Let \(M\) be an oriented smooth manifold with or without boundary and \(\omega\) a smooth \(n\)-form . We define the integral of \(\omega\) on \(M\) in two steps.
First, consider \(\omega\) has compact support on a local chart \((U, \varphi)\). Then
\begin{equation*} \int_{M} \omega = \pm \int_{\varphi(U)} (\varphi^{-1})^*\omega, \end{equation*}where the right hand side is the usual Lebesgue integral. The sign depends on the orientation of the chart [@lee2013IntroductionSmoothManifolds, p.404].
For general \(\omega\), we admit a partition of unity on an atlas of \(M\), which we denote by \((\psi_i)\). We define
\begin{equation*} \int_{M} \omega=\sum_{i} \int_{M} \psi_i\omega. \end{equation*}
Remarks
- The definition does not depend on the choice of the partition of unity or the choice of the atlas [@lee2013IntroductionSmoothManifolds, Proposition 16.5].
- In [1] the integral is motivated by assigning to every point an infinitesimal volume element. Alternating tensors do the job due to their algebraic structure (it mimics volume).
- If the manifold itself is compact, not partition of unity is required.
See also Link to heading
References Link to heading
- J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.