= Integration on an oriented manifold
{title2=$\int_M\alpha$}
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.
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