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Covariant derivative of the gradient

  1. May 9, 2006 #1
    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.
  2. jcsd
  3. May 9, 2006 #2
    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,


    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.
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