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Difference between revisions of "Affine tensor"

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An affine tensor as described above is commonly called simply a tensor.
 
An affine tensor as described above is commonly called simply a tensor.
  
====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) {{ZBL|0369.53012}} Graduate Texts in Mathematics '''130''' (2nd ed.) Springer (1991) ISBN 3-540-52018-X {{ZBL|0732.53002}}</TD></TR>
 
</table>
 
 
====Comments====
 
 
The tensor $\delta^i_j$ is the [[Kronecker delta]] tensor.
 
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.   
+
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 [[discriminant 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]].
 
See also: [[Contravariant tensor]], [[Covariant tensor]].
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====References====
 
====References====
 
<table>
 
<table>
<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>
+
<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) {{ZBL|0369.53012}} Graduate Texts in Mathematics '''130''' (2nd ed.) Springer (1991) {{ISBN|3-540-52018-X}} {{ZBL|0732.53002}}</TD></TR>
 +
<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>
 
</table>
 
</table>
  
 
{{TEX|done}}
 
{{TEX|done}}

Latest revision as of 16:21, 18 November 2023

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.

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 discriminant 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

[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) Zbl 0369.53012 Graduate Texts in Mathematics 130 (2nd ed.) Springer (1991) ISBN 3-540-52018-X Zbl 0732.53002
[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=37523
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article