Orientable smooth manifold

A smooth manifold is orientable when it admits an atlas whose transition maps have positive Jacobian determinant wherever they are defined.

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 $v_1$.

For a smooth $n$-manifold, the following are equivalent: an atlas with positive transition Jacobians, a smooth choice of orientation of every tangent space, and a nowhere-vanishing smooth $n$-form. Positive charts orient their coordinate frames; oriented frames select the positive local coordinate volume forms, which a partition of unity glues; and a nonzero top form declares a frame positive when its value on that frame is positive.

If $M\times N$ is orientable, fix a point of $N$ and an ordered basis of its tangent space. Contracting a product orientation with that basis gives a smooth orientation of $M$. Interchanging the factors gives an orientation of $N$.

An orientation-reversing diffeomorphism reverses the chosen orientation. In oriented local coordinates its derivative has negative determinant.