Question on Einstein Summation convention evaluating certain expression

Question

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?

Answer 1

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.