The line integral of a vector field 
on a curve 
is defined by
 |
(1)
|
where 
denotes a dot product. In Cartesian coordinates, the
line integral can be written
 |
(2)
|
where
![F=[F_1(x); F_2(x); F_3(x)].](/images/equations/LineIntegral/NumberedEquation3.svg) |
(3)
|
For 
complex
and 
a path in the complex
plane parameterized by ![t in [a,b]](/images/equations/LineIntegral/Inline6.svg)
,
 |
(4)
|
Poincaré's theorem states that if 
in a simply connected neighborhood
of a point 
, then in this neighborhood, 
is the gradient of a scalar
field 
,
 |
(5)
|
for 
,
where 
is the gradient operator. Consequently, the gradient
theorem gives
 |
(6)
|
for any path 
located completely within 
, starting at 
and ending at 
.
This means that if 
(i.e., 
is an irrotational field in some region), then
the line integral is path-independent in this region. If desired, a Cartesian path
can therefore be chosen between starting and ending point to give
 |
(7)
|
If 
(i.e., 
is a divergenceless
field, a.k.a. solenoidal field), then there
exists a vector field 
such that
 |
(8)
|
where 
is uniquely determined up to a gradient field (and which can be chosen so that 
).
