Module 16 :  Line Integrals, Conservative fields Green's Theorem and applications
Lecture 47 :  Conservative Vector Fields [Section 47.2]
47.2.6 Theorem (Necessary Condition for  to be conservative):
 

Let be an open connected set and be a continuously differentiable vector-field. If

is conservative, then

  Proof
 

Since is conservative,

Since is continuously differentiable, is twice-continuously differentiable, and we have

Since is twice continuously differentiable, we get

proving the required claim.

The above theorem is useful in verifying that a scalar field is not conservative.

 
 

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