What is submanifold with boundary?

A $k$-submanifold with boundary $M \subseteq \R^n$ is a set covered by regular parametrizations of two kinds: interior charts $\gamma\colon V \to M \cap W$ with $V \subseteq \R^k$ open, and boundary charts $\gamma\colon V \cap H^k \to M \cap W$, where $H^k = \{u \in \R^k : u_k \geq 0\}$ and $\gamma$ extends smoothly and regularly to the open $V$. The boundary $\partial M$ is the set of points reached at $u_k = 0$; it is a $(k-1)$-submanifold without boundary, parametrized by the maps $u' \mapsto \gamma(u', 0)$. An orientation of $M$ induces one on $\partial M$ by the outward-normal-first rule: at $p \in \partial M$, a basis $(w_1, \dots, w_{k-1})$ of $T_p\partial M$ is positive iff $(\nu, w_1, \dots, w_{k-1})$ is a positive basis of $T_pM$, where $\nu \in T_pM \setminus T_p\partial M$ points out of $M$ (in a boundary chart: $\nu = -\partial_k\gamma$, up to adding tangential components — the orientation class does not see them).