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,