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 (see tensor field), except that it is additionally multiplied or weighted by a power W {\displaystyle W} 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 W {\displaystyle W} 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.
Motivation In physics and related fields, it is often useful to work with the components of an algebraic object rather than the object itself. An example would be decomposing a vector into a sum of basis vectors weighted by some coefficients such as
v → = c 1 e → 1 + c 2 e → 2 + c 3 e → 3 {\displaystyle {\vec {v}}=c_{1}{\vec {e}}_{1}+c_{2}{\vec {e}}_{2}+c_{3}{\vec {e}}_{3}} where v → {\displaystyle {\vec {v}}} is a vector in 3-dimensional Euclidean space, c i ∈ R 1 and e → i {\displaystyle c_{i}\in \mathbb {R} ^{1}{\text{ and }}{\vec {e}}_{i}} are the usual standard basis vectors in Euclidean space. This is usually necessary for computational purposes, and can often be insightful when algebraic objects represent complex abstractions but their components have concrete interpretations. However, with this identification, one has to be careful to track changes of the underlying basis in which the quantity is expanded; it may in the course of a computation become expedient to change the basis while the vector v → {\displaystyle {\vec {v}}} remains fixed in physical space. More generally, if an algebraic object represents a geometric object, but is expressed in terms of a particular basis, then it is necessary to, when the basis is changed, also change the representation. Physicists will often call this representation of a geometric object a tensor if it transforms under a sequence of linear maps given a linear change of basis (although confusingly others call the underlying geometric object which hasn't changed under the coordinate transformation a "tensor", a convention this article strictly avoids). In general there are representations which transform in arbitrary ways depending on how the geometric invariant is reconstructed from the representation. In certain special cases it is convenient to use representations which transform almost like tensors, but with an additional, nonlinear factor in the transformation. A prototypical example is a matrix representing the cross product (area of spanned parallelogram) on R 2 . {\displaystyle \mathbb {R} ^{2}.} The representation is given by in the standard basis by
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