= Equivalent formulations of orientability of a smooth manifold
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.
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