tensor density
0.1 Heuristic definition
A tensor density is a quantity whose transformation law under change of basis involves the determinant 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 , we may define a representation of the group on the vector space of tensor arrays of rank as follows:
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.
0.3 Examples
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.
Title | tensor density |
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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 |