- #1
cristina89
- 29
- 0
Homework Statement
My teacher solved this in class but I'm not understanding some parts of tis solution.
Show that [itex]\nabla_i V^i[/itex] is scalar.
Homework Equations
[itex]\nabla_i V^i = \frac{\partial V^{i}}{\partial q^{i}} + \Gamma^{i}_{ik} V^{k}[/itex]
The Attempt at a Solution
To start this, I'll solve this [itex]\Gamma^{i}_{ik}[/itex] first.
[itex]\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}}[/itex]
[itex]\Gamma^{i}_{ik} = \frac{1}{2} g^{il} \frac{\partial g_{il}}{\partial q^{k}} = \frac{1}{2g} \frac{\partial g}{\partial q^{k}}[/itex]
[itex]\Gamma^{i}_{ik} = \frac{1}{\sqrt g}\frac{\partial \sqrt{g}}{\partial q^{k}}[/itex]
(THIS PART: how this [itex]\sqrt{g}[/itex] appeared??)
Continuing...
[itex]\nabla_i V^i = \frac{\partial V^{i}}{\partial q^{i}} + \frac{V^{k}}{\sqrt{g}} \frac{\partial \sqrt{g}}{\partial q^{k}}[/itex]
[itex]\nabla_i V^i = \frac{1}{\sqrt g} \partial_i (\sqrt g V^{i})[/itex]
And this last part... What happened to [itex]\partial q^{i}[/itex] and [itex]V^{k}[/itex]?