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Hometensor density

## Primary tabs

# 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:

$(\rho_{p}(M)T)^{{i_{1},\ldots,i_{n}}}_{{j_{1},\ldots j_{m}}}=(\mathop{\rm det}% (M))^{p}M^{{i_{1}}}_{{l_{1}}}\cdots M^{{i_{n}}}_{{l_{n}}}(M^{{-1}})_{{k_{1}}}^% {{j_{1}}}\cdots(M^{{-1}})_{{k_{m}}}^{{j_{m}}}T^{{i_{1},\ldots,i_{n}}}_{{j_{1},% \ldots j_{m}}}$ |

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

# 0.3 Examples

The simplest example of such a quantity is a scalar density. Under a change of basis $y^{i}=M^{i}_{j}x^{j}$, a scalar density transforms as follows:

$\rho_{p}(S)=(\mathop{\rm det}(M))^{p}S$ |

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,

$(\rho_{1}\epsilon)_{{j_{1},\ldots j_{m}}}=(\mathop{\rm det}(M))(M^{{-1}})_{{k_% {1}}}^{{j_{1}}}\cdots(M^{{-1}})_{{k_{m}}}^{{j_{m}}}\epsilon^{{i_{1},\ldots,i_{% n}}}_{{j_{1},\ldots j_{m}}}=\epsilon_{{k_{1},\ldots k_{m}}}$ |

# 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

$T^{{i_{1},\ldots,i_{n}}}_{{j_{1},\ldots j_{m}}}(x)=(\mathop{\rm det}(M))^{p}M^% {{i_{1}}}_{{l_{1}}}\cdots M^{{i_{n}}}_{{l_{n}}}(M^{{-1}})_{{k_{1}}}^{{j_{1}}}% \cdots(M^{{-1}})_{{k_{m}}}^{{j_{m}}}T^{{i_{1},\ldots,i_{n}}}_{{j_{1},\ldots j_% {m}}}(y)$ |

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

## Mathematics Subject Classification

15A72*no label found*

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