Let be a continuous vector-field, where is an open connected set. If for any curve in , the line-integral , depends only upon the initial and final point of , then there exists a scalar field such that
Proof
Let us fix any point . For any point , let be any smooth curve with initial point and final point at least one such curve exists as is connected. Define
Figure 191. The path in
Further, the value does not depend upon the curve joining to Thus, the function is well-defined. We show that is the required scalar-field. For , since is open, we can select such that
be a continuous vector-field, where 
is an open connected set. If for any curve 
in 
, the line-integral 
, depends only upon the initial and final point of 
, then there exists a scalar field 
such that 
. For any point 
, let 
be any smooth curve with initial point 
and final point 
at least one such curve exists as 
is connected. Define 
in 
does not depend upon the curve joining 
to 
Thus, the function 
is well-defined. We show that 
is the required scalar-field. For 
, since 
is open, we can select 
such that 
