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CHAPTER 15
Vector Fields
19.
Show that the gradient of a function expressed in terms of
polar coordinates in the plane is
∇
φ(
r
,θ)
=
∂φ
∂
r
ˆ
r
+
1
r
∂θ
ˆ
θ
.
(This is a repeat of Exercise 16 in Section 12.7.)
20.
Use the result of Exercise 19 to show that a necessary
condition for the vector ±eld
F
(
r
=
F
r
(
r
ˆ
r
+
F
θ
(
r
ˆ
θ
(expressed in terms of polar coordinates) to be conservative
is that
∂
F
r
−
r
∂
F
θ
∂
r
=
F
θ
.
21.
Show that
F
=
r
sin 2
θ
ˆ
r
+
r
cos 2
θ
ˆ
θ
is conservative, and ±nd
a potential for it.
22.
For what values of the constants
α
and
β
is the vector ±eld
F
=
r
2
cos
θ
ˆ
r
+
α
r
β
sin
θ
ˆ
θ
conservative? Find a potential for
F
if
α
and
β
have these
values.
15.3
Line Integrals
The de±nite integral,
R
b
a
f
(
x
)
dx
, represents the
total amount
of a quantity distributed
along the
x
axis between
a
and
b
in terms of the
line density
,
f
(
x
)
, of that quantity
at point
x
. The amount of the quantity in an
in±nitesimal
interval of length
at
x
is
f
(
x
)
, and the integral adds up these in±nitesimal contributions (or
elements
)
to give the total amount of the quantity. Similarly, the integrals
RR
D
f
(
x
,
y
)
dA
and
RRR
R
f
(
x
,
y
,
z
)
dV
represent the total amounts of quantities distributed over regions
D
in the plane and
R
in 3space in terms of the
areal
or
volume
densities of these
quantities.
It may happen that a quantity is distributed with speci±ed line density along a
curve
in the plane or in 3space, or with speci±ed areal density over a
surface
in 3space.
In such cases we require
line integrals
or
surface integrals
to add up the contributing
elements and calculate the total quantity. We examine line integrals in this section and
the next and surface integrals in Sections 15.5 and 15.6.
Let
C
be a bounded, continuousparametric curve in
3
. Recall (from Section 11.1)
that
C
is a
smooth curve
if it has a parametrization of the form
r
=
r
(
t
)
=
x
(
t
)
i
+
y
(
t
)
j
+
z
(
t
)
k
,
t
in interval
I
,
with “velocity” vector
v
=
d
r
/
dt
continuous and nonzero. We will call
C
a
smooth
arc
if it is a smooth curve with
±nite
parameter interval
I
=
[
a
,
b
].
In Section 11.3 we saw how to calculate the length of
C
by subdividing it into
short arcs using points corresponding to parameter values
a
=
t
0
<
t
1
<
t
2
<
···
<
t
n
−
1
<
t
n
=
b
,
adding up the lengths

1
r
i
=
r
i
−
r
i
−
1

of line segments joining these points, and
taking the limit as the maximum distance between adjacent points approached zero.
The length was denoted
Z
C
ds
and is a special example of a line integral along
C
having integrand 1.
The line integral of a general function
f
(
x
,
y
,
z
)
can be de±ned similarly. We
choose a point
(
x
∗
i
,
y
∗
i
,
z
∗
i
)
on the
i
th subarc and form the Riemann sum
S
n
=
n
±
i
=
1
f
(
x
∗
i
,
y
∗
i
,
z
∗
i
)

1
r
i

.