i.e., the curve that circles the -axis twice, then
Thus, though satisfies conditions of theorem, it is not conservative. To make the condition to be sufficient also for to be conservative, one has to impose more conditions on the domain of .
47.2.9
Definition :
(i)
A subset is said to be simply connected if no simple closed curve in encloses points that are
not in the region . Intuitively, in a set without 'holes' is simply connected.
(ii)
A region is said to be simply connected if for every simple closed curve in there exists a
surface in whose boundary is .
47.2.10
Examples:
(i)
For example, the region enclosed by a circle, ellipse, a rectangular path are all
simply connected sets in
The region is not simply connected. In particular is not simply connected. There are closed curves that enclose points not in , for example origin.
-axis twice, then 
satisfies conditions of theorem, it is not conservative. To make the condition 
to be sufficient also for 
to be conservative, one has to impose more conditions on the domain 
of 
. 
is said to be simply connected if no simple closed curve in 
encloses points that are 
. Intuitively, in 
a set without 'holes' is simply connected. 
is said to be simply connected if 
for every simple closed curve 
in 
there exists a
in 
whose boundary is 
. 
The region
is not simply connected. In particular is not simply connected. There are closed curves 
that enclose points not in 
, for example origin. 
