de Rham complex ($({\Omega }^{\bullet }(M),d)$)

The de Rham complex of a smooth manifold $M$ is the cochain complex of smooth differential forms with exterior derivative:Here $n=\dim M$ and ${\Omega }^0(M)=C^{\infty }(M)$. The identity $d^2=0$ makes every exact differential form closed. The quotient of closed $k$-forms by exact $k$-forms is the de Rham cohomology group $H_{\mathrm{dR}}^k(M)$.^[ref]

In Euclidean space the Poincaré lemma makes this complex exact in positive degrees. On a general manifold it is locally exact in small coordinate balls, but global exactness can fail. For the unit circle, the restriction of $\omega =-ydx+xdy$ has integral $2\pi$ around one positive turn. It is closed, since the circle has no nonzero two-forms. It cannot equal $df$ for a globally defined smooth function on the circle: the integral of a derivative around a closed loop is zero.

The complex therefore separates local differential identities from global obstructions. In degree zero, $df=0$ means that $f$ is locally constant; in degree one, periods around loops can detect a failure to possess a global potential.