both having initial point and final point , the curve can be continuously deformed to , i.e., there exists a continuous function
Figure: Continuous deformation of
We state a necessary and sufficient condition for a vector field to be conservative. We shall prove this in module 18 (theorem 54.1.5)
47.2.12
Theorem (Sufficient condition for a field to be conservative) :
Let be a simply connected open set and be a continuously differentiable vector-field.
If curl , then there exists a scalar-field such that , i.e., is conservative.
and final point 
, the curve 
can be continuously deformed to 
, i.e., there exists a continuous function 
be a simply connected open set and 
be a continuously differentiable vector-field.
, then there exists a scalar-field 
such that 
, i.e., 
is conservative. 
