Variation of Ricci scalar wrt derivative of metric

[jcap]

I understand from the wiki entry on the Einstein-Hilbert action that:
$$\frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}$$
What is the following?
$$\frac{\delta R}{\delta(\partial_\lambda g^{\mu\nu})}$$
Is there a place I could look up such GR expressions on the internet?
Thanks

 

[haushofer]

I'm not sure about your first identity. For the second: I wouldn't know where to find such a monster. It's not a tensor, so you cannot use clever guesswork to derive its form. Maybe someone else knows.

 

[PeterDonis]

the wiki entry on the Einstein-Hilbert action

I assume you mean this?

https://en.wikipedia.org/wiki/Einstein–Hilbert_action

$$\frac{\delta R}{\delta g^{\mu\nu}}=R_{\mu\nu}$$

This is a somewhat sloppy (i.e., many physicists are OK with it but many mathematicians get ulcers from looking at things like it) way of saying that the variation of the Ricci scalar $R$ with respect to the inverse metric $g^{\mu \nu}$ is the Ricci tensor $R_{\mu \nu}$.

What is the following?
$$\frac{\delta R}{\delta(\partial_\lambda g^{\mu\nu})}$$

Where are you seeing this expression? I don't see it in the Wikipedia article.

Is there a place I could look up such GR expressions on the internet?

I have no idea, but even if there is one, I'm not sure how reliable it would be. The best way to learn what such expressions mean is to look at things called "textbooks".

 

[jcap]

It's ok - I've discovered I don't need this expression. I got confused between using an Euler-Lagrange equation and doing the variation by hand.

Thanks! :)