Covariant derivative of the gradient

[eljose]

If we define the Gradient of a function:

[tex] \uparrow u= Gra(f) [/tex]

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

[tex] \nabla _{u}u [/tex]

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,

[tex]\nabla_iu_j=\partial_ju_j-\Gamma^k_{ij}u_k[/tex]

Where [itex]\Gamma[/itex] 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.

 

[tacman]

The covariant derivative of the gradient of a function can be written as:

\nabla _{u}u = \nabla _{u}(\nabla f)

This can be understood as the directional derivative of the gradient vector with respect to the vector u. In other words, it measures how the gradient changes in the direction of the vector u.

To calculate this, we can use the formula for the covariant derivative of a vector field:

\nabla _{u}v = \frac{\partial v}{\partial u} + \Gamma _{ij}^{k}u^{i}\frac{\partial v^{j}}{\partial x^{k}}

where \Gamma _{ij}^{k} are the Christoffel symbols and u^{i} and v^{j} represent the components of the vectors u and v respectively.

In this case, since the gradient vector is defined as \nabla f = \frac{\partial f}{\partial x^{i}}, we can substitute this into the formula above to get:

\nabla _{u}u = \frac{\partial (\frac{\partial f}{\partial x^{i}})}{\partial u} + \Gamma _{ij}^{k}u^{i}\frac{\partial (\frac{\partial f}{\partial x^{j}})}{\partial x^{k}}

Simplifying this, we get:

\nabla _{u}u = \frac{\partial ^{2}f}{\partial u\partial x^{i}}u^{i} + \Gamma _{ij}^{k}u^{i}\frac{\partial ^{2}f}{\partial x^{j}\partial x^{k}}

This is the covariant derivative of the gradient vector u with respect to the vector u. It is a vector field that measures the rate of change of the gradient vector in the direction of u.