Module 16 :  Line Integrals, Conservative fields Green's Theorem and applications
Lecture 47 :  Conservative Vector Fields [Section 47.2]
47.2.4 Theorem (Existence of potential):
  Proof
 

where is the open ball in with center and radius . Let and be any given unit vector in such that

Let be the point and denote the line segment joining to Then,

Using the properties of the line integral, we have

Let for the line segment us choose the arc-length parameterization

Then