Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary

Let $M$ be a smooth manifold with boundary that is compact in the sense of that definition, with chosen smooth atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$. Then there exist $N\in$ $\mathbb{N}$, indices $\alpha_1,\dots,\alpha_N\in A$, and smooth differential $0$-forms $\chi_1,\dots,\chi_N$ on $M$, identified with real-valued functions on $M$ as in that definition, such that the following hold.

1. $0\le\chi_i(p)\le 1$ for every $i\in\{1,\dots,N\}$ and every $p\in M$.

2. For each $i\in\{1,\dots,N\}$, the support of $\chi_i$ — the set of all $p\in M$ such that every open subset of $M$ containing $p$ contains a point $q$ with $\chi_i(q)\ne 0$ — is contained in $U_{\alpha_i}$.

3. For every $p\in M$,

$$\sum_{i=1}^{N}\chi_i(p)=1.$$

A family $\chi_1,\dots,\chi_N$ with these properties is called a smooth partition of unity subordinate to the chart domains $U_{\alpha_1},\dots,U_{\alpha_N}$.