I am studying (Zorich, Mathematical Analysis II, Chapter 13) the integration of differential forms on orientable smooth surfaces in $ \mathbb {R}^n $ and I have some doubts about the definitions. My textbook does not seem to me to give a precise definition (or maybe I don't see it well) of what it means to integrate a differential k-form on an orientable smooth k-surface $S$, when $S$ cannot be described by a single chart, but searching a bit I seem to have understood that the correct definition to be taken into consideration should be the following:
$$\int_S \omega := \sum_i \int_{U_{k(i)}} \alpha_i \cdot \omega$$
where $\{\alpha_i(x)\}_i$ is a family of continuous functions (at most countable) with compact support defined on $S$ such that for every $i$ we can always find a patch $U_{k(i)}\subset S$ in the atlas of $S$ such that $\text{supp }\alpha_i\subset U_{k(i)}$. In other words, $\{\alpha_i(x)\}_i$ is a partition of unity defined on $S$.
First question: is there a theorem that guarantees me that such a family of functions can always be constructed, regardless of the atlas chosen for $ S $?
Second question: could $\sum_i[...]$ be infinite? If yes, how can I justify its convergence?
Thanks in advance.