Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$}. We call a subset $\Omega$ of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$ an \textbf{admissible domain} if either $\Omega$ is an \reftext{def:open-subset-euclidean-space-2026a}{open subset} of $\mathbb{R}^n$, or $\Omega$ is a subset of the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} $H^n$ that is open in $H^n$ in the sense of that definition. In the first case set $D=\mathbb{R}^n$, and otherwise set $D=H^n$; we call $D$ the ambient set of $\Omega$.

A differential $n$-form on $\Omega$ is an assignment $\omega$ which to each point $x\in\Omega$ assigns an \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating $n$-linear form} $\omega_x$ on $\mathbb{R}^n$. For each $i\in\{1,\dots,n\}$ let $e_i\in\mathbb{R}^n$ denote the $i$th standard basis vector, whose $i$th coordinate is $1$ and whose other coordinates are $0$. The \textbf{coefficient function} of $\omega$ is the function

We say that $\omega$ is \textbf{continuous} if $f$ is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point} of $\Omega$. When $\Omega$ is open in $\mathbb{R}^n$, this agrees with \ref{def:continuous-differential-k-form-euclidean-open-set-2026b}, because by \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b} the function $f$ is exactly the coefficient of the top-degree basis form in the coordinate expansion of $\omega$.

The \textbf{support} of $\omega$, denoted $\operatorname{supp}\omega$, is the set of all $x\in D$ such that every \reftext{def:open-subset-euclidean-space-2026a}{open subset} $W\subseteq\mathbb{R}^n$ with $x\in W$ contains a point $y\in W\cap\Omega$ with $\omega_y\ne 0$. We say that $\omega$ is \textbf{compactly supported in $\Omega$} if $\operatorname{supp}\omega$ is a \reftext{def:compact-space-and-subset-2026a}{compact} subset of $\mathbb{R}^n$ and $\operatorname{supp}\omega\subseteq\Omega$.

The \textbf{zero extension} of the coefficient function of $\omega$ is the function $\tilde f:D\to\mathbb{R}$ defined by $\tilde f(x)=f(x)$ for $x\in\Omega$ and $\tilde f(x)=0$ for $x\in D\setminus\Omega$.