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Covariant Tensor first order, and antisymmetric second order

  1. Apr 7, 2012 #1
    Hi there. This is my first time working with tensors, so I have to break the ice I think. I have this exercise, which I don't know how to solve, which says:

    If [tex]V=V_1...V_n[/tex] is a first order covariant tensor, prove that:
    [tex]T_{ik}=\frac{\partial V_i}{\partial x^k}-\frac{\partial V_k}{\partial x^i}[/tex]

    Is a second order covariant antisymmetric tensor.

    Now, in my notes I have this definitions:
    A vector field V is a first order covariant tensor, if under a change of coordinates from [tex]x[/tex] to [tex]\overline x[/tex] it's components are:
    [tex]\overline {V}=\frac{\partial x^r}{\partial \overline {x}^i}V_r[/tex]

    I think I should use this, but as I said, I'm starting with this, and I don't know how to work this out.

    Any help will be appreciated.
     
  2. jcsd
  3. Apr 8, 2012 #2

    tiny-tim

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    Hi Telemachus! :smile:
    Write out the barred version of that equation, then use the chain rule to compare one with the other. :wink:
     
  4. Apr 8, 2012 #3
    Thank you Tim.
     
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