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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∫∫s F · dS; that is, calculate the flux of F across S.

14. F = |r|2r, where r = x i + y j+ z k, S is the sphere with radius R and center the origin

To determine

To calculate: The flux of vector field F=|r|2r across the surface S , where r=xi+yj+zk .

Explanation

Given data:

The vector field is given as follows.

F=|r|2r (1)

Here,

r=xi+yj+zk

The surface S is the sphere with radius R and centered the origin.

Formula used:

Write the expression to find flux of the vector field F(x,y,z) across the surface S .

SFdS=EdivFdV (2)

Here,

E is the solid region.

Write the expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk .

divF=xP+yQ+zR (3)

Write the expression for spherical coordinate system.

Ef(x,y,z)dV=ρ1ρ2ϕ1ϕ2θ1θ2ρ2sinϕf(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)dρdϕdθ (4)

Here,

ρ is the radius of the sphere.

Calculation of vector field F :

Substitute xi+yj+zk for r in equation (1),

F=|xi+yj+zk|2(xi+yj+zk)=(x2+y2+z2)2(xi+yj+zk)=(x2+y2+z2)(xi+yj+zk)=x(x2+y2+z2)i+y(x2+y2+z2)j+z(x2+y2+z2)k

F=(x3+xy2+xz2)i+(x2y+y3+z2y)j+(x2z+y2z+z3)k

Calculation of divF :

Substitute (x3+xy2+xz2) for P , (x2y+y3+z2y) for Q , and (x2z+y2z+z3) for R in equation (3),

divF=[x(x3+xy2+xz2)+y(x2y+y3+z2y)+z(x2z+y2z+z3)]=(3x2+y2+z2)+(x2+3y2+z2)+(x2+y2+3z2)=5(x2+y2+z2)

Calculation of flux of vector field:

Substitute 5(x2+y2+z2) for divF in equation (2),

SFdS=E5(x2+y2+z2)dV (5)

Parameterize the sphere with radius R as follows

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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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