Integration on an oriented manifold (${\int }_M\alpha$)

A compactly supported top-degree differential form on an oriented smooth manifold is integrated using orientation-preserving coordinates and a partition of unity. The coordinate change of variables formula makes the result independent of the choices. More generally, a form is integrated over a lower-dimensional oriented submanifold by its pullback. The Generalized Stokes theorem states ${\int }_{M}d\beta ={\int }_{\partial M}\beta$ with the induced boundary orientation. Integrating a closed differential form over a closed cycle depends only on its homology class, since its integral over a boundary vanishes.