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# Which of the following is equal to the surface

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Unformatted text preview: urface integral D √ D f (Φ (u, v )) · Tu × Tv du dv f (Φ (u, v )) ||Tu × Tv ||du dv − D f (Φ (u, v )) ||Tu × Tv ||du dv 8. Which of the following is equal to the surface integral D √ S F · dS ? F (Φ (u, v )) · Tu × Tv du dv − D D f dS ? f (Φ (u, v )) · Tu × Tv du dv − D S F (Φ (u, v )) · Tu × Tv du dv F (Φ (u, v )) ||Tu × Tv ||du dv − D F (Φ (u, v )) ||Tu × Tv ||du dv For questions 9-10, let S be the elliptical cylinder with outward pointing normal that satisﬁes 2 x2 + y4 = 1 and 1 z 2. 9. Which of the following is a parametrization of S ? Φ : [0, 2π ] × [1, 2] → R3 , Φ (u, v ) = cos u, 1 sin u, v 2 Φ : [0, 2π ] × [1, 2] → R3 , √ Φ (u, v ) = (cosh u, 2 sinh u, v ) Φ : [0, 2π ] × [1, 2] → R3 , Φ (u, v ) = (cos u, 2 sin u, v ) Φ : [0, 2π ] × [1, 2] → R3 , 10. What is the value of √ S 1 Φ (u, v ) = cosh u, 2 sinh u, v F · dS ? where F (x, y, z ) = (xz, yz, x2 − 24yz )? −6π 6π 2π −2π For questions 11-12, let C be the oriented tilted ellipse positively parametrized by c (θ) = y (3 cos θ + sin θ , cos θ + 7 sin θ), θ ∈ [0, 2π ]. Let F (x, y ) = (P, Q) where P = − x2 +y2 and x Q = x 2 +y 2 . 11. Which of the following is equal to x +y x 2 +y 2 √ ∂Q ∂x − ∂P ∂y on R2 \ {0}? −2x2 +2y 2 (x2 +y 2 )2 x −y (x2 +y 2 )2 0 12. What is the value of C F · ds? 0 π √ −2π 2π Hint: Introduce a closed curve inside the ellipse and use the result in question 11. Be careful with how you orient the curves. Page 2...
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