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

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  • $\begingroup$ Have you cheked the wiki article on the summation convention? Those are called dummy indices $\endgroup$ – Jackozee Hakkiuz May 15 at 17:35
  • $\begingroup$ 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 $\endgroup$ – nabzdyczony May 15 at 17:47
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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 $$X^i\frac{\partial Y^j}{\partial x^i}\frac{\partial f}{\partial x^j} = X^j\frac{\partial Y^i}{\partial x^j}\frac{\partial f}{\partial x^i}$$

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