volume form

In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing top-dimensional form (i.e., a differential form of top degree). Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle , that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.