# Covariant Derivative derivation.

1. Aug 20, 2012

### T-chef

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

2. Aug 20, 2012