20E - HW 7 Solutions

# 20E - HW 7 Solutions - Math 20E Homework#8 24 February 2005...

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Unformatted text preview: Math 20E Homework #8 24 February 2005 Section 7.5, Problem 6: Verify that in spherical coordinates, on a sphere of radius R , k T θ × T φ k d φ d θ = R 2 sin φ d φ d θ . Solution. The spherical coordinates for ρ = R are ( x , y , z ) = ( R cos θ sin φ , R sin θ sin φ , R cos φ ) , so we get T θ = (- R sin θ sin φ , R cos θ sin φ , 0 ) and T φ = ( R cos θ cos φ , R sin θ cos φ ,- R sin φ ) . Hence! We have T θ × T φ = (- R 2 cos θ sin 2 φ ,- R 2 sin θ sin 2 φ ,- R 2 sin φ cos φ ) , and so k T θ × T φ k d φ d θ = q R 4 cos 2 θ sin 4 φ + R 4 sin 2 θ sin 4 φ + R 4 sin 2 φ cos 2 φ d φ d θ = R 2 q ( cos 2 θ + sin 2 θ ) sin 4 φ + sin 2 φ cos 2 φ d φ d θ = R 2 q sin 2 φ ( sin 2 φ + cos 2 φ ) d φ d θ = R 2 sin φ d φ d θ . Section 7.5, Problem 15: Let Φ : D ⊂ R 2 → R 3 be a parameterization of a surface S described by x = x ( u , v ) , y = y ( u , v ) , z = z ( u , v ) . (a) Let ∂ Φ ∂ u = ∂ x ∂ u , ∂ y ∂ u , ∂ z ∂ u and ∂ Φ ∂ v = ∂ x ∂ v , ∂ y ∂ v , ∂ z ∂ v , and set E = ∂ Φ ∂ u 2 , F = ∂ Φ ∂ u · ∂ Φ ∂ v , G = ∂ Φ ∂ v 2 . Show that √ EG- F 2 = k T u × T v k , and that the surface area of S is A ( S ) = ZZ D p EG- F 2 du dv . In this notation, how can we express RR S f dS for a general function f ? (b) What does the formula for A ( S ) become if the vectors ∂ Φ / ∂ u and ∂ Φ / ∂ v are orthogonal? (c) Use parts (a) and (b) to compute the surface area of a sphere of radius a . Solution. (a) For the rest of this problem, we will use the shorthand notation x u = ∂ x / ∂ u , and so on. k T u × T v k 2 = ( y u z v- y v z u ) 2 + ( z u x v- z v x u ) 2 + ( x u y v- x v y u ) 2 = x 2 u ( y 2 v + z 2 v ) + y 2 u ( x 2 v + z 2 v ) + z 2 u ( x 2 v + y 2 v )- 2 ( x u x v y u y v + x u x v z u z v + y u y v z u z v ) = ( x 2 u + y 2 u + z 2 u )( x 2 v + y 2 v + z 2 v )- ( x 2 u x 2 v + y 2 u y 2 v + z 2 u z 2 v + 2 x u x v y u y v + 2 x u x v z u z v + 2 y u y v z u z v ) = EG- ( x u x v + y u y v + z u z v ) 2 = EG- F 2 , 1 Math 20E Homework #8 24 February 2005 as desired....
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20E - HW 7 Solutions - Math 20E Homework#8 24 February 2005...

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