Poincare's Lemma and Electromagnetics
If B and A are vector fields such that B = curl A, then it can be verified by direct calculation that div B = 0.
The converse, namely div B = 0 implies the existence of (at least) a vector field A such that B = curl A is stated in Electromagnetics books without proof.
I have seen elsewhere some references stating that the converse is based on Poincare's lemma. That lemma is then stated in modern terminology that is pretty obscure,about contractible manifolds.
Can someone provide a description in simple terms, just like Poincare would have done, also with a reference to the original source?
Also, a counterexample in the case of a "non-contractible" manifold would illuminate the concept better.
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