  A fundamental result of classical electromagnetism is that Maxwell's
equations imply that electric charge is locally conserved. Here we show the
converse: Local charge conservation implies the local existence of fields
satisfying Maxwell's equations. This holds true for any conserved quantity
satisfying a continuity equation. It is obtained by means of a strong form of
the Poincar\'e lemma presented here that states: Divergence-free multivector
fields locally possess curl-free antiderivatives on flat manifolds. The above
converse is an application of this lemma in the case of divergence-free vector
fields in spacetime. We also provide conditions under which the result
generalizes to curved manifolds.
