   Chapter 16, Problem 29RE

Chapter
Section
Textbook Problem

Evaluate the surface integral.29. ∬S F · dS, where F(x, y, z) = xz i − 2y j + 3x k and S is the sphere x2 + y2 + z2 = 4 with outward orientation

To determine

To Evaluate: The surface integral of SFdS , where F(x,y,z)=xzi2yj+3xk

Explanation

Given data:

F(x,y,z)=xzi2yj+3xk , x2+y2+z2=4 .

Formula used:

Write the expression for Divergence Theorem.

SFdS=EdivFdV (1)

Write the expression for volume of sphere.

V=43πr3 (2)

Write the expression for divergence.

divF=Px+Qy+Rz (3)

Consider the expression as follows.

x2+y2+z2=4 (4)

Write the expression for sphere.

x2+y2+z2=r2 (5)

Compare equations (4) and (5).

r2=4r=2

Consider the expression for F(x,y,z) as follows.

F(x,y,z)=xzi2yj+3xk (6)

Write the expression for F(x,y,z) .

F(x,y,z)=Pi+Qj+Rk (7)

Compare equations (6) and (7)

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