Calculus 2nd Edition Rogawski

978 C H A P T E R 17 LINE AND SURFACE INTEGRALS
17.3 SUMMARY
• A vector field F on a domain D is conservative if there exists a function V such that
∇V = F on D. The function V is called a potential function of F.
• Avector field F on a domain D is called path-independent if for any two points P,Q ∈ D,
we have
∫ ∫
F · ds = F · ds
c 1 c 2
for any two paths c 1 and c 2 in D from P to Q.
• The Fundamental Theorem for Conservative Vector Fields: If F = ∇V , then
∫
F · ds = V (Q) − V (P )
c
for any path c from P to Q in the domain of F. This shows that conservative vector fields
are path-independent. In particular, if c is a closed path (P = Q), then
∮
F · ds = 0
c
• The converse is also true: On an open, connected domain, a path-independent vector
field is conservative.
• Conservative vector fields satisfy the cross-partial condition
∂F 1
∂y = ∂F 2
∂x , ∂F 2
∂z = ∂F 3
∂y , ∂F 3
∂x = ∂F 1
∂z
4
• Equality of the cross-partials guarantees that F is conservative if the domain D is simply
connected—that is, if any loop in D can be drawn down to a point within D.
17.3 EXERCISES
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?
2. Which of the following statements are true for all vector fields, and
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
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.
(f) The line integral is equal to the integral of the tangential component
along the curve.
(g) The cross-partials of the components are equal.
3. Let F be a vector field on an open, connected domain D. Which
of the following statements are always true, and which are true under
additional hypotheses on D?
(a) If F has a potential function, then F is conservative.
(b) If F is conservative, then the cross-partials of F are equal.
(c) If the cross-partials of F are equal, then F is conservative.
4. Let C, D, and E be the oriented curves ∫ in Figure 16 and let F = ∇V
be a gradient vector field such that F · ds = 4. What are the values
C
of the following integrals?
∫
∫
(a) F · ds
(b) F · ds
D
E
y
Q
C
D
E
P
x
FIGURE 16