Integration on manifolds

Coming from integration on Rn.
If we consider a submanifold $N \subseteq M$ such that $d i m (N) = k$ and a $k$ -differential form $w$ on $M$ , we can define

\int_{N} w

In a first step, we consider the case where we have a coordinate chart $(U, φ)$ of $N$ such that $N = U$ . Then
we define:

\int_{N} w := \int_{φ (U)} (φ^{- 1})^{*} (w)

This definition is independent of the chart, thanks to the change of variable theorem.

Finally, to integrate a $k$ -form $w$ on an arbitrary submanifold $N \subseteq M$ , we take a partition of unity $p_{i}$ and define

\int_{N} w := \sum_{i} p_{i} \cdot \int_{φ (U_{i})} (φ_{i}^{- 1})^{*} (w)

If we want to integrate functions, we need a global volume form, and then we can convert functions to $n$ -forms.