Manifolds | Differential geometry

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). (Wikipedia).

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Group representation | Lebesgue measure | Differential form | Tensor product | Pseudotensor | Associated bundle | Frame bundle | Tensor density | General linear group | Differentiable manifold | Mathematics | Conformal geometry | Distribution (mathematics) | Parallelepiped | Section (fiber bundle) | Partition of unity | Vector bundle | Integral | Integration by substitution | Atlas (topology) | Differential geometry | Line bundle | Lp space | Radon measure