# Differential Geometry - vector fields Lie bracket

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

• Have you cheked the wiki article on the summation convention? Those are called dummy indices – Jackozee Hakkiuz May 15 at 17:35
• Yes, I know this convention. Still I have no idea why it looks like this. For example, second component looks the same as in the previous line – nabzdyczony May 15 at 17:47

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