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Poincaré's Lemma


Poincaré's lemma says that on a contractible manifold, all closed forms are exact. While d^2=0 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.


See also

Cohomology, Cohomology Class, Closed Form, de Rham Cohomology, Differential k-Form, Exact Form, Exterior Derivative, Manifold, Poincaré's Holomorphic Lemma, Stokes' Theorem, Wedge Product

Portions 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 Web Resource. https://mathworld.wolfram.com/PoincaresLemma.html

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