James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1088 Chapter 16 Vector Calculus
Proof of Theorem 2 Using Definition 16.2.13, we have
y C
=f dr − y b
a
b
− y aS −f
−x
− y b
a
=f srstdd r9std dt
d
dt
dx
dt 1 −f
−y
f srstdd dt
− f srsbdd 2 f srsadd
dy
dt 1 −f
−z
dz
dt
dtD
(by the Chain Rule)
The last step follows from the Fundamental Theorem of Calculus (Equation 1).
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Although we have proved Theorem 2 for smooth curves, it is also true for piecewisesmooth
curves. This can be seen by subdividing C into a finite number of smooth curves
and adding the resulting integrals.
ExamplE 1 Find the work done by the gravitational field
Fsxd − 2 mMG
| x | 3 x
in moving a particle with mass m from the point s3, 4, 12d to the point s2, 2, 0d along a
piecewise-smooth curve C. (See Example 16.1.4.)
SOLUTION From Section 16.1 we know that F is a conservative vector field and, in
fact, F − =f , where
mMG
f sx, y, zd −
sx 2 1 y 2 1 z 2
Therefore, by Theorem 2, the work done is
W − y C
F dr − y C
=f dr
− f s2, 2, 0d 2 f s3, 4, 12d
−
mMG
s2 2 1 2 2 2
mMG
− mMGS 1 2 1 s3 2 1 4 2 1 12 2 2s2 13D
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Independence of Path
Suppose C 1 and C 2 are two piecewise-smooth curves (which are called paths) that have
the same initial point A and terminal point B. We know from Example 16.2.4 that, in
general, y C1
F dr ± y C2
F dr. But one implication of Theorem 2 is that
y =f dr −
C1
y =f dr
C2
whenever =f is continuous. In other words, the line integral of a conservative vector field
depends only on the initial point and terminal point of a curve.
In general, if F is a continuous vector field with domain D, we say that the line integral
y C
F dr is independent of path if y C1
F dr − y C2
F dr for any two paths C 1 and
C 2 in D that have the same initial points and the same terminal points. With this terminology
we can say that line integrals of conservative vector fields are independent of path.
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