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I've somehow gone the past year without paying attention to the order of the indicies when one is upper and one is lower i.e. that in general ##g^{\mu}## ##_{\nu}## ##\neq g_{\nu}## ## ^{\mu}##.

A have a couple of questions :

1)

##g^{u}## ##_{v} x^{v}=x^{u}## [1]

##g _{v} ## ##^{u} x^{v} = x^{u}## [2]

I believe that both of these are mathematically correct to write, since there is a dummy index being summed over in both cases. However ## x^{\mu}## in [1] ##\neq## ##x^{\mu}## in [2] because ##g^{\mu}## ##_{\nu}## ##\neq g_{\nu}## ## ^{\mu}##, in general, is this correct? (i.e I am just confirming that the indices do not need to be next to each other to be summed over, as they are in [1] - this is probably a stupid question but the fact I haven't paid attention to the order of an upper and lower index for so long makes me question?)

2) Given the matrix ##g^{v} ## ##_{u}##, am I correct in thinking that we can obtain given a metric matrix ## g _{v}## ## ^{u} ## from it, but not, solely using the metric matrix, ##g ^{u}## ## _ {v}## because on top of raising and lowering the indices, the order needs to be interchanged?

Many thanks.

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# I Index algebra questions / order of indices

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