A differential form α of degree r on an n-manifold is exact if there exists a form β of degree r — 1 such that α = dβ and is closed if dα = 0. Since d-d = 0 any exact form is closed. The Poincaré lemma asserts that a closed differential form of positive degree is locally exact. There is also a complex form, proved by Cartan-Grothendieck, of the Poincaré lemma in which the operator d has a decomposition into components and .
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