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Difference between revisions of "Contraction of a tensor"

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An operation of tensor algebra that associates with a tensor with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258702.png" />, the tensor
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An operation of tensor algebra that associates with a tensor with components $ a^{i_{1} \ldots i_{p}}_{j_{1} \ldots j_{q}} $, $ p,q \geq 1 $, the tensor
 
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\begin{align}
<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/c/c025/c025870/c0258703.png" /></td> </tr></table>
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b^{i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1}}
 
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& = a^{1 i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} 1} + a^{2 i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} 2} + \cdots + a^{n i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} n} \\
<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/c/c025/c025870/c0258704.png" /></td> </tr></table>
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& = a^{\alpha i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} \alpha}.
 
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\end{align}
<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/c/c025/c025870/c0258705.png" /></td> </tr></table>
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(Here, the contraction is made with respect to the pair of indices $ i_{1} $ and $ j_{q} $). The contraction of a tensor with respect to any pair of upper and lower indices is defined similarly. The $ p $-fold contraction of a tensor that is $ p $-times covariant and $ p $-times contravariant is an invariant. Thus, the contraction of the tensor with components $ a^{i}_{j} $ is an invariant $ a^{i}_{i} $, called the trace of the tensor; it is denoted by $ \text{Sp}(a^{i}_{j}) $, or $ \text{tr}(a^{i}_{j}) $. A contraction of the product of two tensors is a contraction of the product with respect to an upper index of one factor and a lower index of the other.
 
 
(here the contraction is made with respect to the pair of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258706.png" />). The contraction of a tensor with respect to any pair of upper and lower indices is defined similarly. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258707.png" />-fold contraction of a tensor that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258708.png" />-times covariant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258709.png" />-times contravariant is an invariant. Thus, the contraction of the tensor with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c02587010.png" /> is an invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c02587011.png" />, called the trace of the tensor; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c02587012.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c02587013.png" />. A contraction of the product of two tensors is a contraction of the product with respect to an upper index of one factor and a lower index of the other.
 
 
 
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">P.K. [P.K. Rashevskii] Rashewski, “Riemannsche Geometrie und Tensoranalyse”, Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)</TD></TR></table>

Latest revision as of 02:26, 9 September 2015

An operation of tensor algebra that associates with a tensor with components $ a^{i_{1} \ldots i_{p}}_{j_{1} \ldots j_{q}} $, $ p,q \geq 1 $, the tensor \begin{align} b^{i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1}} & = a^{1 i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} 1} + a^{2 i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} 2} + \cdots + a^{n i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} n} \\ & = a^{\alpha i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} \alpha}. \end{align} (Here, the contraction is made with respect to the pair of indices $ i_{1} $ and $ j_{q} $). The contraction of a tensor with respect to any pair of upper and lower indices is defined similarly. The $ p $-fold contraction of a tensor that is $ p $-times covariant and $ p $-times contravariant is an invariant. Thus, the contraction of the tensor with components $ a^{i}_{j} $ is an invariant $ a^{i}_{i} $, called the trace of the tensor; it is denoted by $ \text{Sp}(a^{i}_{j}) $, or $ \text{tr}(a^{i}_{j}) $. A contraction of the product of two tensors is a contraction of the product with respect to an upper index of one factor and a lower index of the other.

Comments

References

[a1] P.K. [P.K. Rashevskii] Rashewski, “Riemannsche Geometrie und Tensoranalyse”, Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
How to Cite This Entry:
Contraction of a tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contraction_of_a_tensor&oldid=18772
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article