# Christoffel symbol or Christoffel symbols?

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

• 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. Jan 17, 2017 at 18:34
• @Arthur well you're answer is what I used to think... but he said a strange argument about "the components of the Christoffel Symbol"...
– Dac0
Jan 17, 2017 at 18:39

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 $a$.
• 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