Stokes' Theorem provides interesting results for special forms

- A form $\alpha$ is said to be
*closed*if its exterior derivative is zero: $\mathrm{d}\alpha=0$ - A form $\alpha$ is said to be
*exact*if it is the exterior derivative of some other form: $\alpha=\mathrm{d}(\beta)$

And since $\mathrm{d}^2=0$, exact forms are immediately closed. Moreover:

A closed form always vanishes in submanifolds that are a boundary

An exact form always vanishes in closed submanifolds, no matter if they are boundary of something else

This is the very beginning of *de Rham Cohomology*