 Closed and exact differential forms

A closed differential form satisfies $d\alpha =0$; an exact differential form has $\alpha =d\beta$. The identity $d^2=0$ makes every exact form closed. Global failure of the converse is measured by de Rham cohomology; the Poincaré lemma gives the local converse in positive degrees.

In differential geometry and calculus, the concepts of closed and exact differential forms are crucial for understanding forms on manifolds, specifically in the context of integration and topology.