Page "Conservative vector field" Paragraph 26
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More abstractly, a conservative vector field is an exact 1-form.
That is, it is a 1-form equal to the exterior derivative of some 0-form ( scalar field ).
An irrotational vector field is a closed 1-form.
Since d < sup > 2 </ sup > = 0, any exact form is closed, so any conservative vector field is irrotational.
The domain is simply connected if and only if its first homology group is 0, which is equivalent to its first cohomology group being 0.
The first de Rham cohomology group is 0 if and only if all closed 1-forms are exact.
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