tensor density

0.1 Heuristic definition

A tensor density is a quantity whose transformation law under change of basis involves the determinantDlmfMathworldPlanetmath of the transformation matrix (as opposed to a tensor, whose transformation law does not involve the determinant).

0.2 Linear Theory

For any real number p, we may define a representation ρp of the group GL(k) on the vector spaceMathworldPlanetmath of tensor arrays of rank m,n as follows:


A tensor density T of rank m,n and weight p is an element of the vector space on which this representation acts.

Note that if the weight equals zero, the concept of tensor density reduces to that of a tensor.

0.3 Examples

The simplest example of such a quantity is a scalar density. Under a change of basis yi=Mjixj, a scalar density transforms as follows:


An important example of a tensor density is the Levi-Civita permutation symbol. It is a density of weight 1 because, under a change of coordinates,


0.4 Tensor Densities on Manifolds

As with tensors, it is possible to define tensor density fields on manifolds. On each coordinatePlanetmathPlanetmath neighborhoodMathworldPlanetmathPlanetmath, the density field is given by a tensor array of functionsMathworldPlanetmath. When two neighborhoods overlap, the tensor arrays are related by the change of variable formula


where M is the Jacobian matrix of the change of variables.

Title tensor density
Canonical name TensorDensity
Date of creation 2013-03-22 14:55:18
Last modified on 2013-03-22 14:55:18
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 12
Author rspuzio (6075)
Entry type Definition
Classification msc 15A72
Synonym density
Related topic tensor