Proof of Transition Maps of Induced Boundary Charts of an Oriented Atlas are Orientation-Preserving

Write $A=\varphi(U\cap V)$, a set open in the closed upper half-space $H^n$.

Claim 1. The value $\det J_F(a)$ does not depend on the chosen local smooth extension $F$, since first-order partial derivatives at points of a set open in $H^n$ are determined by the values of the map on that set (as observed in Smooth Diffeomorphism of Euclidean or Half-Space Domains, Jacobian, and Pullback), and $x\mapsto\det J_F(x)$ is continuous near $a$, being obtained from the continuous partial derivatives of $F$ by finitely many sums and products (Determinant of a Real Square Matrix, Sums and Products of Continuous Real-Valued Functions).

If $a\in A\cap\operatorname{int}(H^n)$, then $a\in\varphi(U\cap V\cap\operatorname{int}(M))$ by claim 3 of Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 (interior points are exactly those mapping into the interior of the half-space), and $\det J_F(a)>0$ is precisely the positive compatibility required of the oriented smooth atlas.

If $a\in A\cap\partial H^n$: since $A$ is open in $H^n$, the points $a+te_n$ (with $e_n$ the last standard basis vector) lie in $A\cap\operatorname{int}(H^n)$ for all sufficiently small $t>0$, so $\det J_F(a+te_n)>0$ there; letting $t\to 0^{+}$ and using continuity, $\det J_F(a)\ge 0$. Moreover $\det J_F(a)\ne 0$: exactly as in Step A of the proof of Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1, a local smooth extension $G$ of the inverse transition satisfies $J_G(F(a))\,J_F(a)=$ identity matrix by the chain rule (the composition equals the identity on a set open in $H^n$ around $a$, which determines the first-order partials at $a$), so $J_F(a)$ is invertible and its determinant is nonzero. Hence $\det J_F(a)>0$.

Claim 2. By claim 3 of Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1, the transition maps $A\cap\partial H^n$ into $\partial H^n$; since $F$ agrees with the transition on $W\cap A$ for its domain $W$, we have $F_n=0$ on $W\cap A\cap\partial H^n$, which is open in $\partial H^n$ and contains $a$. For $j\in\{1,\dots,n-1\}$: the points $a+he_j$ lie in $A\cap\partial H^n$ for small $|h|$, where $F_n$ vanishes; since the partial derivative of the extension at $a$ is determined by these values (the difference quotients along the boundary directions), $\frac{\partial F_n}{\partial x_j}(a)=0$. For $j=n$: $F$ maps $W\cap A$ into $H^n$, so $F_n(a+te_n)\ge 0$ for small $t>0$ while $F_n(a)=0$; the one-sided difference quotients are $\ge 0$, and since $\frac{\partial F_n}{\partial x_n}$ is continuous, the partial derivative at $a$ equals the limit of these one-sided quotients, so $\frac{\partial F_n}{\partial x_n}(a)\ge 0$. Expanding $\det J_F(a)$ along its $n$th row — whose only possibly nonzero entry is $\frac{\partial F_n}{\partial x_n}(a)$ — using Minor, Cofactor, and Adjugate of a Real Square Matrix gives

$$\det J_F(a)=\frac{\partial F_n}{\partial x_n}(a)\cdot\det J'(a),$$

where $J'(a)$ is the $(n-1)\times(n-1)$ matrix with entries $\frac{\partial F_i}{\partial x_j}(a)$ for $i,j\le n-1$. Since $\det J_F(a)>0$ by claim 1 and $\frac{\partial F_n}{\partial x_n}(a)\ge 0$, we conclude $\frac{\partial F_n}{\partial x_n}(a)>0$ (and also $\det J'(a)>0$).

Claim 3. By claim 4 of Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 (and its proof), the boundary transition $\psi_2\circ\psi_1^{-1}$ is a bijection between its images admitting, at every point and in both directions, local smooth extensions of the form

$$G(z)=T_{c_2}\bigl(\pi\bigl(F(T_{c_1}^{-1}(z),0)\bigr)\bigr),$$

with $F$ a local smooth extension of the ambient transition, $\pi$ the projection dropping the last coordinate, and $T_{c_1},T_{c_2}$ the translations of the charts. Both images are open subsets of $\mathbb{R}^{n-1}$, hence admissible domains in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain; so the boundary transition is a smooth diffeomorphism of admissible domains. Its Jacobian matrix at $z$ is computed from $G$: translations have identity Jacobian, so by the chain rule the $(i,j)$ entry of $J_G(z)$, for $i,j\le n-1$, is $\frac{\partial F_i}{\partial x_j}(a)$ with $a=(T_{c_1}^{-1}(z),0)$ — that is, $J_G(z)=J'(a)$ in the notation of claim 2. By claim 2, $\det J'(a)>0$. Hence the boundary transition is orientation-preserving. $\blacksquare$