The fundamental theorem of calculus for definite integration helped us to compute If has an anti-derivative , then
----------(38)
In fact, if is continuous on then and are related by for every i.e., an antiderivative of is given by
----------(39)
We shall extend both these parts of the fundamental theorem of calculus for line integrals.
Let be an open set in and be a continuously differentiable scalar field. Let
and let , be any smooth curve in such that initial point of is and final point of is . Then
Let be a continuously differentiable vector-field such that is conservative, i.e.,
for some continuously differentiable scalar field on . Then, for and for any smooth curve
If 
has an anti-derivative 
, then 
----------(38) 
is continuous on 
then 
and 
are related by 
for every 
i.e., an antiderivative 
of 
is given by 
----------(39) 
be an open set in 
and 
be a continuously
differentiable scalar field. Let 
and let 
, be any smooth curve in 
such that initial point of 
is 
and final point of 
is 
. Then 
be a continuously differentiable vector-field such that 
is
conservative, i.e., 
on 
. Then, for 
and for any smooth curve 
