Scalar field lagrangian in curved spacetime

[resaypi]

Homework Statement

I am studying inflation theory for a scalar field $\phi$ in curved spacetime. I want to obtain Euler-Lagrange equations for the action:
$I\left[\phi\right] = \int \left[\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi + V\left(\phi\right) \right]\sqrt{-g} d^4x$

Homework Equations

Euler-Lagrange equations for a scalar field is given by
$\partial_\mu \frac{\partial L}{\partial\left(\partial_\mu\phi\right)} - \frac{\partial L}{\partial \phi} = 0$

The Attempt at a Solution

$\partial_\mu \frac{\partial L}{\partial\left(\partial_\mu\phi\right)} = \frac{1}{2}\partial_\mu\left(\sqrt{-g}g^{\mu\nu}\partial_nu\phi \right)$
$\frac{\partial L}{\partial \phi} = \frac{\partial \left[\sqrt{-g}V\left(\phi\right)\right]}{\partial \phi}$

But according to the book the resulting equation is
$\frac{1}{\sqrt{-g}}\partial_\mu\left(\sqrt{-g}g^{\mu\nu}\partial_\nu\phi\right) = \frac{\partial V\left(\phi\right)}{\partial \phi}$

What am I doing wrong?

 

[Oxvillian]

Hi resaypi!

Looks right to me except for:

1. typo with $\partial_nu$ instead of $\partial_{\nu}$

2. no factor of 1/2 when you take the $\frac{\partial}{\partial (\partial_{\mu} \phi)}$ derivative

3. $\frac{\partial \left[\sqrt{-g}V\left(\phi\right)\right]}{\partial \phi} = \sqrt{-g}\frac{\partial V\left(\phi\right)}{\partial \phi}$