   Chapter 16.9, Problem 24E

Chapter
Section
Textbook Problem

Use the Divergence Theorem to evaluate ∬ S ( 2 x   +   2 y   +   z 2 )   d S where S is the sphere x2 + y2 + z2 = 1.

To determine

To evaluate: The integral S(2x+2y+z2)dS .

Explanation

Given Data:

Write the given integral as follows.

S(2x+2y+z2)dS (1)

Here,

S is the sphere.

Write the given equation of the sphere as follows.

x2+y2+z2=1

Formula used:

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

SFdS=SFndS (2)

Write the expression to find SFndS .

SFndS=BdivFdV (3)

Here,

B is the boundary of the unit ball.

Write the expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk .

divF=xP+yQ+zR (4)

Write the expression to find volume of the sphere with the radius r .

V=43πr3 (5)

From equation (1) and (2), it is clear that the expression (Fn) is equals to the expression (2x+2y+z2) .

Fn=2x+2y+z2 (6)

For the surface S , the normal n is considered as follows.

n=xi+yj+zkx2+y2+z2

As the surface is a sphere with the equation x2+y2+z2=1 , substitute 1 for x2+y2+z2 .

n=xi+yj+zk1=xi+yj+zk

If the normal n is xi+yj+zk , the field vector F must be 2i+2j+zk in order to satisfy the equation (6)

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