Module 16 : Line Integrals, Conservative fields Green's Theorem and applications
Lecture 47 : Fundamental Theorems of Calculus for Line integrals [Section 47.1]
We observe that the given property of the vector field tells us that , as given by equation , is well-defined since the line-integral does not depend upon the choice of the curve joining to . However, there is one problem:
That this may not be always possible for some domains is illustrated in the next example.
47.1.3
Example :
Consider the domain the plane minus the shaded annulus region, i.e.,
Let and
Figure: The region
tells us that 
, as given by equation 
, is well-defined since the line-integral does not depend upon the choice of the curve 
joining 
to 
. However, there is one problem: 
the plane 
minus the shaded annulus region, i.e., 
and
