(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is not homework but more like self-study - thought I'd post it here anyway.

I'm taking the variation of the determinant of the metric tensor:

[itex]\delta(det[g_{\mu}_{\nu}])[/itex].

2. Relevant equations

The answer is

[itex]\delta(det[g_{\mu\nu}])[/itex] =det[g_{\mu\nu}] g^{[itex]\mu\nu[/itex]}[itex]\delta[/itex]g_{[itex]\mu\nu[/itex]}

Here, g_{[itex]\mu\nu[/itex]}is the metric tensor, [g_{[itex]\mu\nu[/itex]}] is the matrix of the components of the metric tensor, and [itex]\delta[/itex] is a variation.

3. The attempt at a solution

I have managed to getcloseto the answer, I hope, with

[itex]\delta(det[g_{\mu\nu}])[/itex]

=det[g_{\mu\nu}] tr([g_{\mu\nu}]^{-1}[itex]\delta[/itex]([g_{\mu\nu}]))

The problem, in my view, is the trace. I cannot see how to remove it.

Also, if someone could kindly describe how to tidy the LaTeX up, I will do that.

Thank you!

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# Homework Help: Variation of the metric tensor determinant

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