Could anybody help to spot the inconsistency in the following reasoning?(adsbygoogle = window.adsbygoogle || []).push({});

When calculating the normal derivative of the metric tensor I get:

[tex] \partial_\mu g^{\rho \sigma} = g^{\rho \lambda} g^{\sigma \gamma} \partial_\mu g_{\lambda \gamma} + 2 \partial_\mu g^{\rho \sigma}, [/tex] (1)

which means that:

[tex] g^{\rho \lambda} g^{\sigma \gamma} \partial_\mu g_{\lambda \gamma} = -\partial_\mu g^{\rho \sigma}. [/tex] (2)

And I don't see how this could be.

That's how I get this result:

[tex]

\partial_\mu g^{\rho \sigma} =

\partial_\mu (g^{\rho \lambda} g^{\sigma \gamma} g_{\lambda \gamma}) =

g^{\rho \lambda} g^{\sigma \gamma} \partial_\mu g_{\lambda \gamma} + g^{\rho \lambda} g_{\lambda \gamma} \partial_\mu g^{\sigma \gamma} + g_{\lambda \gamma} g^{\sigma \gamma} \partial_\mu g^{\rho \lambda} =

g^{\rho \lambda} g^{\sigma \gamma} \partial_\mu g_{\lambda \gamma} + \delta^\rho_\gamma \partial_\mu g^{\sigma \gamma} + \delta^\sigma_\lambda \partial_\mu g^{\rho \lambda} =

[/tex]

[tex]

= g^{\rho \lambda} g^{\sigma \gamma} \partial_\mu g_{\lambda \gamma} + 2 \partial_\mu g^{\rho \sigma}.

[/tex] (3)

Could anybody show how to get directly the right hand side of equation (2) from the left hand side, or show where the mistake in the equation (3) is?

Thanks a lot.

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# Derivative of the metric tensor

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