Tensor: Difference between revisions

Revision as of 17:53, 18 June 2025

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=== Tensor densities ===

{{Main|Tensor density}}

Suppose that a homogeneous medium fills {{math|<math>\mathbb{'''R}</math>'''³}}, so that the density of the medium is described by a single [[scalar (physics)|scalar]] value {{math|''ρ''}} in {{math|kg⋅m⁻³}}. 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>

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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'') ∈ <math>\mathbb{'''R}</math>'''²}} with the transformation law

:<math>(x, y) \mapsto (x + y\log \left|\det R\right|, y).</math>