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:

(ρp(M)T)j1,jmi1,,in=(det(M))pMl1i1Mlnin(M-1)k1j1(M-1)kmjmTj1,jmi1,,in

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:

ρp(S)=(det(M))pS

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,

(ρ1ϵ)j1,jm=(det(M))(M-1)k1j1(M-1)kmjmϵj1,jmi1,,in=ϵk1,km

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

Tj1,jmi1,,in(x)=(det(M))pMl1i1Mlnin(M-1)k1j1(M-1)kmjmTj1,jmi1,,in(y)

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