A conservative vector field 
on an open set 
is a vector field for which
there exists a scalar potential
function 
such that 
,
where 
denotes the gradient of 
. The following conditions are equivalent:
1. For any oriented simple closed curve 
, the line integral 
.
2. For any two oriented simple curves 
and 
with the same endpoints, 
.
3. The potential function 
exists.
The curves in the first two conditions must be contained in 
, and 
must hold at every point of 
. If the components of 
have continuous first partial derivatives,
then a conservative vector field has
 |
Here 
denotes the curl of 
, so every conservative field of this regularity is an irrotational
field. Conversely, if 
is simply connected, every
continuously differentiable vector
field with zero curl is conservative. The distinct condition
defines a solenoidal field; a conservative field
need not be solenoidal.
Whether a vector field is conservative depends on the domain. A vector field
may be conservative on each of two domains 
and 
but not on their union 
.
