Tensor density

Definition

In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another, except that it is additionally multiplied or weighted by a power of the Jacobian determinant of the coordinate transition function or its absolute value. A tensor density with a single index is called a vector density. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight are called tensor capacity. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.

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