Let $M\subset\mathbb R^n$ be an embedded m-dimensional manifold with boundary. Then $M$ is orientable if there exists an open set $U \subset\mathbb R^n$ containing $M$ and an m-form $\phi$ on $U$ such that for every local parametrization $f:W\to M$ with $W \subset H^+$ open in $H^+$, the pullback $f^*\phi$ is an m-form on $W$ which is nowhere zero.
Wikidata ID: Q2748415