Smooth Diffeomorphism of Euclidean or Half-Space Domains, Jacobian, and Pullback

Let $n\in$ $\mathbb{N}$ and let $\Omega,\Omega'$ be admissible domains in Euclidean space $\mathbb{R}^n$ in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain.

A smooth diffeomorphism $F:\Omega\to\Omega'$ is a bijection with the following two properties.

1. For every $x\in\Omega$ there exist open sets $W,W'\subseteq\mathbb{R}^n$ with $x\in W$ and $F(x)\in W'$, and a 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 Jacobian matrix $J_F(x)$ of $F$ at $x$ is the Jacobian matrix $DG(x)$ of a local smooth extension $G$ of $F$ at $x$ as in property 1; by Jacobian Matrix of a Local Smooth Extension on an Admissible Domain this matrix exists and does not depend on the choice of $G$.

We say that $F$ is orientation-preserving if for every $x\in\Omega$ the determinant of $J_F(x)$ satisfies

$$\det J_F(x)>0.$$

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 alternating $k$-linear form $\omega_{x'}$ on $\mathbb{R}^n$. The pullback $F^{*}\omega$ is the assignment which to each $x\in\Omega$ assigns the alternating $k$-linear form given by

$$(F^{*}\omega)_x(v_1,\dots,v_k)=\omega_{F(x)}\bigl(J_F(x)v_1,\dots,J_F(x)v_k\bigr)$$

for all vectors $v_1,\dots,v_k\in\mathbb{R}^n$, where $J_F(x)v_r$ is the matrix-vector product. For $k=0$ we use the convention, consistent with Differential k-Form on an Open Subset of Euclidean Space, 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

$$(F^{*}\omega)(x)=\omega(F(x))$$

for every $x\in\Omega$.