Density on a manifold

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the density bundle. An element of the density bundle at x is a function that assigns a volume for the parallelotope spanned by the n given tangent vectors at x. From the operational point of view, a density is a collection of functions on coordinate charts which become multiplied by the absolute value of the Jacobian determinant in the change of coordinates. Densities can be generalized into s-densities, whose coordinate representations become multiplied by the s-th power of the absolute value of the jacobian determinant. On an oriented manifold, 1-densities can be canonically identified with the n-forms on M. On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor product of the orientation bundle of M and the n-th exterior product bundle of T^∗M (see pseudotensor). In general, there does not exist a natural concept of a "volume" for a parallelotope generated by vectors v1, ..., vn in a n-dimensional vector space V. However, if one wishes to define a function μ : V × ... × V → R that assigns a volume for any such parallelotope, it should satisfy the following properties: If any of the vectors vk is multiplied by λ ∈ R, the volume should be multiplied by |λ|. If any linear combination of the vectors v1, ..., vj−1, vj+1, ..., vn is added to the vector vj, the volume should stay invariant. These conditions are equivalent to the statement that μ is given by a translation-invariant measure on V, and they can be rephrased as Any such mapping μ : V × ... × V → R is called a density on the vector space V. Note that if (v1, ..., vn) is any basis for V, then fixing μ(v1, ..., vn) will fix μ entirely; it follows that the set Vol(V) of all densities on V forms a one-dimensional vector space. Any n-form ω on V defines a density on V by The set Or(V) of all functions o : V × .