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In these notes, I am trying to derive the equation of motion (1.21) corresponding to the Nambu-Goto action:

[itex]\partial_\alpha ( \sqrt{ - \text{det} \gamma} \gamma^{\alpha \beta} \partial_\beta X^\mu ) =0[/itex]

I have found and proved all the stuff he says we need in order to do it in the above paragraph but just cannot get it to work out.

Thanks to anybody who can help!