   Chapter 16.9, Problem 6E

Chapter
Section
Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∫∫s F · dS; that is, calculate the flux of F across S.6. F(x, y, z) = x2yz i + xy2z j + xyz2 k, S is the surface of the box enclosed by the planes x = 0, x = a, y = 0, y = b, z = 0, and z = c, where a, b, and c are positive numbers

To determine

To calculate: The flux of vector field F(x,y,z)=x2yzi+xy2zj+xyz2k across the surface of the box S which is enclosed by the planes x=0,x=a,y=0,y=b,z=0 , and z=c .

Explanation

Given data:

The vector field is F(x,y,z)=x2yzi+xy2zj+xyz2k .

The surface of the box S is enclosed by the planes x=0,x=a,y=0,y=b,z=0 , and z=c .

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)

As the surface S is enclosed by the planes x=0,x=a,y=0,y=b,z=0 , and z=c , the limits of x, y, and z are written as follows.

0xa0yb0zc

Calculation of divF :

Substitute x2yz for P , xy2z for Q , and xyz2 for R in equation (2),

divF=x(x2yz)+y(xy2z)+z(xyz2)=yzx(x2)+xzy(y2)+xyz(z2)=yz(2x)+xz(2y)+xy(2z)=2xyz+2xyz+2xyz

divF=6xyz

Calculation of flux of vector field:

Substitute 6xyz for divF in equation (1),

SFdS=E(6xyz)dV

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

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