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.

Portions of this entry contributed by Todd Rowland

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Rowland, Todd and Weisstein, Eric W. "Poincaré's Lemma." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PoincaresLemma.html