= Solution
The <equivalent formulations of orientability of a smooth manifold> are:
* an atlas whose transition maps have positive <Jacobian determinant>;
* a smooth choice of one of the two orientations of every <tangent space>;
* a nowhere-zero smooth top-degree <differential form>.
A positive chart orients its coordinate frame. Conversely, smoothly oriented frames determine local positive coordinate volume forms. A <partition of unity> subordinate to an oriented atlas glues these positive forms: at each point they are positive multiples of one another, so their weighted sum cannot vanish. Finally, a nowhere-zero top form $\omega$ declares $(v_1,\ldots,v_n)$ positive exactly when $\omega(v_1,\ldots,v_n)>0$. These constructions are inverse at the level of orientations.
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