Consider the following expression: $$X^k\frac{\partial g_{ij}}{\partial x^k} + g_{jk}\frac{\partial X^k}{\partial x^i} + g_{ik}\frac{\partial X^k}{\partial x^j} = 0$$ Am I correct, that this is the same as $$\sum_k \left(X^k\frac{\partial g_{ij}}{\partial x^k} + g_{jk}\frac{\partial X^k}{\partial x^i} + g_{ik}\frac{\partial X^k}{\partial x^j}\right) = 0$$ using the Einstein Summation convention?
Asked
Active
Viewed 270 times
2
-
1Yes, it is correct. – Emilio Novati Apr 07 '17 at 08:32
-
@EmilioNovati Thanks. I just wasn't sure about $i,j$ but then they seem to be free parameters. – TheGeekGreek Apr 07 '17 at 08:33
1 Answers
2
Yes you are correct. The suffix $k$ appears twice in each term in your first expression so according to the convention there is indeed an implied summation over k.
$i$ and $j$ only appear once in each term so there is no implied summation, they are as you suspected free indices. There is no problem with multiple free indices.
PM.
- 5,249