In this note Analysis Tools with Examples
On page 285, it defines the integration on an embedded hypersurface as follows:
The idea is straightforward: Chop a hypersurface into pieces that are parametrized. However, I am a little bit lost with the technical details here. It seems to me that the right way to do it is the following: 1).Find a finite cover of the support of $f$, say$\{U_i\}$ 2).Produce a partition of unity subordinate to this cover 3).My guess it that those $U_i$ will serve as the desired parametrized pieces.
However, how are them parametrized according to the note's definition?(page 284).
Perhaps I can choose $U_i$ to be the inverse image of open balls in $\mathbb R^n$ by charts of $M$, then I will have a natural parametrization of them.
Using only what the notes give: instead of finding an arbitrary finite cover of the support of $f$, use the given parameterizations. That is, given a point $x \in M$, you know that by definition there is a map $\psi: D \to B(x, \varepsilon)$ which, when restricted to $D \cap \mathbb R^{n-1}$, has the image $B(x, \varepsilon) \cap M$. This restricted map satisfies the two conditions for a parameterization of prop 26.3: first, it is a homeomorphism with its image, and second, the derivative is injective because the derivative of the unrestricted map is injective (by virtue of being a diffeomorphism).
So about each point in the support of $f$, pick one of these parameterizations. Since the support of $f$ is compact, throw out all but finitely many. Then produce the partition of unity.
This illustrates a motif in the general theory of manifolds: your open sets as a whole are not very good. They lack structure and it typically isn't clear what you can say about them. However, each point has a neighborhood with excellent structure. Therefore, start with the nice given structure and build up.