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I didn't get if there's a difference between "Christoffel symbol" or "Christoffel symbols" being the second some sort of components of the first? I've always used the term Christoffel symbols, but a professor of my university made an unclear argument pointing out the difference. Is that actually true or something used to be true? Thanks in advance

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  • $\begingroup$ I would guess the difference is grammatical rather than mathematical. So you would say "Christoffel symbols can be difficult to compute", but on the other hand "You fix the non-tensorial nature of partial differentiation by adding a Christoffel symbol". I might be wrong, however. $\endgroup$
    – Arthur
    Jan 17, 2017 at 18:34
  • $\begingroup$ @Arthur well you're answer is what I used to think... but he said a strange argument about "the components of the Christoffel Symbol"... $\endgroup$
    – Dac0
    Jan 17, 2017 at 18:39

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I'm used to seeing it in the plural, Christoffel symbols. I've never really questioned why, but my thoughts are:

  1. Not all of the indices on the entity make a tensor. For example, if you have Chirstoffel symbols of the second kind $\Gamma^a_{bc}$, this is a rank-2 tensor for each fixed value of $a$. The full $\Gamma^a_{bc}$ is not, however, a rank-3 tensor. In this interpretation, you get one symbol for each value of $a$.
  2. Christoffel symbols are related to connection coefficients - usually the term "Christoffel symbols" is used for connection coefficients in a coordinate basis. The plural "symbols" then aligns with the plural "coefficients." You wouldn't say (singular) "coefficient" in that context.
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  • $\begingroup$ I think you point 1 is what he was trying to say, i.e. $\Gamma^a_{bc}$ is a 2-rank tensor for each $a$ fixed... thank you $\endgroup$
    – Dac0
    Jan 17, 2017 at 19:03

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