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 .
In order to be able to use theorem effectively, i.e., given a vector-field to be able to check whether it is conservative or not, one has to verify the condition that for all points and all curves joining to , the line integral is independent of the curve This condition seems difficult to check. One aim is to find some verifiable necessary and sufficient conditions for to be conservative. simple necessary condition for to be conservative is given by our next theorem.
47.2.6
Theorem (Necessary Condition for to be conservative):
Let be an open connected set and be a continuously differentiable vector-field. If
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 
. 
effectively, i.e., given a vector-field 
to be able to check whether it is conservative or not, one has to verify the condition that for all points 
and all curves 
joining 
to 
, the line integral 
is independent of the curve 
This condition seems difficult to check. One aim is to find some verifiable necessary and sufficient conditions for 
to be conservative. 
simple necessary condition for 
to be conservative is given by our next theorem. 
to be conservative): 
be an open connected set and 
be a continuously differentiable vector-field. If 
