= de Rham complex
{c}
{title2=$(\Omega^\bullet(M),d)$}
The <de Rham complex> of a <smooth manifold> $M$ is the <cochain complex> of smooth <differential forms> with <exterior derivative>:
$$
0\longrightarrow\Omega^0(M)\xrightarrow{d}\Omega^1(M)\xrightarrow{d}\cdots\xrightarrow{d}\Omega^n(M)\longrightarrow0.
$$
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 $H^k_{\mathrm{dR}}(M)$.<Introduction to Smooth Manifolds, second edition by John M. Lee>[Chapters 14 and 17, exterior differentiation and de Rham cohomology]{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 $\omega=-y\,dx+x\,dy$ 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.
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