0.1 Heuristic definition
0.2 Linear Theory
A tensor density of rank and weight 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.
The simplest example of such a quantity is a scalar density. Under a change of basis , 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 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 coordinate neighborhood, the density field is given by a tensor array of functions. When two neighborhoods overlap, the tensor arrays are related by the change of variable formula
where is the Jacobian matrix of the change of variables.
|Date of creation||2013-03-22 14:55:18|
|Last modified on||2013-03-22 14:55:18|
|Last modified by||rspuzio (6075)|