James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

section 16.9 The Divergence Theorem 1143
Notice that the method of proof of the
Divergence Theorem is very similar to
that of Green’s Theorem.
Comparison with Equation 5 shows that
y
S
y R k n dS − y y
E
y −R
−z dV
Equations 2 and 3 are proved in a similar manner using the expressions for E as a
type 2 or type 3 region, respectively.
ExamplE 1 Find the flux of the vector field Fsx, y, zd − z i 1 y j 1 x k over the unit
sphere x 2 1 y 2 1 z 2 − 1.
SOLUtion First we compute the divergence of F:
div F − − −x szd 1 − −y syd 1 − −z sxd − 1
■
The solution in Example 1 should
be compared with the solution in
Example 16.7.4.
z
The unit sphere S is the boundary of the unit ball B given by x 2 1 y 2 1 z 2 < 1. Thus
the Divergence Theorem gives the flux as
y
S
y F dS − y yy div F dV − y y 1 dV − VsBd −
B
By 4 3 s1d3 − 4 3
ExamplE 2 Evaluate yy S
F dS, where
■
(0, 0, 1)
0
(1, 0, 0)
x
FIGURE 2
y=2-z
(0, 2, 0) y
z=1-≈
Fsx, y, zd − xy i 1 (y 2 1 e xz2 ) j 1 sinsxyd k
and S is the surface of the region E bounded by the parabolic cylinder z − 1 2 x 2 and
the planes z − 0, y − 0, and y 1 z − 2. (See Figure 2.)
SOLUtion It would be extremely difficult to evaluate the given surface integral
directly. (We would have to evaluate four surface integrals corresponding to the four
pieces of S.) Furthermore, the divergence of F is much less complicated than F itself:
div F − − −x sxyd 1 − −y (y2 1 e ) xz2 1 − ssin xyd − y 1 2y − 3y
−z
Therefore we use the Divergence Theorem to transform the given surface integral into a
triple integral. The easiest way to evaluate the triple integral is to express E as a type 3
region:
Then we have
y
S
E − hsx, y, zd | 21 < x < 1, 0 < z < 1 2 x 2 , 0 < y < 2 2 zj
y F dS − y yy div F dV − y y
E
E
y 3y dV
− 3 y 1 y 22z
y dy dz dx − 3 y 1 s2 2 zd 2
dz dx
21 y12x2 0 0 21 y12x2 0 2
− 3 2 y21F2 1 12x
s2 2 2 zd3
dx − 2 1 2 y 1 fsx
3 21
G0
2 1 1d 3 2 8g dx
− 2y 1
sx 6 1 3x 4 1 3x 2 2 7d dx − 184
0 35
■
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.