I have seen many different notations to denote contravariant/covariant and mixed tensors. For example, I think the notation $\omega^{v}_{\,\,\,\mu}$ stands for a mixed tensor, where one index transforms contravariantly and the other covariantly. What then would the notation $\omega^{v}_{\mu}$ denote?

A covariant vector transforms like $a'_{\mu} = \beta_{\mu}^{\,\,\,v}a_v$, but what is the difference between this and writing $a'_{\mu} = \beta_{\mu}^{v}a_v$ for example?

The only instance where I have seen an equivalence between the two notations is for the Kronecker delta, $\delta^v_{\,\,\,\mu} = \delta_{\mu}^{\,\,\,v} = \delta^v_{\mu}$, but the reason for this is not quite clear.

Many thanks.

  • $\begingroup$ As far as I know it means the same thing. $\endgroup$ – Qiaochu Yuan May 28 '14 at 16:14
  • $\begingroup$ It depends on whether you consider the isomorphic tensor product spaces $V\otimes V^*\simeq Hom(V)$ and $V^*\otimes V\simeq Hom(V^*)$ as identical or not. $\endgroup$ – LutzL May 28 '14 at 17:57
  • $\begingroup$ with the notation ${\omega^{\nu}}_{\mu}$ you don't loose which is row and which is column in the arrangement. $\endgroup$ – janmarqz May 30 '14 at 2:27

In Einstein summation convention, repeated indices represent summation. Depending on the range of indices ($\nu=1$ to n), you would have a sum of n terms, and the contravariant index would cancel out the covariant index, leaving just a covariant $a'_\mu$

  • $\begingroup$ Hi Jonathan. Thanks for your reply, but I understand that. My question is about the placement of the indices. In all cases, one of the indices is shifted to the left e.g $\omega_{v}^{\,\,\,\mu}$ vs $\omega_v^{\mu}$. I haven't seen the latter notation used anywhere, so I wanted to know if there are any differences. $\endgroup$ – CAF May 28 '14 at 13:11
  • $\begingroup$ Im not an expert this by any means, but I believe that it has the same meaning, but hopefully someone else can comfirm this... $\endgroup$ – Jonathan Wheeler May 28 '14 at 13:13
  • $\begingroup$ I think it arises because you want to be able to refer to a specific index. With $\omega^{v}_{\,\,\,\mu}$ you can refer to the first and the second, but with $\omega_v^{\mu}$ what would you do ? $\endgroup$ – Tom Collinge May 28 '14 at 13:22
  • $\begingroup$ It makes sense - the notation is such to label a component of the tensor. There is no ambiguity in the Kronecker delta (since it is diagonal) so either notation is fine when dealing with that. Thanks Tom and Jonathan. $\endgroup$ – CAF May 28 '14 at 13:29

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