# Covariant Derivative

1. Mar 8, 2017

### MattRob

1. The problem statement, all variables and given/known data
Take the Covariant Derivative

$∇_{c} ({∂}_b X^a)$

2. Relevant equations

$∇_{c} (X^a) = ∂_c X^a + Γ_{bc}^a X^b$
$∇_{c} (X^a_b) = ∂_c X^a_b + Γ_{dc}^a X^d_b - Γ^d_{bc} X^a_d$

3. The attempt at a solution

Looking straight at
$∇_{c} ({∂}_b X^a)$
I'm seeing two indices. However, the b is nothing more than a reference to the derivative with respect to what, so I'm not sure whether this counts as a one or two-index object. Namely, do I solve it as

$∇_{c} ({∂}_b X^a) = ∂_c {∂}_b X^a + Γ_{dc}^a {∂}_b X^d$

since the index in the partial differential operator isn't the same as an index on a tensor, or is it?

So I'm thinking since the indices merely indicate you're working with a variety of objects ($x^a = k => x^1 = k, x^2 = k, x^3 = k, ...$), then the index on the differential operator should be treated the same, so I can treat ${∂}_b X^a$ as a two-index tensor $T^a_b = {∂}_b X^a$

$∇_{c} ({∂}_b X^a) = ∂_c {∂}_b X^a + Γ_{dc}^a {∂}_b X^d - Γ^d_{bc} {∂}_d X^a$

So is it the latter case, former, or something else entirely?

I suppose I have every reason to think it should be the latter case, but I guess I'm just uncertain and want some clarification.

Thanks!

2. Mar 8, 2017

### strangerep

Treat it as a 2-index object.

3. Mar 9, 2017

### MattRob

Second formula, then?

Thanks!