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Difference between revisions of "Affine tensor"

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An element of the [[Tensor product|tensor product]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111201.png" /> copies of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111202.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111204.png" /> copies of the dual vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111205.png" />. Such a tensor is said to be of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111206.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111207.png" /> defining the valency, or degree, of the tensor. Having chosen a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111208.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a0111209.png" />, one defines an affine tensor of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a01112010.png" /> with the aid of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a01112011.png" /> components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a01112012.png" /> which transform as a result of a change of basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a01112013.png" /> according to the formula
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An element of the [[tensor product]] of $p$ copies of an $n$-dimensional vector space $E$ and $q$ copies of the dual vector space $E^*$. Such a tensor is said to be of type $(p,q)$, the number $p+q$ defining the valency, or degree, of the tensor. Having chosen a basis $\{e_i\}$ in $E$, one defines an affine tensor of type $(p,q)$ with the aid of $n^{p+q}$ components $T^{i_1\ldots i_p}_{j_1\ldots j_p}$ which transform as a result of a change of basis $e'_i = A_i^s e_s$ according to the formula
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$$
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T'^{i_1\ldots i_p}_{j_1\ldots j_p} = A'^{i_1}_{s_1} \cdots A'^{i_p}_{s_p} A^{t_1}_{j_1} \cdots  A^{t_q}_{j_q} T^{i_1\ldots i_p}_{j_1\ldots j_p}
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$$
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where $A^s_j A'^i_s = \delta^i_j$. It is usually said that the tensor components undergo a ''contravariant'' transformation with respect to the upper indices, and a ''covariant'' transformation with respect to the lower.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a01112014.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011120/a01112015.png" />. It is usually said that the tensor components undergo a contravariant transformation with respect to the upper indices, and a covariant transformation with respect to the lower.
 
  
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====Comments====
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An affine tensor as described above is commonly called simply a tensor.
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====References====
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. Dubrovin,  A.T. Fomenko,  S.P. Novikov,  "Modern geometry - methods and applications" , Springer  (1984)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  W.H. Greub,  "Multilinear algebra" , Springer  (1967)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  C.T.J. Dodson,  T. Poston,  "Tensor geometry" , Pitman  (1977)</TD></TR>
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</table>
  
  
 
====Comments====
 
====Comments====
An affine tensor as described above is commonly called simply a tensor.
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The tensor $\delta^i_j$ is the [[Kronecker delta]] tensor.
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An ''isotropic tensor'' is one for which the components are unchanged under change of basis.  The Kronecker delta tensor is isotropic; in dimension $n=3$ the tensor $\epsilon_{ijk}$ defined by $\epsilon_{123} = \epsilon_{231} = \epsilon_{312} = 1$, $\epsilon_{321} = \epsilon_{213} = \epsilon_{132} = -1$, all other values zero, of order 3, is isotropic. 
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See also: [[Contravariant tensor]], [[Covariant tensor]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. Dubrovin,  A.T. Fomenko,  S.P. Novikov,  "Modern geometry - methods and applications" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.H. Greub,  "Multilinear algebra" , Springer  (1967)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.T.J. Dodson,  T. Poston,  "Tensor geometry" , Pitman  (1977)</TD></TR></table>
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<table>
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  H. Jeffreys ''Cartesian tensors'' (7th imp.) Cambridge University Press [1931] (1969) ISBN 0-521-09191-8 {{ZBL|57.0974.01}}</TD></TR>
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</table>
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{{TEX|done}}

Revision as of 19:27, 13 January 2016

An element of the tensor product of $p$ copies of an $n$-dimensional vector space $E$ and $q$ copies of the dual vector space $E^*$. Such a tensor is said to be of type $(p,q)$, the number $p+q$ defining the valency, or degree, of the tensor. Having chosen a basis $\{e_i\}$ in $E$, one defines an affine tensor of type $(p,q)$ with the aid of $n^{p+q}$ components $T^{i_1\ldots i_p}_{j_1\ldots j_p}$ which transform as a result of a change of basis $e'_i = A_i^s e_s$ according to the formula $$ T'^{i_1\ldots i_p}_{j_1\ldots j_p} = A'^{i_1}_{s_1} \cdots A'^{i_p}_{s_p} A^{t_1}_{j_1} \cdots A^{t_q}_{j_q} T^{i_1\ldots i_p}_{j_1\ldots j_p} $$ where $A^s_j A'^i_s = \delta^i_j$. It is usually said that the tensor components undergo a contravariant transformation with respect to the upper indices, and a covariant transformation with respect to the lower.


Comments

An affine tensor as described above is commonly called simply a tensor.

References

[a1] B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, "Modern geometry - methods and applications" , Springer (1984) (Translated from Russian)
[a2] W.H. Greub, "Multilinear algebra" , Springer (1967)
[a3] C.T.J. Dodson, T. Poston, "Tensor geometry" , Pitman (1977)


Comments

The tensor $\delta^i_j$ is the Kronecker delta tensor.

An isotropic tensor is one for which the components are unchanged under change of basis. The Kronecker delta tensor is isotropic; in dimension $n=3$ the tensor $\epsilon_{ijk}$ defined by $\epsilon_{123} = \epsilon_{231} = \epsilon_{312} = 1$, $\epsilon_{321} = \epsilon_{213} = \epsilon_{132} = -1$, all other values zero, of order 3, is isotropic.

See also: Contravariant tensor, Covariant tensor.

References

[b1] H. Jeffreys Cartesian tensors (7th imp.) Cambridge University Press [1931] (1969) ISBN 0-521-09191-8 Zbl 57.0974.01
How to Cite This Entry:
Affine tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_tensor&oldid=37522
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article