Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$} with $n\ge 2$, let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary} of dimension $n$ with nonempty \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} $\partial M$, and equip $\partial M$ with the smooth manifold structure of dimension $n-1$ from \ref{thm:boundary-smooth-manifold-structure-2026a} (if $M$ is oriented, one may equally use the atlas of the \reftext{def:induced-orientation-boundary-manifold-2026a}{induced orientation}, whose chart maps differ from induced boundary charts at most by composition with the reflection $\rho$ appearing there). Let $k\in\mathbb{N}\cup\{0\}$ and let $\omega=(\omega_\alpha)_{\alpha\in A}$ be a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential $k$-form} on $M$, relative to the chosen atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$ of $M$.

Let $(U'\cap\partial M,\psi)$ be a chart of the chosen atlas of $\partial M$, induced as in \ref{thm:boundary-smooth-manifold-structure-2026a} from a chart $(U_\alpha,\varphi_\alpha)$ of $M$ (possibly composed with the reflection $\rho$), and let $\Omega_\psi$ denote its image. Between the \reftext{def:euclidean-space-rn-2026a}{Euclidean spaces} $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$, let

be the unique affine map satisfying $j(y)=\varphi_\alpha(\psi^{-1}(y))$ for every $y\in\Omega_\psi$; it is the composition of the inverse translation (and, where applicable, the reflection $\rho$) with the inclusion of $\mathbb{R}^{n-1}$ into the boundary hyperplane of the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} $H^n$. Since $j$ is affine, it is a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map} and its \reftext{def:differentiable-map-at-point-euclidean-2026a}{Jacobian matrix} $J_j$ is a constant $n\times(n-1)$ matrix.

The \textbf{restriction of $\omega$ to $\partial M$}, denoted $\iota^{*}\omega$ where $\iota:\partial M\to M$ is the inclusion map, is the family whose representative in the chart $(U'\cap\partial M,\psi)$ assigns to each $y\in\Omega_\psi$ the \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating $k$-linear form} on $\mathbb{R}^{n-1}$ given by

for all vectors $v_1,\dots,v_k\in\mathbb{R}^{n-1}$, where $J_j v_r$ is the \reftext{def:matrix-vector-product-2026a}{matrix-vector product}; this is the pullback formula of \ref{def:pullback-differential-form-c1-euclidean-2026a} applied to the affine map $j$.

The family of these representatives is a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential $k$-form} on $\partial M$.