The following conditions are equivalent for a conservative vector field on a particular domain 
:
1. For any oriented simple closed curve 
, the line integral 
.
2. For any two oriented simple curves 
and 
with the same endpoints, 
.
3. There exists a scalar potential function 
such that 
,
where 
is the gradient.
4. If 
is simply connected, then curl 
.
The domain 
is commonly assumed to be the entire two-dimensional plane or three-dimensional space.
However, there are examples of fields that are conservative in two finite domains
and 
but are not conservative in their union 
.
Note that conditions 1, 2, and 3 are equivalent for any vector field 
defined in any open set 
, with the understanding that the curves 
, 
, and 
are contained in 
and that 
holds at every point of 
.
In general, condition 4 is not equivalent to conditions 1, 2 and 3 (and counterexamples are known in which 4 does not imply the others and vice versa), although if the first
derivatives of the components of 
are continuous, then these conditions do imply 4. In order
for condition 4 to imply the others, 
must be simply connected.
