is to say that a volume form gives rise to -structure on More reduction is clearly possible by considering frames that have Thus a volume form gives rise to an -structure as well. Conversely, given an -structure, one can recover a volume form by imposing () for the special linear frames and then solving for the required -form by requiring homogeneity in its arguments. A manifold is orientable if and only if it has a nowhere-vanishing volume form. Indeed, is a deformation retract since where the positive reals are embedded as scalar matrices. Thus every -structure is reducible to an -structure, and -structures coincide with orientations on More concretely, triviality of the determinant bundle is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section. Thus, the existence of a volume form is equivalent to orientability. Relation to measures Given a volume form on an oriented manifold, the density is a volume pseudo-form on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds. Any volume pseudo-form (and therefore also any volume form) defines a measure on the Borel sets by The difference is that while a measure can be integrated over a (Borel) subset, a volume form can only be integrated over an oriented cell. In single variable calculus, writing considers as a volume form, not simply a measure, and indicates "integrate over the cell with the opposite orientation, sometimes denoted ". Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form need not be absolutely continuous. Divergence Given a volume form on one can define the divergence of a vector field as the unique scalar-valued function, denoted by satisfying where denotes the Lie derivative along and denotes the interior product or the left contraction of along If is a compactly supported vector field and is a manifold with boundary, then Stokes' theorem implies which is a generalization of the divergence theorem. The solenoidal vector fields are those with It follows from the definition of the Lie derivative that the volume form is preserved under the flow of a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving flows. This fact is well-known, for instance, in fluid mechanics