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I was studying a little of tensorial calculus and came up with this problem:

Given a tensor with a rank of (0,2), $T_{\alpha\beta}$. Calculate the rank of this tensor $T_{\alpha\beta}T_{\gamma}^{\sigma}T^{\beta\gamma}$

P.D. I´m self-studying tensorial calculus, but the reference book that I have been using is not that good, so I have been trying to find the method to solve the problem but I have been unable to do so. I searched in the book first (as that is where the problem is mentioned) but there is nothing that can help me. Also, I have been trying to find an answer on the internet but the only answers are from programing... Maybe it is really simple and I cant see it, and that is why I can't find an answer. Any help would be really appreciated.

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  • $\begingroup$ yes @d_b , it is Rank, it is an error of translation, I'm so sorry $\endgroup$
    – AdrinMI49
    Commented Apr 27, 2022 at 1:22
  • $\begingroup$ @AFG, I double-checked and I already updated it to the correct index. What a blunder. $\endgroup$
    – AdrinMI49
    Commented Apr 27, 2022 at 1:26

2 Answers 2

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Just count unpaired indices. You have $\alpha$ down and $\sigma$ up, so you have a (1,1) tensor. (In the same way, you know that $T_{\alpha\beta}$ is (0,2).)

(FYI unless $T_\gamma^\sigma$ is symmetric you should never notate it like that, and even then I'd consider it annoying. Write ${T_\gamma}^\sigma$ or ${T^\sigma}_\gamma,$ since in general these could be different.)

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You can easily find the rank of a tensor by looking at the number of free indices. To find dummy indices, look for where you have the same index on the top (contravariant), and the bottom (covariant), which means you're summing over this index (hence it's not free).

See if you can find how many indices you're not summing over. That should give you the number of free indices, which is the rank of the tensor.

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