Two covariant derivatives of a vector field

[Deadstar]

$$V_{a;b} = V_{a,b} - \Gamma^d_{ad}V_d$$

Now take the second derivative...

$$V_{a;b;c} = (V_{a;b})_{,c} - \Gamma^f_{ac}V_{f;b} - \Gamma^f_{bc}V_{a;f}$$

But I have no idea how to get the parts with the Christoffel symbols.

$$V_{a;b;c} = (V_{a;b})_{,c} - \Gamma^f_{(a;b)c}V_{af} = (V_{a;b})_{,c} - \Gamma^f_{ac}V_{af;b}$$..?

Yes the above is clearly wrong but how is this supposed to be done?

 

[arkajad]

You should start with the following general definition:

$$V_{a_1\ldots a_k;b}= V_{a_1\ldots a_k,b}-\sum_{i=1}^k \Gamma^f_{ba_i}V_{a_1\ldots a_{i-1}fa_{i+1}\ldots a_k}$$

(with understanding that there is no i-1 for i=1)

Simply take it as a definition of the covariant derivative of a covariant tensor of order k. This definition is a natural extension of the covariant derivative to tensors, but it does not follow from the definition of the covariant derivative of vectors. It will follow as a unique extension satisfying certain natural requirements (sometimes explicit, sometimes implicit, depending on the author), but it does not follow from the formula alone.