1064
CHAPTER 17
LINE AND SURFACE INTEGRALS
(ET CHAPTER 16)
x
y
D
P
=
(
a
,
b
)
R
(
c
b
)
Q
(
c
d
)
Since
∂
ϕ
∂
x
(
x
,
y
)
=
0in
D
, in particular
∂
∂
x
(
x
,
b
)
=
0for
a
≤
x
≤
c
. Therefore, for
a
≤
x
≤
c
we have
(
x
,
b
)
=
Z
x
a
∂
∂
u
(
u
,
b
)
du
+
(
a
,
b
)
=
Z
x
a
0
+
(
a
,
b
)
=
k
+
(
a
,
b
)
Substituting
x
=
a
determines
k
=
0. Hence,
(
x
,
b
)
=
(
a
,
b
)
for
a
≤
x
≤
c
In particular,
(
c
,
b
)
=
(
a
,
b
)
⇒
(
R
)
=
(
P
)
(1)
Similarly, since
∂
∂
y
(
x
,
y
)
=
D
,wehave
∂
∂
y
(
c
,
y
)
=
b
≤
y
≤
d
. Therefore for
b
≤
y
≤
d
we have
(
c
,
y
)
=
Z
y
b
∂
∂v
(
c
,v)
d
v
+
(
c
,
b
)
=
Z
y
b
0
d
v
+
(
c
,
b
)
=
k
+
(
c
,
b
)
Substituting
y
=
b
gives
(
c
,
b
)
=
k
+
(
c
,
b
)
or
k
=
0. Therefore,
(
c
,
y
)
=
(
c
,
b
)
for
b
≤
y
≤
d
In particular,
(
c
,
d
)
=
(
c
,
b
)
⇒
(
Q
)
=
(
R
)
(2)
Combining (1) and (2) we obtain the desired equality
(
P
)
=
(
Q
)
.S
ince
P
and
Q
are any two points in
D
,we
conclude that
is constant on
D
.
17.2 Line Integrals
(ET Section 16.2)
Preliminary Questions
1.
What is the line integral of the constant function
f
(
x
,
y
,
z
)
=
10 over a curve
C
of length 5?
SOLUTION
Since the length of
C
is the line integral
R
C
1
ds
=
5, we have
Z
C
10
=
10
Z
C
1
=
10
·
5
=
50
2.
Which of the following have a zero line integral over the vertical segment from
(
0
,
0
)
to
(
0
,
1
)
?
(a)
f
(
x
,
y
)
=
x
(b)
f
(
x
,
y
)
=
y
(c) F
= h
x
,
0
i
(d) F
= h
y
,
0
i
(e) F
= h
0
,
x
i
(f) F
= h
0
,
y
i
The vertical segment from
(
0
,
0
)
to
(
0
,
1
)
has the parametrization
c
(
t
)
=
(
0
,
t
),
0
≤
t
≤
1
Therefore,
c
0
(
t
)
= h
0
,
1
i
and
k
c
0
(
t
)
k=
1. The line integrals are thus computed by
Z
C
f
(
x
,
y
)
=
Z
1
0
f
(
c
(
t
))
k
c
0
(
t
)
k
dt
(1)
Z
C
F
·
=
Z
1
0
F
(
c
(
t
))
·
c
0
(
t
)
(2)
(a)
We have
f
(
c
(
t
))
=
x
=
0. Therefore by (1) the line integral is zero.