View source for Two-point 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 [[Relativity theory|theory of relativity]].

====References====
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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)</TD></TR>
</table>