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| This is a '''glossary of tensor theory'''. For expositions of tensor theory from different points of view, see:
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| * [[Tensor]]
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| * [[Tensor (intrinsic definition)]]
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| * [[Application of tensor theory in engineering science]]
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| For some history of the abstract theory see also [[Multilinear algebra]].
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| ==Classical notation==
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| ;[[Ricci calculus]]
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| The earliest foundation of tensor theory - tensor index notation.<ref>{{citation|title=Méthodes de calcul différentiel absolu et leurs applications|last=Ricci|first=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|url=http://www.springerlink.com/content/u21237446l22rgg7/fulltext.pdf}}</ref>
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| ;[[Tensor#Formulation|Tensor order]]
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| A tensor written in component form is an indexed array. The ''order'' of a tensor is the number of indices required. (The rank of tensor used to mean the order, but now it means something different)
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| ; Rank
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| The rank of the tensor is the minimal number of rank-one tensor that you need to sum up to obtain this higher-rank tensor. Rank-one tensors are given the generalization of outer product to ''m''-vectors where ''m'' is the order of the tensor.
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| ;[[Dyadic tensor]]
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| A ''dyadic'' tensor has order two, and may be represented as a square [[matrix (math)|matrix]]. The conventions ''a<sub>ij</sub>'' , ''a<sup>i</sup><sub>j</sub>'' , and ''a<sup>ij</sup>'' , do have different meanings (the position of the index determines its ''valence'' (variance), in that the first may represent a [[quadratic form]], the second a [[linear transformation]], and the distinction is important in contexts that require tensors that aren't ''orthogonal'' (see below). A ''dyad'' is a tensor such as ''a<sub>i</sub>b<sup>j</sup>'', product component-by-component of rank one tensors. In this case it represents a linear transformation, of [[rank of a linear transformation|rank]] one in the sense of [[linear algebra]] - a clashing terminology that can cause confusion.
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| ;[[Einstein notation]]
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| This notation is based on the understanding that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all 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 generally used in physics and engineering texts, to the extent that if summation is not to be applied it is normal to note that explicitly.
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| ;[[Kronecker delta]]
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| ;[[Levi-Civita symbol]]
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| ;[[Covariance and contravariance of vectors|Covariant]] tensor
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| ;[[Covariance and contravariance of vectors|Contravariant]] tensor
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| 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.
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| ;[[Mixed tensor]]
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| This refers to any tensor with lower and upper indices.
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| '''Cartesian tensor'''
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| 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.
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| ;[[Tensor contraction|Contraction of a tensor]]
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| ;[[Raising and lowering indices]]
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| ;[[Symmetric tensor]]
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| ;[[Antisymmetric tensor]]
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| ;[[Multiple cross products]]
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| ==Algebraic notation==
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| This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.
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| ===Tensor product===
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| If ''v'' and ''w'' are vectors in [[vector space]]s ''V'' and ''W'' respectively, then
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| :<math>v \otimes w \, </math> | |
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| is a tensor in
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| :<math>V \otimes W. \, </math>
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| That is, the ⊗ operation is a [[binary operation]], but it takes values into a fresh space (it is in a strong sense ''external''). The ⊗ operation is a [[bilinear map]]; but no other conditions are applied to it.
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| ===Pure tensor===
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| A pure tensor of ''V'' ⊗ ''W'' is one that is of the form ''v'' ⊗ ''w''
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| 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> ⊗ '''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]].
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| ===Tensor algebra===
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| In the tensor algebra ''T''(''V'') of a vector space ''V'', the operation
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| :<math> \otimes </math>
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| becomes a normal (internal) [[binary operation]]. This is at the cost of ''T''(''V'') being of 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.
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| ===Hodge star operator===
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| ===Exterior power===
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| The [[wedge product]] is the anti-symmetric form of the ⊗ 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''. | |
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| ===Symmetric power, symmetric algebra===
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| This is the invariant way of constructing [[polynomial algebra]]s.
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| ==Applications== | |
| ;[[Metric tensor]]
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| ;[[Strain tensor]]
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| ;[[Stress-energy tensor]]
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| ==Tensor field theory==
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| ;[[Jacobian matrix]]
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| ;[[Tensor field]]
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| ;[[Tensor density]]
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| ;[[Lie derivative]]
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| ;[[Tensor derivative]]
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| ;[[Differential geometry]]
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| ==Abstract algebra==
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| ;[[Tensor product of fields]]
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| This is an operation on fields, that does not always produce a field.
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| ;[[Tensor product of R-algebras]]
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| ;[[Clifford module]]
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| A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.
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| ;[[Tor functor]]s'''
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| 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.
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| ;[[Symbolic method of invariant theory]]
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| ;[[Derived category]]
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| ;[[Grothendieck's six operations]]
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| These are ''highly'' abstract approaches used in some parts of geometry.
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| ==Spinors==
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| See:
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|
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| ;[[Spin group]]
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| ;[[Spin-c group]]
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| ;[[Spinor]]
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| ;[[Pin group]]
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| ;[[Pinor]]s
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| ;[[Spinor field]]
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| ;[[Killing spinor]]
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| ;[[Spin manifold]]
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| ==References==
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| {{reflist}}
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| ==Books==
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| * {{citation | last1=Bishop|first1=R.L.|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}}
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| *{{cite book
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| | last = Danielson
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| | first = Donald A.
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| | title = Vectors and Tensors in Engineering and Physics
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| | edition = 2/e
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| | year= 2003
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| | publisher = Westview (Perseus)
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| | isbn = 978-0-8133-4080-7
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| }}
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| *{{cite book
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| | last = Dimitrienko
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| | first = Yuriy
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| | title = Tensor Analysis and Nonlinear Tensor Functions
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| | year= 2002
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| | publisher = Kluwer Academic Publishers (Springer)
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| | url = http://books.google.com/books?as_isbn=140201015X
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| | isbn = 1-4020-1015-X
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| }}
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| *{{cite book
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| | last = Lovelock
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| | first = David
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| | coauthors = Hanno Rund
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| | title = Tensors, Differential Forms, and Variational Principles
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| | year= 1989
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| | publisher = Dover
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| | isbn = 978-0-486-65840-7
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| | origyear = 1975
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| }}
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| * {{cite book |author=Synge J.L., Schild A. |title=Tensor Calculus |publisher=first Dover Publications 1978 edition |year= 1949
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| |isbn=978-0-486-63612-2}}
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| {{tensors}}
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| {{DEFAULTSORT:Glossary Of Tensor Theory}}
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| [[Category:Glossaries of mathematics|Tensor theory]]
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| [[Category:Tensors| ]]
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| [[Category:Multilinear algebra| ]]
| |
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