Editing Glossary of tensor theory

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This is a '''glossary of tensor theory'''. For expositions of [[tensor theory]] from different points of view, see:

* [[Tensor]]
* [[Tensor (intrinsic definition)]]
* [[Application of tensor theory in engineering science]]

For some history of the abstract theory see also [[multilinear algebra]].

==Classical notation==

;[[Ricci calculus]]
:The earliest foundation of tensor theory – tensor index notation.<ref>{{citation|title=Méthodes de calcul différentiel absolu et leurs applications|last1=Ricci|first1=Gregorio|author-link=Gregorio Ricci-Curbastro|last2=Levi-Civita|first2=Tullio|journal=Mathematische Annalen|publisher=Springer|volume=54|issue=1–2|date=March 1900|pages=125–201|doi=10.1007/BF01454201|s2cid=120009332|url=https://zenodo.org/record/1428270|language=fr|trans-title=Absolute differential calculation methods & their applications}}</ref>

;[[Tensor order|Order of a tensor]]
:The components of a tensor with respect to a basis is an indexed array. The ''order'' of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term ''degree'' or ''rank''.

;[[Tensor rank|Rank of a tensor]]
:The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor.  A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order.

;[[Dyadic tensor]]
:A ''dyadic'' tensor is a tensor of order two, and may be represented as a square [[matrix (mathematics)|matrix]]. In contrast, a ''dyad'' is specifically a dyadic tensor of rank one.

;[[Einstein notation]]
:This notation is based on the understanding that whenever a multidimensional array  contains a repeated index letter, the default interpretation is that the product is summed over all permitted values of the index. For example, if ''a<sub>ij</sub>'' is a matrix, then under this convention ''a<sub>ii</sub>'' is its [[trace (matrix)|trace]]. The Einstein convention is widely used in physics and engineering texts, to the extent that if summation is not to be applied, it is normal to note that explicitly.

;[[Kronecker delta]]

;[[Levi-Civita symbol]]

;[[Covariance and contravariance of vectors|Covariant]] tensor

;[[Covariance and contravariance of vectors|Contravariant]] tensor
:The classical interpretation is by components. For example, in the differential form ''a<sub>i</sub>dx<sup>i</sup>'' the '''components''' ''a<sub>i</sub>'' are a covariant vector. That means all indices are lower; contravariant means all indices are upper.

;[[Mixed tensor]]
:This refers to any tensor that has both lower and upper indices.

;Cartesian tensor
:Cartesian tensors are widely used in various branches of [[continuum mechanics]], such as [[fluid mechanics]] and [[Elasticity (physics)|elasticity]]. In classical [[continuum mechanics]], the space of interest is usually 3-dimensional [[Euclidean space]], as is the tangent space at each point.  If we restrict the local coordinates to be [[Cartesian coordinates]] with the same scale centered at the point of interest, the [[metric tensor]] is the [[Kronecker delta]].  This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and [[Tensor density|tensor densities]].  All [[Cartesian tensor|Cartesian-tensor]] indices are written as subscripts. [[Cartesian tensor]]s achieve considerable computational simplification at the cost of generality and of some theoretical insight.

;[[Tensor contraction|Contraction of a tensor]]

;[[Raising and lowering indices]]

;[[Symmetric tensor]]

;[[Antisymmetric tensor]]

;[[Multiple cross products]]

==Algebraic notation==
This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.

;Tensor product
:If ''v'' and ''w'' are vectors in [[vector space]]s ''V'' and ''W'' respectively, then
::<math>v \otimes w </math>
:is a tensor in
::<math>V \otimes W. </math>
:That is, the &otimes; operation is a [[binary operation]], but it takes values into a fresh space (it is in a strong sense ''external''). The &otimes; operation is a [[bilinear map]]; but no other conditions are applied to it.

;Pure tensor
:A pure tensor of ''V'' &otimes; ''W'' is one that is of the form ''v'' &otimes; ''w''.
:It could be written dyadically ''a<sup>i</sup>b<sup>j</sup>'', or more accurately ''a<sup>i</sup>b<sup>j</sup>'' '''e'''<sub>''i''</sub> &otimes; '''f'''<sub>''j''</sub>, where the '''e'''<sub>''i''</sub> are a basis for ''V'' and the '''f'''<sub>''j''</sub> a basis for ''W''. Therefore, unless ''V'' and ''W'' have the same dimension, the array of components need not be square. Such ''pure'' tensors are not generic: if both ''V'' and ''W'' have dimension greater than 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure. For more see [[Segre embedding]].

;Tensor algebra
:In the tensor algebra ''T''(''V'') of a vector space ''V'', the operation <math> \otimes </math> becomes a normal (internal) [[binary operation]]. A consequence is that ''T''(''V'') has infinite dimension unless ''V'' has dimension 0. The [[free algebra]] on a set ''X'' is for practical purposes the same as the tensor algebra on the vector space with ''X'' as basis.

;Hodge star operator

;Exterior power

:The [[wedge product]] is the anti-symmetric form of the &otimes; operation. The quotient space of ''T''(''V'') on which it becomes an internal operation is the ''[[exterior algebra]]'' of ''V''; it is a [[graded algebra]], with the graded piece of weight ''k'' being called the ''k''-th '''exterior power''' of ''V''.

;Symmetric power, symmetric algebra
:This is the invariant way of constructing [[polynomial algebra]]s.

==Applications==
;[[Metric tensor]]

;[[Strain tensor]]

;[[Stress–energy tensor]]

==Tensor field theory==
;[[Jacobian matrix]]

;[[Tensor field]]

;[[Tensor density]]

;[[Lie derivative]]

;[[Tensor derivative]]

;[[Differential geometry]]

==Abstract algebra==
;[[Tensor product of fields]]
:This is an operation on fields, that does not always produce a field.

;[[Tensor product of R-algebras]]

;[[Clifford module]]
:A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.

;[[Tor functor]]s
:These are the [[derived functor]]s of the tensor product, and feature strongly in [[homological algebra]]. The name comes from the [[torsion subgroup]] in [[abelian group]] theory.

;[[Symbolic method of invariant theory]]

;[[Derived category]]

;[[Grothendieck's six operations]]
:These are ''highly'' abstract approaches used in some parts of geometry.

==Spinors==

See:
 
;[[Spin group]]

;[[Spin-c group]]

;[[Spinor]]

;[[Pin group]]

;[[Pinor]]s

;[[Spinor field]]

;[[Killing spinor]]

;[[Spin manifold]]

==References==
{{reflist}}

==Books==
* {{citation|last1=Bishop|first1=R.L.|author1-link=Richard L. Bishop|last2=Goldberg|first2=S.I.|title=Tensor Analysis on Manifolds|publisher=The Macmillan Company|year=1968|edition=First Dover 1980|isbn=0-486-64039-6|url=https://archive.org/details/tensoranalysison00bish}}
*{{cite book
 | last = Danielson
 | first = Donald A.
 | author-link1 = Donald A. Danielson
 | title = Vectors and Tensors in Engineering and Physics
 | edition = 2/e
 | year=  2003
 | publisher = Westview (Perseus)
 | isbn = 978-0-8133-4080-7
 }}
*{{cite book
 | last = Dimitrienko
 | first = Yuriy
 | title = Tensor Analysis and Nonlinear Tensor Functions
 | year=  2002
 | publisher = Kluwer Academic Publishers (Springer)
 | url = https://books.google.com/books?as_isbn=140201015X
 | isbn = 1-4020-1015-X
 }}
*{{cite book
 | last = Lovelock
 | first = David
 |author2=Hanno Rund
 | title = Tensors, Differential Forms, and Variational Principles
 | year=  1989
 | publisher = Dover
 | isbn = 978-0-486-65840-7
 | orig-year = 1975
 }}
* {{cite book |first1=John L|last1=Synge|author-link1=John L. Synge|first2=Alfred|last2=Schild|author-link2=Alfred Schild| title=Tensor Calculus |publisher=[[Dover Publications]] 1978 edition |year=1949 |isbn=978-0-486-63612-2 |url=https://archive.org/details/tensorcalculus00syng }}

{{tensors}}

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[[Category:Glossaries of mathematics|Tensor theory]]
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