Volume form

It is a differential form of degree equal to the differentiable manifold dimension. A manifold admits a nowhere-vanishing volume form if and only if it is an oriented manifold . On non-orientable manifolds, one may instead define the weaker notion of a density.

A volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. Keep an eye: not every measure comes from a volume form (see here)