Exterior derivative ($d$)

The exterior derivative is the unique graded derivation $d:{\Omega }^p(M)\to {\Omega }^{p+1}(M)$ that agrees with the differential on functions and satisfies $d^2=0$.

A differential form $\alpha$ is closed when $d\alpha =0$.

A differential form is exact when it is the exterior derivative of a form of one lower degree. Since $d^2=0$, every exact form is closed.

The $p$th de Rham cohomology group is the quotient of closed $p$-forms by exact $p$-forms.