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} $\partial M$. Equip $\partial M$ with the smooth manifold structure of dimension $n-1$ from \ref{thm:boundary-smooth-manifold-structure-2026a}, built from the induced boundary charts arising from charts of the chosen oriented atlas.

Let $\rho:\mathbb{R}^{n-1}\to\mathbb{R}^{n-1}$ denote the reflection of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} given by

When $n$ is odd we have $n-1\ge 2$, so $\rho$ leaves the last coordinate unchanged and maps the region of points of $H^{n-1}$ with positive last coordinate into itself; hence composing a chart map with $\rho$ again yields a \reftext{def:chart-upper-half-space-2026a}{chart}.

The \textbf{induced orientation} on $\partial M$ is the orientation of $\partial M$, in the sense of \ref{def:oriented-smooth-manifold-boundary-2026a}, given by the following \reftext{def:oriented-smooth-atlas-manifold-boundary-2026a}{oriented smooth atlas}, where \reftext{def:even-odd-natural-numbers-2026a}{even and odd} refer to the parity of $n$.

1. If $n$ is even: the atlas of all induced boundary charts $(U'\cap\partial M,\psi)$ from \ref{thm:boundary-smooth-manifold-structure-2026a} arising from charts of the chosen oriented atlas of $M$. This is an oriented smooth atlas by \ref{lem:positive-jacobian-boundary-transition-2026a}.

2. If $n$ is odd: the atlas of all charts $(U'\cap\partial M,\rho\circ\psi)$, where $(U'\cap\partial M,\psi)$ ranges over the induced boundary charts as in case 1. This is an oriented smooth atlas because each transition map of this atlas is the conjugate by $\rho$ of the corresponding transition map of the atlas in case 1, and conjugation by $\rho$ does not change the Jacobian determinant of the transition, which is positive by \ref{lem:positive-jacobian-boundary-transition-2026a}.

This sign convention is chosen so that the Stokes identity holds without an extraneous sign; it corresponds to the outward-normal-first convention for orienting the boundary.