Editing Tensor (section)

=== Tensor densities ===
{{Main|Tensor density}}
Suppose that a homogeneous medium fills {{math|'''R'''<sup>3</sup>}}, so that the density of the medium is described by a single [[scalar (physics)|scalar]] value {{math|''ρ''}} in {{math|kg⋅m<sup>−3</sup>}}.  The mass, in kg, of a region {{math|Ω}} is obtained by multiplying {{math|''ρ''}} by the volume of the region {{math|Ω}}, or equivalently integrating the constant {{math|''ρ''}} over the region:
:<math>m = \int_\Omega \rho\, dx\,dy\,dz ,</math>

where the Cartesian coordinates {{math|''x''}}, {{math|''y''}}, {{math|''z''}} are measured in {{math|m}}.  If the units of length are changed into {{math|cm}}, then the numerical values of the coordinate functions must be rescaled by a factor of 100:
:<math>x' = 100 x,\quad y' = 100y,\quad z' = 100 z .</math>

The numerical value of the density {{math|''ρ''}} must then also transform by {{math|100<sup>−3</sup> m<sup>3</sup>/cm<sup>3</sup>}} to compensate, so that the numerical value of the mass in kg is still given by integral of <math>\rho\, dx\,dy\,dz</math>.  Thus <math>\rho' = 100^{-3}\rho</math> (in units of {{math|kg⋅cm<sup>−3</sup>}}).

More generally, if the Cartesian coordinates {{math|''x''}}, {{math|''y''}}, {{math|''z''}} undergo a linear transformation, then the numerical value of the density {{math|''ρ''}} must change by a factor of the reciprocal of the absolute value of the [[determinant]] of the coordinate transformation, so that the integral remains invariant, by the [[change of variables formula]] for integration. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a [[scalar density]].  To model a non-constant density, {{math|''ρ''}} is a function of the variables {{math|''x''}}, {{math|''y''}}, {{math|''z''}} (a [[scalar field]]), and under a [[curvilinear coordinates|curvilinear]] change of coordinates, it transforms by the reciprocal of the [[Jacobian matrix and determinant|Jacobian]] of the coordinate change. For more on the intrinsic meaning, see ''[[Density on a manifold]]''.

A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of the determinant of the coordinate transition:<ref>{{citation|first=Jan Arnoldus|last=Schouten|author-link=Jan Arnoldus Schouten|chapter-url={{google books |plainurl=y |id=WROiC9st58gC}}|title=Tensor analysis for physicists |chapter=§II.8: Densities}}</ref>
:<math>
  T^{i'_1\dots i'_p}_{j'_1\dots j'_q}[\mathbf{f} \cdot R] =
    \left|\det R\right|^{-w}\left(R^{-1}\right)^{i'_1}_{i_1} \cdots \left(R^{-1}\right)^{i'_p}_{i_p}
  T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}]
  R^{j_1}_{j'_1}\cdots R^{j_q}_{j'_q} .
</math>

Here {{math|''w''}} is called the weight.  In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor.<ref>{{cite book|title=Applications of tensor analysis|first=A.J. |last=McConnell|url={{google books |plainurl=y |id=ZCP0AwAAQBAJ}}|publisher=Dover|orig-year=1957 |isbn=9780486145020 |date=2014|page=28}}</ref>{{sfn|Kay|1988|p=27}} An example of a tensor density is the [[current density]] of [[electromagnetism]].

Under an affine transformation of the coordinates, a tensor transforms by the linear part of the transformation itself (or its inverse) on each index.  These come from the [[rational representation]]s of the general linear group.  But this is not quite the most general linear transformation law that such an object may have: tensor densities are non-rational, but are still [[semisimple]] representations.  A further class of transformations come from the logarithmic representation of the general linear group, a reducible but not semisimple representation,<ref>{{citation|first=Peter |last=Olver|title=Equivalence, invariants, and symmetry|url={{google books |plainurl=y |id=YuTzf61HILAC|page=77}}|page=77|publisher=Cambridge University Press|year=1995 |isbn=9780521478113}}</ref> consisting of an {{math|(''x'', ''y'') ∈ '''R'''<sup>2</sup>}} with the transformation law
:<math>(x, y) \mapsto (x + y\log \left|\det R\right|, y).</math>