Difference between revisions of "Two-point tensor"
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| − | A tensor | + | 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 |
| − | + | $$ | |
| − | + | [T_{ij'}] =\lim_{x' \rightarrow x} \nabla_i \nabla_{j'} T(x,x') | |
| − | + | $$ | |
| − | are employed in the calculus of variations and in the theory of relativity. | + | 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 & Interscience (1960)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> J.L. Synge, "Relativity: the general theory" , North-Holland & Interscience (1960)</TD></TR> | ||
| + | </table> | ||
Revision as of 18:11, 21 August 2013
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) |
How to Cite This Entry:
Two-point tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-point_tensor&oldid=13796
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