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Tensor help

  1. Nov 3, 2014 #1
    1. The problem statement, all variables and given/known data
    A tensor and vector have components Tαβγ, and vα respectively in a coordinate system xμ. There is another coordinate system x'μ. Show that Tαβγvβ = Tαβγvβ

    2. Relevant equations
    umm not sure...

    αvβ = ∂vβ/∂xα - Γγαβvγ

    3. The attempt at a solution
    Tαβγvβ = (∂xα/∂xi*∂xj/∂xβ*∂xγ/∂xk*Tijk)(∂xβ/∂xa*va)

    Tαβγvβ = (∂xα/∂xi*∂xβ/∂xj*∂xγ/∂xk*Tijk)(∂xa/∂xβ*va)

    and the two are not equal which they should be. I really don't know where I've went wrong...
     
  2. jcsd
  3. Nov 3, 2014 #2

    Orodruin

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    How do you relate Tijk with Tijk and va with va?
     
  4. Nov 3, 2014 #3
    A previous part asked to express Tαβγ in terms of Tαβγ and I put my answer as ∂xα/∂xi*∂xβ/∂xj*∂xγ/∂xk*Tijj or by using the metric tensor

    however does that also apply to vectors? if not, I don't know how vα relates to vα...

    edit: wait is vα related to vα by vα = ∂xβ/∂xα*v'β?
     
    Last edited: Nov 3, 2014
  5. Nov 3, 2014 #4

    Orodruin

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    You are confusing the concepts of how a vector transforms under coordinate transformations to how a contravariant tensor relates to a covariant one. A contravariant index can be turned into a covariant one by contraction with the metric. Covariant and contravariant indices transform differently under coordinate transformations.

    A vector is a tensor of rank one and its indices therefore follow precisely the same rules as any tensor indices - what distinguishes a vector is that it only has one.
     
  6. Nov 3, 2014 #5
    Okay so if I take it one step at a time: expressing Tαβγ in terms of Tαβγ without taking into account the vector, I use the metric tensor. gαμ(Tμβγ) would then equal gγμ(Tαβμ) = Tαβγ. Is that right?
     
  7. Nov 3, 2014 #6

    Orodruin

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    In order to raise a covariant index, you need to contract it with one of the contravariant indices in gij. In the same way, in order to lower a contravariant index, you need to contract it with one of the indices in gij. Remember that you cannot contract covariant indices with each other but must always contract a covariant index with a contravariant one and vice versa.
     
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