Poincaré's lemma says that on a contractible manifold, all closed forms
are exact. While 
implies that all exact forms are closed, it is not always
true that all closed forms are exact. The Poincaré lemma is used to show that
closed forms represent cohomology classes.
Poincaré's Lemma
See also
Cohomology, Closed Form, de Rham Cohomology, Differential k-Form, Exact Form, Exterior Derivative, Manifold, Poincaré's Holomorphic Lemma, Stokes' Theorem, Wedge ProductPortions of this entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd and Weisstein, Eric W. "Poincaré's Lemma." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoincaresLemma.html