Covariant derivative in coordinate basis

[LCSphysicist]

Homework Statement

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Relevant Equations

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I need to evaluate $\nabla_{\mu} A^{\mu}$ at coordinate basis. Indeed, i should prove that $\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt(|g|)}\partial_{\mu}(|g|^{1/2} A^{\mu})$.

So, $$\nabla_{\mu} A^{\mu} = \partial_{\mu} A^{\mu} + A^{\beta} \Gamma^{\mu}_{\beta \mu}$$

The first and third terms of Christoffel will cancel, so $$ = \partial_{\mu}A^{\mu} + A^{\beta} \frac{g^{\mu x}}{2}(\partial_{\beta}g_{x \mu})$$

Now, using the fact that $\delta g = g g^{\mu v} \delta g_{\mu v}$, we can easily find that $$\frac{g^{\mu x}}{2}(\partial_{\beta}g_{x \mu}) = \frac{\partial_{\beta}(|g|^{1/2})}{|g|^{1/2}}$$

After substitute this at our main expression, we can recover $\nabla_{\mu} A^{\mu} = \frac{1}{\sqrt(|g|)}\partial_{\mu}(|g|^{1/2} A^{\mu})$.

The problem is, i have no idea what assumption i have made so that my result applies only to coordinate basis! That is, the problem ask for prove it at coordinate basis , so i guess it should be true only at these type of basis. But i haven't assumed nothing, just manipulate the terms and got the result. What am i missing? Is this expression really general like i have found?

 

[Orodruin]

You have used the coordinate expression for the Christoffel symbols.

 

[LCSphysicist]

You have used the coordinate expression for the Christoffel symbols.

I know there are another expressions for Christoffel symbol, like:
$$\Gamma^{i}_{jk} = - \partial e^{i}/\partial x^{j} e_{k}$$
But i didn't know that this expression i have used is not the most general. What is the most general expression for it so? (namely, the one that makes no reference to any frame)