Lebesgue–Stieltjes integration: Difference between revisions
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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|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> | |||
;[[Tensor#Formulation|Tensor order]] | |||
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) | |||
; Rank | |||
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. | |||
;[[Dyadic tensor]] | |||
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. | |||
;[[Einstein notation]] | |||
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. | |||
;[[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 with 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 ⊗ 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. | |||
===Pure tensor=== | |||
A pure tensor of ''V'' ⊗ ''W'' is one that is of the form ''v'' ⊗ ''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> ⊗ '''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]]. 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. | |||
===Hodge star operator=== | |||
===Exterior power=== | |||
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''. | |||
===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.|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}} | |||
*{{cite book | |||
| last = Danielson | |||
| first = Donald A. | |||
| 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 = http://books.google.com/books?as_isbn=140201015X | |||
| isbn = 1-4020-1015-X | |||
}} | |||
*{{cite book | |||
| last = Lovelock | |||
| first = David | |||
| coauthors = Hanno Rund | |||
| title = Tensors, Differential Forms, and Variational Principles | |||
| year= 1989 | |||
| publisher = Dover | |||
| isbn = 978-0-486-65840-7 | |||
| origyear = 1975 | |||
}} | |||
* {{cite book |author=Synge J.L., Schild A. |title=Tensor Calculus |publisher=first Dover Publications 1978 edition |year= 1949 | |||
|isbn=978-0-486-63612-2}} | |||
{{tensors}} | |||
{{DEFAULTSORT:Glossary Of Tensor Theory}} | |||
[[Category:Glossaries of mathematics|Tensor theory]] | |||
[[Category:Tensors| ]] | |||
[[Category:Multilinear algebra| ]] | |||
Revision as of 05:41, 1 January 2014
This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:
For some history of the abstract theory see also Multilinear algebra.
Classical notation
The earliest foundation of tensor theory - tensor index notation.[1]
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)
- Rank
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.
A dyadic tensor has order two, and may be represented as a square matrix. The conventions aij , aij , and aij , 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 aibj, product component-by-component of rank one tensors. In this case it represents a linear transformation, of rank one in the sense of linear algebra - a clashing terminology that can cause confusion.
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 aij is a matrix, then under this convention aii is its 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.
- Covariant tensor
- Contravariant tensor
The classical interpretation is by components. For example in the differential form aidxi the components ai are a covariant vector. That means all indices are lower; contravariant means all indices are upper.
This refers to any tensor with lower and upper indices.
Cartesian tensor
Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and 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 densities. All Cartesian-tensor indices are written as subscripts. Cartesian tensors achieve considerable computational simplification at the cost of generality and of some theoretical insight.
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 spaces V and W respectively, then
is a tensor in
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.
Pure tensor
A pure tensor of V ⊗ W is one that is of the form v ⊗ w
It could be written dyadically aibj, or more accurately aibj ei ⊗ fj, where the ei are a basis for V and the fj 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
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.
Hodge star operator
Exterior power
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.
Symmetric power, symmetric algebra
This is the invariant way of constructing polynomial algebras.
Applications
Tensor field theory
Abstract algebra
This is an operation on fields, that does not always produce a field.
A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.
These are the derived functors of the tensor product, and feature strongly in homological algebra. The name comes from the torsion subgroup in abelian group theory.
These are highly abstract approaches used in some parts of geometry.
Spinors
See:
References
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Books
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
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