 Outward-normal-first boundary orientation

The boundary of an oriented manifold with boundary is oriented by the outward-normal-first convention: an ordered basis $(v_2,\dots ,v_n)$ of $T_p\partial X$ is positive when $(v_1,v_2,\dots ,v_n)$ is positive in $T_{p}X$ for an outward-pointing normal $v_1$.