Modul
e
16 : Line Integrals, Conservative fields Green's Theorem and applications
Lecture
47 :
Conservative Vector Fields
[Section 47.2]
47.2.5
Theorem :
Let
be an open connected subset of
and
be a continuous vector-field. Then the following statements are equivalent :
(i)
There exists a scalar-field
such that
i.e.,
is conservative.
(ii)
For any two points
, the line integral
is independent of the path
joining
and
.
(iii)
, for every closed smooth curve
in
.
Proof
If
, then by theorem
for any curve
joining
,
Hence
holds.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
be an open connected subset of 
and 
be a continuous vector-field. Then the following statements are equivalent : 
such that 
is conservative. 
, the line integral 
is independent of the path 
joining 
and 
.
, for every closed smooth curve 
in 
. 
, then by theorem 
for any curve 
joining 
, 
holds. 
