1094
C H A P T E R
17
LINE AND SURFACE INTEGRALS
(ET CHAPTER 16)
=
t
1
t
0
(
y
(
t
)
x
(
t
)
+
x
(
t
)
y
(
t
)
)
dt
=
t
1
t
0
d
dt
(
x
(
t
)
y
(
t
))
dt
The last equality follows from the Product Rule for differentiation. We now use the Fundamental Theorem of Calculus
to obtain:
c
F
·
d
s
=
x
(
t
)
y
(
t
)
t
1
t
=
t
0
=
x
(
t
1
)
y
(
t
1
)
−
x
(
t
0
)
y
(
t
0
)
=
cd
−
ab
17.3 Conservative Vector Fields
(ET Section 16.3)
Preliminary Questions
1.
The following statement is false.
If
F
is a gradient vector field, then the line integral of
F
along every curve is zero.
Which single word must be added to make it true?
SOLUTION
The missing word is “closed” (curve). The line integral of a gradient vector field along every closed curve
is zero.
2.
Which of the following statements are true for all vector fields, which are true only for conservative vector fields?
(a)
The line integral along a path from
P
to
Q
does not depend on which path is chosen.
(b)
The line integral over an oriented curve
C
does not depend on how the
C
is parametrized.
(c)
The line integral around a closed curve is zero.
(d)
The line integral changes sign if the orientation is reversed.
(e)
The line integral is equal to the difference of a potential function at the two endpoints.
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 Winter '08
 Rogawski
 Product Rule, Vector Calculus, Line integral, Vector field, Gradient, Gradient Vector Fields

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