Covariant derivative of the gradient

[eljose]

If we define the Gradient of a function:

$$\uparrow u= Gra(f)$$

which is a vector then what would be the covariant derivative:

$$\nabla _{u}u$$

where the vector u has been defined above...i know the covariant derivative is a vector but i don,t know well how to calculate it...thank you.

 

[Perturbation]

If u is defined as the gradient of a scalar, then u is a one-form. The components of the covariant derivative of u is, in a coordinate basis,

$$\nabla_iu_j=\partial_ju_j-\Gamma^k_{ij}u_k$$

Where $\Gamma$ is your connection (Levi-Civita or whatever). To actually work it out you need (1) your components in some coordinate system and (2) a connection.

The covariant derivative adds one to your covariant valence. The covariant derivative of a (1 0) tensor, a vector, its covariant derivative is a (1 1) tensor.