# 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 $p$, we may define a representation ${\rho}_{p}$ of the group $GL({\mathbb{R}}^{k})$ on the vector space^{} of tensor arrays of rank $m,n$ as follows:

$${({\rho}_{p}(M)T)}_{{j}_{1},\mathrm{\dots}{j}_{m}}^{{i}_{1},\mathrm{\dots},{i}_{n}}={(det(M))}^{p}{M}_{{l}_{1}}^{{i}_{1}}\mathrm{\cdots}{M}_{{l}_{n}}^{{i}_{n}}{({M}^{-1})}_{{k}_{1}}^{{j}_{1}}\mathrm{\cdots}{({M}^{-1})}_{{k}_{m}}^{{j}_{m}}{T}_{{j}_{1},\mathrm{\dots}{j}_{m}}^{{i}_{1},\mathrm{\dots},{i}_{n}}$$ |

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 ${y}^{i}={M}_{j}^{i}{x}^{j}$, a scalar density transforms as follows:

$${\rho}_{p}(S)={(det(M))}^{p}S$$ |

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,

$${({\rho}_{1}\u03f5)}_{{j}_{1},\mathrm{\dots}{j}_{m}}=(det(M)){({M}^{-1})}_{{k}_{1}}^{{j}_{1}}\mathrm{\cdots}{({M}^{-1})}_{{k}_{m}}^{{j}_{m}}{\u03f5}_{{j}_{1},\mathrm{\dots}{j}_{m}}^{{i}_{1},\mathrm{\dots},{i}_{n}}={\u03f5}_{{k}_{1},\mathrm{\dots}{k}_{m}}$$ |

## 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

$${T}_{{j}_{1},\mathrm{\dots}{j}_{m}}^{{i}_{1},\mathrm{\dots},{i}_{n}}(x)={(det(M))}^{p}{M}_{{l}_{1}}^{{i}_{1}}\mathrm{\cdots}{M}_{{l}_{n}}^{{i}_{n}}{({M}^{-1})}_{{k}_{1}}^{{j}_{1}}\mathrm{\cdots}{({M}^{-1})}_{{k}_{m}}^{{j}_{m}}{T}_{{j}_{1},\mathrm{\dots}{j}_{m}}^{{i}_{1},\mathrm{\dots},{i}_{n}}(y)$$ |

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 |