Let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{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 $N\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$}, indices $\alpha_1,\dots,\alpha_N\in A$, and \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{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 \textbf{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 $p\in M$,

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