Let $n\in$ \reftext{def:natural-numbers-2026a}{$\mathbb{N}$} with $n\ge 2$, and let $M$ be an \reftext{def:oriented-smooth-manifold-boundary-2026a}{oriented smooth manifold with boundary} of dimension $n$ with nonempty \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} $\partial M$, the orientation being given by the chosen \reftext{def:oriented-smooth-atlas-manifold-boundary-2026a}{oriented smooth atlas}. Then the following hold.

1. Let $(U,\varphi)$ and $(V,\psi)$ be charts of the oriented atlas with $U\cap V\ne\varnothing$, let $a\in\varphi(U\cap V)$, and let $F$ be any local smooth extension of the transition map $\psi\circ\varphi^{-1}$ at $a$ furnished by \ref{def:smooth-compatible-charts-upper-half-space-2026a}. Then the \reftext{def:jacobian-determinant-euclidean-open-set-2026a}{Jacobian determinant} satisfies $\det J_F(a)>0$. (For $a$ in the image of the interior of $M$ this is the \reftext{def:positive-compatibility-charts-interior-manifold-boundary-2026a}{positive compatibility} required of the oriented atlas; the content of this claim is that positivity extends to all points of $\varphi(U\cap V)$, including boundary points.)

2. In the setting of claim 1, suppose additionally that $a$ lies in the boundary hyperplane of the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} $H^n$, and write $F=(F_1,\dots,F_n)$ in coordinates. Then the \reftext{def:partial-derivative-coordinate-map-2026a}{partial derivatives} of the last coordinate function satisfy

3. Let $\psi_1$ and $\psi_2$ be induced boundary charts of $\partial M$ in the sense of \ref{thm:boundary-smooth-manifold-structure-2026a}, arising from charts of the oriented atlas, with overlapping domains. Then the transition map $\psi_2\circ\psi_1^{-1}$ between the corresponding images is a \reftext{def:smooth-diffeomorphism-euclidean-half-space-domain-2026a}{smooth diffeomorphism} of admissible domains in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^{n-1}$, and it is orientation-preserving in the sense of that definition; that is, the determinant of its Jacobian matrix is positive at every point of its domain.