1208
CHAPTER 14
.
.
Vector Calculus
14-94
25.
Show that
±
C
(
f
∇
f
)
·
d
r
=
0 for any simple closed curve
C
and differentiable function
f
.
26.
Show that
±
C
(
f
∇
g
+
g
∇
f
)
·
d
r
=
0 for any simple closed
curve
C
and differentiable functions
f
and
g
.
27.
Let
F
(
x
,
y
)
=±
M
(
x
,
y
)
,
N
(
x
,
y
)
²
be a vector ﬁeld whose com-
ponents
M
and
N
have continuous ﬁrst partial derivatives in
all of
R
2
. Show that
∇·
F
=
0i
f and only if
²
C
F
·
n
ds
=
0
for all simple closed curves
C
. (Hint: Use a vector form of
Green’s Theorem.)
28.
Under
the
assumptions
of
exercise
27,
show
that
²
C
F
·
n
=
0 for all simple closed curves
C
if and only
if
²
C
F
·
n
is path-independent.
29.
Under the assumptions of exercise 27, show that
F
=
0
if and only if
F
has a
stream function
g
(
x
,
y
) such that
M
(
x
,
y
)
=
g
y
(
x
,
y
) and
N
(
x
,
y
)
=−
g
x
(
x
,
y
).
30.
Combine the results of exercises 27–29 to state a two-variable
theorem analogous to Theorem 8.3.
31.
If
S
1
and
S
2
are surfaces that satisfy the hypotheses of Stokes’
Theorem and that share the same boundary curve, under what
circumstances can you conclude that
³³
S
1
(
∇×
F
)
·
n
dS
=
S
2
(
F
)
·
n
dS
?
32.
Give an example where the two surface integrals of exercise 31
are not equal.
33.
Use Stokes’ Theorem to verify that
´
C
(
f
∇
g
)
·
d
r
=
S
(
∇
f
×∇
g
)
·
n
dS
,
where
C
is the positively oriented boundary of the surface
S
.
34.
Use Stokes’ Theorem to verify that
´
C
(
f
∇
g
+
g
∇
f
)
·
d
r
=
0,
where
C
is the positively oriented boundary of some surface
S
.
EXPLORATORY EXERCISES
1.
The
circulation
of a vector ﬁeld
F
around the curve
C
is de-
ﬁned by
²
C
F
·
d
r
. Show that the curl
F
(0
,
0
,
0) is in the
same direction as the normal to the plane in which the circula-
tion per unit area around the origin is a maximum as the area
around the origin goes to 0. Relate this to the interpretation of
the curl given in section 14.5.
2.
The Fundamental Theorem of Calculus can be viewed as re-
lating the values of the function on the boundary of a region
(interval) to the sum of the derivative values of the function
within the region. Explain what this statement means and
then explain why the same statement can be applied to The-
orem 3.2, Green’s Theorem, the Divergence Theorem and
Stokes’ Theorem. In each case, carefully state what the “re-
gion” is, what its boundary is and what type derivative is
involved.
14.9
APPLICATIONS OF VECTOR CALCULUS
Through the past eight sections, we have developed a powerful set of tools for analyzing
vector quantities. You can now compute ﬂux integrals and line integrals for work and
circulation, and you have the Divergence Theorem and Stokes’ Theorem to relate these
quantities to one another. To this point in the text, we have emphasized the conceptual and
computational aspects of vector analysis. In this section, we present a small selection of
applications from ﬂuid mechanics and electricity and magnetism. As you work through the
examples in this section, notice that we are using vector calculus to derive general results
that can be applied to any speciﬁc vector ﬁeld that you may run across in an application.