Smooth Differential k-Form on a Smooth Manifold with Boundary

Let $n\in$ $\mathbb{N}$, and let $M$ be a 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 closed upper half-space $H^n$ of Euclidean space $\mathbb{R}^n$, hence an admissible domain in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain. Let $k\in\mathbb{N}\cup\{0\}$.

A 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 alternating $k$-linear form $\omega_{\alpha,x}$ on $\mathbb{R}^n$. The assignment $\omega_\alpha$ is called the 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 open set $W\subseteq\mathbb{R}^n$ with $a\in W$ and a differential $k$-form $\eta$ on $W$ whose coefficient functions in the coordinate expansion are 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 smooth diffeomorphism of admissible domains, its local smooth extensions being furnished by the smooth compatibility of the charts. The requirement is that the 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 smooth functions on $M$.