   Chapter 16.7, Problem 44E

Chapter
Section
Textbook Problem

Seawater has density 1025 kg/m3 and flows in a velocity field v = y i + x j, where .x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the hemisphere x2 + y2 + z2 = 9, z ≥ 0.

To determine

To find: The rate of flow outward through the hemisphere x2+y2+z2=9 , z0 .

Explanation

Given:

Density is 1025kgm3 , v=yi+xjms and hemisphere with x2+y2+z2=9 , z0 .

Formula used:

SρvndS=SρvdS (1)

rϕ=xϕi+yϕj+zϕk (2)

rθ=xθi+yθj+zθk (3)

By using the spherical coordinates to parameterize the hemisphere, consider r(ϕ,θ)=3sinϕcosθi+3sinϕsinθj+3cosϕk with limits are 0θ2π and 0ϕπ2 .

Find rϕ .

Substitute 3sinϕcosθ for x , 3sinϕsinθ for y and 3cosϕ for z in equation (2),

rϕ=ϕ(3sinϕcosθ)i+ϕ(3sinϕsinθ)j+ϕ(3cosϕ)k=(3cosθ)ϕ(sinϕ)i+(3sinθ)ϕ(sinϕ)j+(3)ϕ(cosϕ)k=3cosθcosϕi+3sinθcosϕj3sinϕk

Find rθ .

Substitute 3sinϕcosθ for x , 3sinϕsinθ for y and 3cosϕ for z in equation (3),

rθ=θ(3sinϕcosθ)i+θ(3sinϕsinθ)j+θ(3cosϕ)k=(3sinϕ)θ(cosθ)i+(3sinϕ)θ(sinθ)j+(3cosϕ)θ(1)k=(3sinϕ)(sinθ)i+(3sinϕ)(cosθ)j+(3cosϕ)(0)k=3sinϕsinθi+3sinϕcosθj

Find rϕ×rθ .

rϕ×rθ=(3cosθcosϕi+3sinθcosϕj3sinϕk)×(3sinϕsinθi+3sinϕcosθj)=|ijk3cosθcosϕ3sinθcosϕ3sinϕ3sinϕsinθ3sinϕcosθ0|={(0+9sin2ϕcosθ)i+(9sin2ϕsinθ+0)j+(9cos2θcosϕsinϕ+9sin2θcosϕsinϕ)k}=9sin2ϕcosθi+9sin2ϕsinθj+9cosϕsinϕ(cos2θ+sin2θ)k

rϕ×rθ=9sin2ϕcosθi+9sin2ϕsinθj+9cosϕsinϕk {cos2θ+sin2θ=1}

Find SρvdS

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