Covariant Tensor first order, and antisymmetric second order

[Telemachus]

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.

 

[tiny-tim]

Hi Telemachus!

[tex]T_{ik}=\frac{\partial V_i}{\partial x^k}-\frac{\partial V_k}{\partial x^i}[/tex]

Write out the barred version of that equation, then use the chain rule to compare one with the other.

 

[Telemachus]

Thank you Tim.