   Chapter 16.9, Problem 5E

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.5. F(x, y, z) = xye2 i + xy2z3 j - yez k, S is the surface of the box bounded by the coordinate planes and the planes x = 3, y = 2, and z = 1

To determine

To calculate: The flux of vector field F(x,y,z)=xyezi+xy2z3jyezk across the surface S which is bounded by the coordinate planes x=3,y=2 , and z=1 .

Explanation

Given data:

The vector field is F(x,y,z)=xyezi+xy2z3jyezk .

The surface S is bounded by the coordinate planes x=3,y=2 , and z=1 .

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 bounded by the coordinate planes x=3,y=2 , and z=1 , the limits of x, y, and z are written as follows.

0x30y20z1

Calculation of divF :

Substitute xyez for P , xy2z3 for Q , and (yez) for R in equation (2),

divF=x(xyez)+y(xy2z3)+z(yez)=yezx(x)+xz3y(y2)yz(ez)=yez(1)+xz3(2y)y(ez)=yez

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