Differential Geometry - vector fields Lie bracket

Question

Can anyone tell me why (in the last line) $i$ changes to $j$ in the first component of the sum?

Answer 1

Repeated indices are interchangeable as long as you don't have more than two of the same letter in a single term. E.g. $$a_ib^i = a_jb^j = a_kb^k$$

In your case $$X^i\frac{\partial Y^j}{\partial x^i}\frac{\partial f}{\partial x^j} = X^j\frac{\partial Y^i}{\partial x^j}\frac{\partial f}{\partial x^i}$$