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I know that the Kronecker delta is used to raise and lower indices, but I am not certain if the order in which it is placed in an equation matters, what I mean is, for example:

Are the following two any different?

$$\delta^\mu _\nu x_\mu\hspace{5mm} and \hspace{5mm}x_\mu \delta^\mu _\nu $$

I know the first one would give me $x_\nu$ but I don't know if changing the order they are written in would give me the same answer.

Would there be any difference in a situation in which instead of using a Kronecker delta, a metric tensor, $\eta_{\mu\nu}$, was used?

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No, the order doesn't matter. In general, things with indices are just components of tensors, so they are just numbers, so they commute (as far as the indices are of concern)

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  • $\begingroup$ In addition, note that if the metric tensor is $\mathbf{\eta}$, then $\eta^\mu{}_\nu = \delta^\mu{}_\nu$: what you are using is the metric tensor in fact. $\endgroup$ – tfb Apr 8 at 13:53

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