A 1-form can be integrated along a path giving :
If is a loop encircling a surface , by Stokes theorem, we have :
Now, if , the integral of F around any (null homotopic) closed path vanishes. If and are two path with the same end points, then we can build a loop by first following and then in the reverse order and we get (if and are homotopic) :
Therefore the integral of F is depending only on the path. On the other hand, if , then this integral is path dependent.
Another way, maybe more familiar in physics is the following : using the Poincaré lemma, if then, there is (at least locally) a 0-form V (ie a function) such that dV. In that case :
In other words, if then F does not derive from a potential.
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