Poincaré's lemma says that on a contractible manifold, all closed forms
of positive degree 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
Closed Form, Cohomology, 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:
Weisstein, Eric W., with contributions by Todd Rowland. "Poincaré's Lemma." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoincaresLemma.html