Let us say we are dealing with a compact, orientable,differentiable manifold of dimension $k$, which is in ambient space $R^n$. Now, I know that this manifold can be covered with an atlas, and then from this atlas we can extract a finite cover of images of charts that we use to cover the chart (each of the patches is a differentiable homemorphism).
Now, I would like to integrate some $k$ form over this surface. The integral of the form over any single region that is covered by a patch is given by: $$ \int_{\phi(U)} \omega = \int_U \phi^* \omega $$ where $\phi$ is our mapping and $U$ is the open set that it maps from.
Now, we want to integrate over the entire manifold, and we use partitions of unity to do so. This is where I am lost. The book I am using(diff forms and applications by Do Carmo), tells me that if $V_i$ forms the collection of coordinate neighborhoods of our compact manifold, then a partition of unity $\theta_i$ subordinate to this has the following properties:
- $\sum_{i = 1}^{n}\theta_i(p) = 1$ for any $p$ on the surface.
- $\mathrm{supp} \ \theta_i \subset V_i$ for some coordinate neighborhood
- Each $\theta_i$ is smooth with codomain $[0,1]$
The integral over the entire manifold is then defined as: $$ \sum_{i}\int_{S \cap V_i}\theta_i \omega $$
I have a couple questions about this definition. First I will explain my understanding. Let us say we integrate over a region where patches overlap. The maximum number of patches that overlap is however many patches we have. Then, the fact that the sum of the partitions is $1$, guarantees thjat the integral is not being scaled up in any way. This perhaps is the only part that seems necessary to me. What issues do we have if the support of one of partitions is in multiple disjoint neighborhoods? Isn't that issue solved by everything adding to $1$? And why is the smoothness necessary?