Module 16 :  Line Integrals, Conservative fields Green's Theorem and applications
Lecture 47 :  Conservative Vector Fields [Section 47.2]
47 .2 Conservative Vector Fields
 

The independence of the line integral over the path joining two points and can also be described in terms of closed paths as follows:

47.2.1 Theorem:
  Let be any scalar-field. Then the following are equivalent:
(i)

For any two points ,the line integral    does not depend upon the path joining and .
(ii)
 for every closed path in .
 
   
47.2.2 Definition :
  Let be a region in . We say is connected if any two points in can be joined by a piecewise smooth curve completely in .
   
47.2.3 Examples :
(i)
In , the only connected sets are intervals.
(ii)
 In , examples of connected sets are:
 

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