It is easy to see that in , every convex set is connected. In fact, by definition, any two points in a convex
set can be joined by a line segment.
For connected regions, we can answer our question: which vector fields have a potentials?
47.2.4
Theorem (Existence of potential):
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
, every convex set is connected. In fact, by definition, any two points in a convex
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 
