Covariant Derivative derivation.

[T-chef]

Homework Statement

Using the Leibniz rule and:
$$\nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b$$
$$\nabla_{a}\Phi=\partial\Phi$$

Show that $\nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b}$.
The question is from Ray's Introducing Einsteins relativity,

My attempt:
$$\nabla_c(X^aX_a)=\nabla_c(X^a)X_a+X^a\nabla_c(X_a)$$
$$= (\partial_{c}X^a+\Gamma_{bc}^{a}X^b)X_a+X^a\nabla_c(X_a)$$

From here I'm not sure how to introduce the scaler field phi, or how doing so would help. Cheers for any help!

 

[clamtrox]

You already introduced it!

 

[T-chef]

Of course! The left hand side is exactly the scaler field I need. All comes out nicely after that! Thank you sir,