A2 b 2 42 31 10 42 32 15 15 3 5 25 21

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Unformatted text preview: 2 θdρ dφ dθ = 0 0 0 192 π 5 26. Let G1 be the region u2 + v 2 + w2 ≤ 1, let x = au, y = bv, z = cw, (y 2 + z 2 )dx dy dz = Ix = G (b2 v 2 + c2 w2 )du dv dw G1 2π π 1 abc(b2 sin2 φ sin2 θ + c2 cos2 φ)ρ4 sin φ dρ dφ dθ = 0 0 2π = 0 0 4 abc 2 2 (4b sin θ + 2c2 )dθ = πabc(b2 + c2 ) 15 15 27. Let u = y − 4x, v = y + 4x, then x = 1 u dAuv = v 8 1 8 S 5 2 2 0 1 2 uv dAuv = − 2 1 2 1 ∂ (x, y ) 1 1 (v − u), y = (v + u) so =− ; 8 2 ∂ (u, v ) 8 15 u du dv = ln v 42 28. Let u = y + x, v = y − x, then x = − 1 ∂ (x, y ) 1 1 (u − v ), y = (u + v ) so =; 2 2 ∂ (u, v ) 2 1 uv du dv = − 0 S 0 29. Let u = x − y, v = x + y , then x = 1 2 1 ∂ (x, y ) 1 1 (v + u), y = (v − u) so = ; the boundary curves of 2 2 ∂ (u, v ) 2 the region S in the uv -plane are u = 0, v = u, and v = π/4; thus 1 sin u dAuv = cos v 2 1 2 S ∂ (x, y, z ) = abc; ∂ (u, v, w) π /4 0 v 0 √ 1 sin u du dv = [ln( 2 + 1) − π/4] cos v 2 Exercise Set 16.8 602 1 ∂ (x, y ) 1 1 (v − u), y = (u + v ) so = − ; the boundary 2 2 ∂ (u, v ) 2 curves of the region S in the uv -plane are v = −u, v = u, v = 1, and v = 4; thus 30. Let u = y − x, v = y + x, then x = 1 2 v 4 1 2 eu/v dAuv = eu/v du dv = −v 1 S 15 (e − e−1 ) 4 31. Let u = y/x, v = x/y 2 , then x = 1/(u2 v ), y = 1/(uv ) so 4 1 dAuv = u4 v 3 2 1 S 1 1 ∂ (x, y ) = 4 3; ∂ (u, v ) uv 1 du dv = 35/256 u4 v 3 ∂ (x, y ) = 6; S is the region in the uv -plane enclosed by the circle u2 + v 2 = 1 ∂ (u, v ) 32. Let x = 3u, y = 2v , 2π (9 − x − y )dA = so R S 2 1 3 0 0 1 ∂ (x, y, z ) =− ; ∂ (u, v, w) u 33. x = u, y = w/u, z = v + w/u, 4 (9 − 3r cos θ − 2r sin θ)r dr dθ = 54π 0 S v2 w dVuvw = u 1 6(9 − 3u − 2v )dAuv = 6 1 v2 w du dv dw = 2 ln 3 u 34. u = xy, v = yz, w = xz, 1 ≤ u ≤ 2, 1 ≤ v ≤ 3, 1 ≤ w ≤ 4, x= uw/v, y = V= uv/w, z = dV = 2 G 1 3 1 4 1 1 ∂ (x, y, z ) =√ ∂ (u, v, w) 2 uvw √ √ dw dv du = 4( 2 − 1)( 3 − 1) v w/u, 1 2 uvw √ 35. (b) If x = x(u, v ), y = y (u, v ) where u = u(x, y ), v = v (x, y ), then by the chain rule ∂x ∂x ∂u ∂x ∂v ∂x ∂x ∂u ∂x ∂v + = = 1, + = =0 ∂u ∂x ∂v ∂x ∂x ∂u ∂y ∂v ∂y ∂y ∂y ∂y ∂u ∂y ∂v ∂y ∂y ∂u ∂y ∂v + = = 0, + = =1 ∂u ∂x ∂v ∂x ∂x ∂u ∂y ∂v ∂y ∂y ∂ (x, y ) = ∂ (u, v ) 1−v v ∂ (u, v ) = ∂ (x, y ) 36. (a) −u u 1 −y/(x + y )2 = u; u = x + y, v = 1 x/(x + y )2 = y , x+y 1 y 1 x =; + = (x + y )2 (x + y )2 x+y u ∂ (u, v ) ∂ (x, y ) =1 ∂ (x, y ) ∂ (u, v ) (b) ∂ (x, y ) = ∂ (u, v ) ∂ (u, v ) = ∂ (x, y ) √ √ u = 2v 2 ; u = x/ y, v = y, 2v √ 1 ∂ (u, v ) ∂ (x, y ) 1 1/ y −x/(2y −3/2 ) √ = 2; =1 = 0 1/(2 y ) 2y 2v ∂ (x, y ) ∂ (u, v ) v 0 603 Chapter 16 (c) ∂ (x, y ) = ∂ (u, v ) ∂ (u, v ) = ∂ (x, y ) 37. √ √ v = −2uv ; u = x + y, v = x − y, −v √ √ 1/(2 x + y ) 1/(2 x + y ) 1 ∂ (u, v ) ∂ (x, y ) 1 ; =1 =− =− √ √ 2 − y2 2uv ∂ (x, y ) ∂ (u, v ) 1/(2 x − y ) −1/(2 x − y ) 2x u u 11 ∂ (x, y ) ∂ (u, v ) = 3xy 4 = 3v so = ; ∂ (x, y ) ∂ (u, v ) 3v 3 1 S 38. ∂ (x, y ) 1 ∂ (x, y ) ∂ (u, v ) = 8xy so = ; xy = xy ∂ (x, y ) ∂ (u, v ) 8xy ∂ (u, v ) 1 8 dAuv = 1 8 S 39. 16 1 8xy 2π 2 1 sin u dAuv = v 3 = π 2 sin u du dv = − ln 2 v 3 1 so 8 4 du dv = 21/8 9 1 1 ∂ (x, y ) ∂ (u, v ) = −2(x2 + y 2 ) so =− ; ∂ (x, y ) ∂ (u, v ) 2(x2 + y 2 ) (x4 − y 4 )exy 1 2 x4 − y 4 xy 1 1 ∂ (x, y ) = e = (x2 − y 2 )exy = veu so ∂ (u, v ) 2(x2 + y 2 ) 2 2 veu dAuv = 4 1 2 3 veu du dv = 3 S 1 73 (e − e) 4 40. Set u = x + y + 2z, v = x − 2y + z, w = 4x + y + z , then 6 2 3 −6 −2 −3 dx dy dz = V= R ∂ (u, v, w) = ∂ (x, y, z ) 1 12 1 −2 1 4 11 = 18, and 1 ∂ (x, y, z ) du dv dw = 6(4)(12) = 16 ∂ (u, v, w) 18 41. (a) Let u = x + y, v = y , then the triangle R with vertices (0, 0), (1, 0) and (0, 1) becomes the triangle in the uv -plane with vertices (0, 0), (1, 0), (1, 1), and u 1 f (x + y )dA = f (u) 0 R 1 ueu du = (u − 1)eu (b) 0 42. (a) (b) ∂ (x, y ) = ∂ (r, θ) ∂ (x, y, z ) = ∂ (ρ, θ, φ) cos θ sin θ 0 ∂ (x, y ) dv du = ∂ (u, v ) 1 uf (u) du 0 1 =1 0 −r sin θ r cos θ sin φ cos θ sin φ sin θ cos φ = r, ∂ (x, y ) =r ∂ (r, θ) −ρ sin φ sin θ ρ sin φ cos θ 0 ρ cos φ cos θ ρ cos φ sin θ −ρ sin φ = −ρ2 sin φ; ∂ (x, y, z ) = ρ2 sin φ ∂ (ρ, θ, φ) Chapter 16 Supplementary Exercises 604 CHAPTER 16 SUPPLEMENTARY EXERCISES 3. (a) dA (b) dV R G 2 ∂z ∂x (c) + ∂z ∂y 2 dA R 4. (a) x = a sin φ cos θ, y = a sin φ sin θ, z = ρ cos φ, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π (b) x = a cos θ, y = a sin θ, z = z, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ h 1 1+ 7. 1− 0 √ √ 1−y 2 2x 2 f (x, y ) dx dy 8. 3 6−x f (x, y ) dy dx + 1−y 2 x 0 f (x, y ) dy dx 2 x 9. (a) (1, 2) = (b, d), (2, 1) = (a, c), so a = 2, b = 1, c = 1, d = 2 1 1 dA = (b) 0 R 10. 0 < sin √ 0 π 0= π 0 0 1 2x cos(πx2 ) dx = 11. 1/2 2 0 π 0 dy dx < 0 x2 y3 e 2 1 1 1 3du dv = 3 0 0 xy < 1 for 0 < x,y &...
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