 2 Sin Φ 2 r 2 Cos Φ . Therefore, by the change of variables formula, we have that 2 1xyz E 4 4 Π3 Π J r , Θ, Φ 0 0 0 0 2 r 2 Cos Φ 0 4 Π3 Π ΦΘr 0 0 4 Π 3 r2 Θ r ΦΘr 0 0 3 Π r2 r 64 Π 3 ....
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