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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, where F(x, y, z) = x3 i + y3 j + z3 k and S is the surface of the solid bounded by the cylinder x2 + y2 = 1 and the planes z = 0 and z = 2.

To determine

To calculate: The surface integral SFdS for F(x,y,z)=x3i+y3j+z3k across the surface of the solid S, which is bounded by the cylinder x2+y2=1 , and the planes z=0 and z=2 .

Explanation

Given data:

The vector field is F(x,y,z)=x3i+y3j+z3k .

The surface of solid S is bounded by the cylinder x2+y2=1 , and the planes z=0 and z=2 .

Formula used:

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

SFdS=EdivFdV (1)

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 (2)

Write the equation of cylinder as follows.

x2+y2=1

From the equation of cylinder, the parameters are considered as follows.

x=rcosθ,y=rsinθ,0r1,0θ2π

As the surface S is bounded by the cylinder x2+y2=1 , and the planes z=0 and z=2 , the limits of x, r, and θ are written as follows.

0z20r10θ2π

Find the value of divF .

Substitute x3 for P , y3 for Q , and z3 for R in equation (2),

divF=x(x3)+y(y3)+z(z3)=3x2+3y2+3z2=3(x2+y2+z2)

Substitute rcosθ for x and rsinθ for y ,

divF=3((rcosθ)2+(rsinθ)2+z2)=3(r2cos2θ+r2sin2θ+z2)=3(r2+z2)

Substitute 3(r2+z2) for divF in equation (1),

SFdS=E3(r2+z2)dV

Apply the limits of bounded region of surface and rewrite the expression 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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