2
$\begingroup$

A tensor exercise in a text reads: If $T_i$ are the components of a covariant vector $T$, show that $S_{ij}:=T_iT_j-T_jT_i$ is an order 2 covariant tensor $S$.

Am I missing something or is $S$ uniformly zero?

  • 1
    Dear student: it would be helpful if you made explicit what text this is, as the source can make it easier to make sense of it (the tiny detail of whether this is a text intended for mathematicians or physicists would already be a useful piece of information!)2012-01-13

1 Answers 1

1

No, but it is certainly antisymmetric. consider the two 1-tensors, $\partial_i$, and $A_j$, then look at $F_{ij}=\partial_i A_j - \partial_j A_i$. This has zero components, but they are not all zero... Maybe go through them component by component until that's clear :)

  • 0
    Ah, indeed, silly me. Is this the same Ben Crowell from PF? if so, cheers.2012-01-14