Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$}, and let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary} of dimension $n$, with chosen smooth atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$; for each $\alpha\in A$ write $\Omega_\alpha=\varphi_\alpha(U_\alpha)$. Each $\Omega_\alpha$ is open in the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} $H^n$ of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$, hence an admissible domain in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}. Let $k\in\mathbb{N}\cup\{0\}$.

A \textbf{smooth differential $k$-form} $\omega$ on $M$ is a family $(\omega_\alpha)_{\alpha\in A}$ with the following three properties.

1. For each $\alpha\in A$, $\omega_\alpha$ assigns to each point $x\in\Omega_\alpha$ an \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating $k$-linear form} $\omega_{\alpha,x}$ on $\mathbb{R}^n$. The assignment $\omega_\alpha$ is called the \textbf{chart representative} of $\omega$ in the chart $(U_\alpha,\varphi_\alpha)$.

2. (Local smoothness.) For each $\alpha\in A$ and each $a\in\Omega_\alpha$ there exist an \reftext{def:open-subset-euclidean-space-2026a}{open} set $W\subseteq\mathbb{R}^n$ with $a\in W$ and a \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential $k$-form} $\eta$ on $W$ whose coefficient functions in the \reftext{thm:coordinate-expansion-differential-forms-euclidean-2026b}{coordinate expansion} are \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth}, such that $\eta_x=\omega_{\alpha,x}$ for every $x\in W\cap\Omega_\alpha$.

3. (Compatibility.) For all $\alpha,\beta\in A$ with $U_\alpha\cap U_\beta\ne\varnothing$, consider the transition map $\tau_{\beta\alpha}=\varphi_\beta\circ\varphi_\alpha^{-1}$ from $\varphi_\alpha(U_\alpha\cap U_\beta)$ to $\varphi_\beta(U_\alpha\cap U_\beta)$; it is a \reftext{def:smooth-diffeomorphism-euclidean-half-space-domain-2026a}{smooth diffeomorphism} of admissible domains, its local smooth extensions being furnished by the \reftext{def:smooth-compatible-charts-upper-half-space-2026a}{smooth compatibility} of the charts. The requirement is that the \reftext{def:smooth-diffeomorphism-euclidean-half-space-domain-2026a}{pullback} satisfies

on $\varphi_\alpha(U_\alpha\cap U_\beta)$, where $\omega_\beta$ is restricted to $\varphi_\beta(U_\alpha\cap U_\beta)$.

When $k=0$, each $\omega_\alpha$ is a real-valued function on $\Omega_\alpha$ and property 3 states that $\omega_\alpha=\omega_\beta\circ\tau_{\beta\alpha}$ on chart overlaps. Consequently there is a well-defined function $\omega:M\to\mathbb{R}$ given by $\omega(p)=\omega_\alpha(\varphi_\alpha(p))$ for any chart with $p\in U_\alpha$, and we identify smooth differential $0$-forms on $M$ with such functions, called \textbf{smooth functions} on $M$.