Tensor density

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A tensor density transforms as a tensor, except that it is additionally multiplied or weighted by a power of the Jacobian determinant.

For example, a rank-3 tensor density of weight W transforms as:

$A_{ijk}^\prime = \begin{vmatrix} \alpha \end{vmatrix}^W \cdot \alpha_i^l \cdot \alpha_j^m \cdot \alpha_k^n \cdot A_{lmn}$

where A is the rank-3 tensor density, A′ is the transformed tensor density, and α is the transformation matrix. Here $\begin{vmatrix} \alpha \end{vmatrix}$ is the Jacobian determinant.

A tensor density of weight zero is an ordinary tensor.

A distinction is made between odd tensor densities, in which (as here) the term attributable to the determinant may be negative, and even tensor densities which have a power of the absolute value of the determinant, or an even power of it, in the transformation rule.