Christoffel Symbol / Covariant derivative

1. May 1, 2012

cristina89

1. The problem statement, all variables and given/known data
My teacher solved this in class but I'm not understanding some parts of tis solution.

Show that $\nabla_i V^i$ is scalar.

2. Relevant equations
$\nabla_i V^i = \frac{\partial V^{i}}{\partial q^{i}} + \Gamma^{i}_{ik} V^{k}$

3. The attempt at a solution
To start this, I'll solve this $\Gamma^{i}_{ik}$ first.

$\Gamma^{i}_{ik} = \frac{1}{2} g^{il} (\frac{\partial g_{lk}}{\partial q^{i}} + \frac{\partial g_{il}}{\partial q^{k}} - \frac{\partial g_{ki}}{\partial q^{l}}$

$\Gamma^{i}_{ik} = \frac{1}{2} g^{il} \frac{\partial g_{il}}{\partial q^{k}} = \frac{1}{2g} \frac{\partial g}{\partial q^{k}}$

$\Gamma^{i}_{ik} = \frac{1}{\sqrt g}\frac{\partial \sqrt{g}}{\partial q^{k}}$

(THIS PART: how this $\sqrt{g}$ appeared??)

Continuing...

$\nabla_i V^i = \frac{\partial V^{i}}{\partial q^{i}} + \frac{V^{k}}{\sqrt{g}} \frac{\partial \sqrt{g}}{\partial q^{k}}$

$\nabla_i V^i = \frac{1}{\sqrt g} \partial_i (\sqrt g V^{i})$

And this last part... What happened to $\partial q^{i}$ and $V^{k}$?

2. May 1, 2012

lanedance

take this with a grain of salt as I haven't played with tensors in a while... anyway here I go... I think both these are tricks that can be understood by working back

I think the first part is just normal chain rule differentiation, what is the standard derivative of sqrt(g)?

then try expanding the very last expression using the standard product rule, what do you end up with?