Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$} and let $\Omega,\Omega'$ be admissible domains in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$ in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}.

A \textbf{smooth diffeomorphism} $F:\Omega\to\Omega'$ is a \reftext{def:bijection-sets-2026a}{bijection} with the following two properties.

1. For every $x\in\Omega$ there exist \reftext{def:open-subset-euclidean-space-2026a}{open} sets $W,W'\subseteq\mathbb{R}^n$ with $x\in W$ and $F(x)\in W'$, and a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map} $G:W\to W'$ such that $G(y)=F(y)$ for every $y\in W\cap\Omega$.
2. The analogous local smooth extension condition holds for the inverse map $F^{-1}:\Omega'\to\Omega$ at every point of $\Omega'$.

For $x\in\Omega$, the \textbf{Jacobian matrix} $J_F(x)$ of $F$ at $x$ is defined to be the \reftext{def:differentiable-map-at-point-euclidean-2026a}{Jacobian matrix} $J_G(x)$ of any local smooth extension $G$ as in property 1. This does not depend on the chosen extension: any two such extensions agree on the intersection of their domains with $\Omega$, which contains a set open in the ambient set of $\Omega$ around $x$. Explicitly, if $x$ is interior to $\Omega$ in $\mathbb{R}^n$ that shared set contains an open ball around $x$, on which equal maps have equal partial derivatives; and if $\Omega$ has ambient set the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} and $x$ lies in its boundary hyperplane, the shared set contains a half-ball around $x$, the first-order \reftext{def:partial-derivative-coordinate-map-2026a}{partial derivatives} of each smooth extension exist two-sidedly and are continuous, and agreement on the half-ball determines their values at $x$ (at interior points of the half-ball directly, and at $x$ by continuity); hence the Jacobian matrices of any two extensions coincide at $x$.

We say that $F$ is \textbf{orientation-preserving} if for every $x\in\Omega$ the \reftext{def:determinant-real-square-matrix-2026a}{determinant} of $J_F(x)$ satisfies

Finally, let $k\in\mathbb{N}\cup\{0\}$. For $k\ge 1$, let $\omega$ be an assignment which to each point $x'\in\Omega'$ assigns an \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating $k$-linear form} $\omega_{x'}$ on $\mathbb{R}^n$. The \textbf{pullback} $F^{*}\omega$ is the assignment which to each $x\in\Omega$ assigns the alternating $k$-linear form given by

for all vectors $v_1,\dots,v_k\in\mathbb{R}^n$, where $J_F(x)v_r$ is the \reftext{def:matrix-vector-product-2026a}{matrix-vector product}. For $k=0$ we use the convention, consistent with \ref{def:differential-k-form-euclidean-open-set-2026a}, that an assignment of alternating $0$-linear forms on $\Omega'$ is a real-valued function $\omega:\Omega'\to\mathbb{R}$, and the pullback is defined by

for every $x\in\Omega$. When $\Omega$ and $\Omega'$ are open in $\mathbb{R}^n$, both cases agree exactly with the pullback from \ref{def:pullback-differential-form-c1-euclidean-2026a}, whose statement covers $k=0$ by the same convention.