Explanation Given: Density is 1025 k g m 3 , v = y i + x j m s and hemisphere with x 2 + y 2 + z 2 = 9 , z ≥ 0 . Formula used: ∬ S ρ v ⋅ n d S = ∬ S ρ v ⋅ d S (1) r ϕ = ∂ x ∂ ϕ i + ∂ y ∂ ϕ j + ∂ z ∂ ϕ k (2) r θ = ∂ x ∂ θ i + ∂ y ∂ θ j + ∂ z ∂ θ k (3) By using the spherical coordinates to parameterize the hemisphere, consider r ( ϕ , θ ) = 3 sin ϕ cos θ i + 3 sin ϕ sin θ j + 3 cos ϕ k with limits are 0 ≤ θ ≤ 2 π and 0 ≤ ϕ ≤ π 2 . Find r ϕ . Substitute 3 sin ϕ cos θ for x , 3 sin ϕ sin θ for y and 3 cos ϕ for z in equation (2), r ϕ = ∂ ∂ ϕ ( 3 sin ϕ cos θ ) i + ∂ ∂ ϕ ( 3 sin ϕ sin θ ) j + ∂ ∂ ϕ ( 3 cos ϕ ) k = ( 3 cos θ ) ∂ ∂ ϕ ( sin ϕ ) i + ( 3 sin θ ) ∂ ∂ ϕ ( sin ϕ ) j + ( 3 ) ∂ ∂ ϕ ( cos ϕ ) k = 3 cos θ cos ϕ i + 3 sin θ cos ϕ j − 3 sin ϕ k Find r θ . Substitute 3 sin ϕ cos θ for x , 3 sin ϕ sin θ for y and 3 cos ϕ for z in equation (3), r θ = ∂ ∂ θ ( 3 sin ϕ cos θ ) i + ∂ ∂ θ ( 3 sin ϕ sin θ ) j + ∂ ∂ θ ( 3 cos ϕ ) k = ( 3 sin ϕ ) ∂ ∂ θ ( cos θ ) i + ( 3 sin ϕ ) ∂ ∂ θ ( sin θ ) j + ( 3 cos ϕ ) ∂ ∂ θ ( 1 ) k = ( 3 sin ϕ ) ( − sin θ ) i + ( 3 sin ϕ ) ( cos θ ) j + ( 3 cos ϕ ) ( 0 ) k = − 3 sin ϕ sin θ i + 3 sin ϕ cos θ j Find r ϕ × r θ . r ϕ × r θ = ( 3 cos θ cos ϕ i + 3 sin θ cos ϕ j − 3 sin ϕ k ) × ( − 3 sin ϕ sin θ i + 3 sin ϕ cos θ j ) = | i j k 3 cos θ cos ϕ 3 sin θ cos ϕ − 3 sin ϕ − 3 sin ϕ sin θ 3 sin ϕ cos θ 0 | = { ( 0 + 9 sin 2 ϕ cos θ ) i + ( 9 sin 2 ϕ sin θ + 0 ) j + ( 9 cos 2 θ cos ϕ sin ϕ + 9 sin 2 θ cos ϕ sin ϕ ) k } = 9 sin 2 ϕ cos θ i + 9 sin 2 ϕ sin θ j + 9 cos ϕ sin ϕ ( cos 2 θ + sin 2 θ ) k r ϕ × r θ = 9 sin 2 ϕ cos θ i + 9 sin 2 ϕ sin θ j + 9 cos ϕ sin ϕ k { ∵ cos 2 θ + sin 2 θ = 1 } Find ∬ S ρ v ⋅ d S

9th Edition

ISBN: 9780357771105

Author: Stewart

Publisher: CENGAGE L

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Question

Chapter 16.7, Problem 44E

To determine

To find: The rate of flow outward through the hemisphere x2+y2+z2=9 , z≥0 .

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Chapter 16.7, Problem 43E

Chapter 16.7, Problem 45E

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Chapter 16 Solutions

CALCULUS,EARLY TRANS LCPO

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The rate of flow outward through the hemisphere x 2 + y 2 + z 2 = 9 , z ≥ 0 .

Solution Summary: The author explains that the rate of flow outward through the hemisphere is underset_0kgm

To find: The rate of flow outward through the hemisphere x2+y2+z2=9 , z≥0 .

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Chapter 16 Solutions