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A tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945801.png" /> which depends on a pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945802.png" /> in a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945803.png" />, i.e. a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945804.png" /> defined on the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945805.png" />. As an example, covariant derivatives of the [[World function|world function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945806.png" /> and, in general, of an arbitrary invariant depending on two points are two-point tensors. The properties of such a tensor, in particular the limits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945807.png" /> and its derivatives as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094580/t0945808.png" />, such as
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A tensor $T$ which depends on a pair of points $x,x'$ in a manifold $X$, i.e. a tensor field $T(x,x')$ defined on the product $X \times X$. As an example, covariant derivatives of the [[World function|world function]] $\Omega(x,x')$ and, in general, of an arbitrary invariant depending on two points are two-point tensors. The properties of such a tensor, in particular the limits of $T$ and its derivatives as $x' \rightarrow x$, such as
 
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$$
<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/t/t094/t094580/t0945809.png" /></td> </tr></table>
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[T_{ij'}] =\lim_{x' \rightarrow x} \nabla_i \nabla_{j'} T(x,x')
 
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$$
are employed in the calculus of variations and in the theory of relativity.
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are employed in the [[calculus of variations]] and in the [[Relativity theory|theory of relativity]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Synge,  "Relativity: the general theory" , North-Holland &amp; Interscience  (1960)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Synge,  "Relativity: the general theory" , North-Holland &amp; Interscience  (1960) {{ZBL|0090.18504}}</TD></TR>
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</table>

Latest revision as of 22:12, 29 November 2014

A tensor $T$ which depends on a pair of points $x,x'$ in a manifold $X$, i.e. a tensor field $T(x,x')$ defined on the product $X \times X$. As an example, covariant derivatives of the world function $\Omega(x,x')$ and, in general, of an arbitrary invariant depending on two points are two-point tensors. The properties of such a tensor, in particular the limits of $T$ and its derivatives as $x' \rightarrow x$, such as $$ [T_{ij'}] =\lim_{x' \rightarrow x} \nabla_i \nabla_{j'} T(x,x') $$ are employed in the calculus of variations and in the theory of relativity.

References

[1] J.L. Synge, "Relativity: the general theory" , North-Holland & Interscience (1960) Zbl 0090.18504
How to Cite This Entry:
Two-point tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-point_tensor&oldid=13796
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article