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

Let be any closed path in  with parameterizations

Choose and consider the curves

 
 
Figure 189. Caption text.
   
 

Then, is the union of the two curves and . Further, is the curve with initial point and final point . Thus, by

 

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