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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find ∬S F · n dS, where F(x, y, z) = x i + y j + z k and S is the outwardly oriented surface shown in the figure (the boundary surface of a cube with a unit corner cube removed).

images

To determine

To find: The value of SFndS , where F(x,y,z)=xi+yj+zk and S is the outwardly oriented surface.

Explanation

Given data:

The vector field is F(x,y,z)=xi+yj+zk , where S is the outwardly oriented surface.

Formula used:

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

SFndS=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 expression for volume of cube.

V=a3 (3)

Here,

a is side of the cube.

Find the value of divF .

Substitute x for P , y for Q , and z for R in equation (2),

divF=x(x)+y(y)+z(z)=1+1+1=3

Substitute 3 for divF in equation (1),

SFdS=E3dV

SFdS=3V (4)

Consider the side of full cube is represented by afull .

afull=2

Consider the side of removed portion of the cube is represented by aremoved

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Chapter 16 Solutions

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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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