# Dual vector is the covariant derivative of a scalar?

1. Apr 21, 2012

### NSH

1. The problem statement, all variables and given/known data

In Wald's text on General Relativity he makes an assertion that I'm not sure why it is allowed mathematically. Here's the basic setup:

Let $\omega_{b}$ be a dual vector, $\nabla_{b}$ and $\tilde{\nabla}_{b}$ be two covariant derivatives and $f\in\mathscr{F}$. Then we may let $\omega_{b}=\nabla_{b}f=\tilde{\nabla}_{b}f$

This is in chapter 3 on curvature between equations 3.1.7 and 3.1.8...

2. Relevant equations

If it is relevant he is using this assertion to show:

$\nabla_{a}\omega_{b}=\tilde{\nabla}_{a}\omega_{b}-C^{c}_{ab}\nabla_{c}f$

implies
$\nabla_{a}\nabla_{b}f=\tilde{\nabla}_{a}\tilde{ \nabla}_{b}f-C^{c}_{ab}\nabla_{c}f$

3. The attempt at a solution

I know how to plug in his assertion I just don't get why the heck it is allowed? I've tried reading up on dual vector spaces but I haven't found what I'm looking for... any help would be appreciated.