Poincaré's theorem states that if 
(i.e., 
is an irrotational field)
in a simply connected neighborhood 
of a point 
, then in this neighborhood, 
is the gradient of a scalar
field 
,
 |
for 
,
where 
is the gradient operator. Consequently, the gradient
theorem gives
 |
for any path 
located completely within 
, starting at 
and ending at 
.
This means that if 
,
the line integral of 
is path-independent.
