Tensor density

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A geometric object described in a coordinate system by components , , transforming under a change of coordinates according to the formula

where . The number is called the weight of the tensor density. When , the tensor density is a tensor (cf. Tensor on a vector space). Concepts such as type, valency, covariance, contravariance, etc. are introduced similar to the corresponding tensor concepts. Tensor densities of types and are called vector densities. Tensor densities of type are called scalar densities.


A tensor density as defined above is also called a relative tensor. One distinguishes between odd relative tensors of weight , which transform as above, and even relative tensors, which transform according to the same formula except that is replaced by its absolute value . In [a2] an even tensor density is simply called a "tensor density" and an odd one is called a tensor -density.


[a1] M. Spivak, "A comprehensive introduction to differential geometry" , I , Publish or Perish (1970) pp. 437ff
[a2] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. 12 (Translated from German)
How to Cite This Entry:
Tensor density. L.P. Kuptsov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098