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MATH1211/LSweek32/TNK/1011(2)
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1211: Multivariable Calculus
Lecture Summary – Week 32
April 4, 2011
We noted that
Theorem 3.2
of
Section 6.3
(p.391) says that:
F
has pathindependent line integrals
is the same as
H
C
F
¢
d
s
=0
for all
C
1
, simple, closed curves
C
in the domain of
F ,
and the
requirement
for this to hold is that
F
is a
continuous
vector field.
Theorem 3.3
(p.392
−
393) says that:
F
has pathindependent line integrals
is the same as
F
=
r
f
(i.e.,
F
is a
gradient field
(
conservative vector field
)) for some
f
of class
C
1
,
and the
requirement
for this to hold is that
F
is a
continuous
vector field defined on its domain of
definition which is a
connected
,
open
region of
R
n
. Moreover, when
F
=
r
f
, then for any
piecewise
C
1
curve
C
in the domain of
F
with initial point
A
and terminal point
B
, we have
R
C
F
¢
d
s
=
f
(
B
)
¡
f
(
A
)
. [A side note: in fact if
F
=
r
f
for some scalar function
f
of class
C
1
then it follows that
F
has pathindependent line integrals, irrespective of whether the domain of
F
is
connected or not (see bottom of p.392, before Theorem 3.3). It is the
converse
of this that requires
that the domain of
F
is open and connected.]
We remarked that the property of having pathindependent line integrals is very nice, but it is
impractical to check this directly for
every possible path
from
every possible initial point
to
every
possible terminal point.
In this respect, Theorem 3.2 does not offer much help also, because there are
infinitely many simple closed curves, and hence it is impossible to check whether
H
C
F
¢
d
s
=0
for
all
simple closed
C
1
curves
C
(yet, if we found that
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 Spring '11
 wang
 Integrals, Multivariable Calculus, Vector Calculus, Vector Space, 0 K, UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS, pathindependent line integrals

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