Editing Tensor (section)

== Generalizations ==

=== Tensor products of vector spaces ===
The vector spaces of a [[tensor product]] need not be the same, and sometimes the elements of such a more general tensor product are called "tensors".  For example, an element of the tensor product space {{math|''V'' ⊗ ''W''}} is a second-order "tensor" in this more general sense,<ref name="Maia2011">{{cite book|first=M. D. |last=Maia|title=Geometry of the Fundamental Interactions: On Riemann's Legacy to High Energy Physics and Cosmology|url={{google books |plainurl=y |id=wEWw_vGBDW8C|page=48}}
|year=2011|publisher=Springer |isbn=978-1-4419-8273-5|page=48}}</ref> and an order-{{math|''d''}} tensor may likewise be defined as an element of a tensor product of {{math|''d''}} different vector spaces.<ref name="Hogben2013">{{cite book|url={{google books |plainurl=y |id=Er7MBQAAQBAJ|page=7}}|title=Handbook of Linear Algebra |publisher=CRC Press|year=2013|isbn=978-1-4665-0729-6|editor-last=Hogben|editor-first=Leslie|editor-link= Leslie Hogben |edition=2nd|pages=15–7}}</ref>  A type {{math|(''n'', ''m'')}} tensor, in the sense defined previously, is also a tensor of order {{math|''n'' + ''m''}} in this more general sense.  The concept of tensor product [[tensor product of modules|can be extended]] to arbitrary [[module over a ring|modules over a ring]].

=== Tensors in infinite dimensions ===
The notion of a tensor can be generalized in a variety of ways to [[Dimension (vector space)|infinite dimensions]].  One, for instance, is via the [[tensor product of Hilbert spaces|tensor product]] of [[Hilbert space]]s.<ref>{{cite journal | last1 = Segal | first1 = I. E. | date=January 1956 | title = Tensor Algebras Over Hilbert Spaces. I | journal = [[Transactions of the American Mathematical Society]] | volume = 81 | issue = 1 | pages = 106–134 | jstor = 1992855 | doi = 10.2307/1992855 | doi-access = free }}</ref>  Another way of generalizing the idea of tensor, common in [[Nonlinear system|nonlinear analysis]], is via the [[#As multilinear maps|multilinear maps definition]] where instead of using finite-dimensional vector spaces and their [[algebraic dual]]s, one uses infinite-dimensional [[Banach space]]s and their [[continuous dual]].<ref>{{cite book
|last1=Abraham |first1=Ralph
|last2=Marsden |first2=Jerrold E.
|last3=Ratiu |first3=Tudor S.
|chapter-url={{google books |plainurl=y |id=dWHet_zgyCAC}}
|title=Manifolds, Tensor Analysis and Applications
|edition=2nd |series=Applied Mathematical Sciences |volume=75
|date= February 1988
|publisher=Springer
|isbn=978-0-387-96790-5
|oclc= 18562688
|pages=338–9
|chapter=5. Tensors
|quote=Elements of T<sup>r</sup><sub>s</sub> are called tensors on E, [...].
}}</ref>  Tensors thus live naturally on [[Banach manifold]]s<ref>{{cite book | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Differential manifolds |url={{google books |plainurl=y |id=dn7rBwAAQBAJ}}| publisher=[[Addison-Wesley]] | year=1972 |isbn= 978-0-201-04166-8 }}</ref> and [[Fréchet manifold]]s.

=== 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>

=== Geometric objects ===
The transformation law for a tensor behaves as a [[functor]] on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as [[local diffeomorphism]]s).  This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes.<ref>{{cite journal |last1=Haantjes |first1=J. |author2-link=Gerard Laman |last2=Laman |first2=G.  |title=On the definition of geometric objects. I |journal=Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen: Series A: Mathematical Sciences |volume=56 |issue=3 |pages=208–215 |date=1953 }}</ref>  Examples of objects obeying more general kinds of transformation laws are [[jet (mathematics)|jets]] and, more generally still, [[natural bundle]]s.<ref>{{citation|first=Albert|last=Nijenhuis|author-link=Albert Nijenhuis|chapter-url=http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0463.0469.ocr.pdf|chapter=Geometric aspects of formal differential operations on tensor fields|title=Proc. Internat. Congress Math.(Edinburgh, 1958)|year=1960|publisher=Cambridge University Press|pages=463–9|access-date=2017-10-26|archive-date=2017-10-27|archive-url=https://web.archive.org/web/20171027025011/http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0463.0469.ocr.pdf|url-status=dead}}.</ref><ref>{{citation|url=https://projecteuclid.org/download/pdf_1/euclid.jdg/1214430830|title=On the theory of geometric objects|first=Sarah |last=Salviori|journal=[[Journal of Differential Geometry]]|year=1972|volume=7|issue=1–2|pages=257–278|doi=10.4310/jdg/1214430830|doi-access=free}}.</ref>

=== Spinors ===
{{Main|Spinor}}
When changing from one [[orthonormal basis]] (called a ''frame'') to another by a rotation, the components of a tensor transform by that same rotation.  This transformation does not depend on the path taken through the space of frames.  However, the space of frames is not [[simply connected]] (see [[orientation entanglement]] and [[plate trick]]): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other.  It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1.<ref>{{cite book|title=The road to reality: a complete guide to the laws of our universe|url={{google books |plainurl=y |id=VWTNCwAAQBAJ|page=203}}|first=Roger|last=Penrose|author-link=Roger Penrose|publisher=Knopf|year=2005|pages=203–206}}</ref>  A [[spinor]] is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.<ref>{{cite book|first=E. |last=Meinrenken|title=Clifford Algebras and Lie Theory|chapter=The spin representation|series=Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics|volume=58|pages=49–85|doi=10.1007/978-3-642-36216-3_3|publisher=Springer |year=2013|isbn=978-3-642-36215-6}}</ref><ref>{{citation|first=S. H.|last=Dong |title=Wave Equations in Higher Dimensions|chapter=2. Special Orthogonal Group SO(''N'')|publisher=Springer|year=2011|pages=13–38}}</ref>

Spinors are elements of the [[spin representation]] of the rotation group, while tensors are elements of its [[tensor representation]]s. Other [[classical group]]s have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.