Modul
e
15 : Vector fields, Gradient, Divergence and Curl
Lecture
44 : Gradient Divergence and Curl [Section 44.1]
Then
Thus,
is irrotational. Hence, we have shown that every vector-field which has a potential is irrotational.
We state next some properties of the curl operator which show that it behaves like a differential operator.
44.1.13
Theorem:
Let
be continuously differentiable vector-fields and
a continuously differentiable Scalar-fields. Then the following hold:
(i)
(ii)
(iii)
where
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is irrotational. Hence, we have shown that every vector-field which has a
potential is irrotational. 
be continuously differentiable vector-fields and 
a continuously differentiable Scalar-fields. Then the following hold: 
where 
