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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.

For smooth vector fields $X,Y$, their Lie bracket of vector fields vanishes if and only if their local flows commute locally. Writing their flows as ${\Phi }_t$ and ${\Psi }_s$, every point has a neighborhood and sufficiently small time intervals on whichMore generally, if one composition is defined on an entire product $I\times J$ of open time intervals containing zero, the other is defined there too and the equality holds. If both fields are complete vector fields, it holds for all real $s,t$. For incomplete fields, existence of the two compositions at isolated endpoint choices alone does not guarantee equality.^[ref]

The derivative of the pullback of $Y$ by the $X$ flow is the pullback of $[X,Y]$. If the bracket is zero, this makes $Y$ invariant under that flow. Uniqueness for the associated ordinary differential equation then identifies the two compositions on the stated common time rectangles. Conversely, differentiating local flow commutation first in one time and then the other gives the zero bracket. Neither field needs to be complete for the local statement.