Editing Tensor field (section)

== Generalizations ==
=== Tensor densities ===
{{main|Tensor density}}
The concept of a tensor field can be generalized by considering objects that transform differently.  An object that transforms as an ordinary tensor field under coordinate transformations, except that it is also multiplied by the determinant of the [[Jacobian matrix and determinant|Jacobian]] of the inverse coordinate transformation to the ''w''th power, is called a tensor density with weight ''w''.<ref>{{Springer|id=T/t092390|title=Tensor density}}</ref>  Invariantly, in the language of multilinear algebra, one can think of tensor densities as [[multilinear map]]s taking their values in a [[density bundle]] such as the (1-dimensional) space of ''n''-forms (where ''n'' is the dimension of the space), as opposed to taking their values in just '''R'''.  Higher "weights" then just correspond to taking additional tensor products with this space in the range.

A special case are the scalar densities.  Scalar 1-densities are especially important because it makes sense to define their integral over a manifold.  They appear, for instance, in the [[Einstein–Hilbert action]] in general relativity.  The most common example of a scalar 1-density is the [[volume element]], which in the presence of a metric tensor ''g'' is the square root of its [[determinant]] in coordinates, denoted <math>\sqrt{\det g}</math>.  The metric tensor is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition:
: <math>\det(g') = \left(\det\frac{\partial x}{\partial x'}\right)^2\det(g),</math>
which is the transformation law for a scalar density of weight&nbsp;+2.

More generally, any tensor density is the product of an ordinary tensor with a scalar density of the appropriate weight.  In the language of [[vector bundle]]s, the determinant bundle of the [[tangent bundle]] is a [[line bundle]] that can be used to 'twist' other bundles ''w'' times.  While locally the more general transformation law can indeed be used to recognise these tensors, there is a global question that arises, reflecting that in the transformation law one may write either the Jacobian determinant, or its absolute value.  Non-integral powers of the (positive) transition functions of the bundle of densities make sense, so that the weight of a density, in that sense, is not restricted to integer values.  Restricting to changes of coordinates with positive Jacobian determinant is possible on [[orientable manifold]]s, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of ''n''-forms are distinct.  For more on the intrinsic meaning, see ''[[Density on a manifold]]''.