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Covariant differentiation commutes with contraction?

  1. Jun 4, 2010 #1
    1. The problem statement, all variables and given/known data
    I've been reading a textbook on tensor analysis for a while. The book uses the conclusion of "covariant differentiation commutes with contraction" directly and I searched around and found most people just use the conclusion without proof.


    2. Relevant equations
    For example, [tex]\nabla_{i}T^{jk}_{kl}[/tex].


    3. The attempt at a solution
    I believe it can be interpreted in two ways. First, form the variant [tex]T^{jk}_{kl}[/tex] with two free indices j, l and apply [tex]\nabla_{i}[/tex] to that tensor; Or, apply [tex]\nabla_{i}[/tex] to the tensor [tex]T^{jk}_{ml}[/tex] and contract m and k. If the two interpretations lead to the same result, it should then be proved.

    Any help is appreciated!
     
    Last edited: Jun 4, 2010
  2. jcsd
  3. Jun 4, 2010 #2

    diazona

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    Homework Helper

    Whenever you contract something, there's a delta tensor involved. Try writing that out explicitly and using the product rule. What do you know about [itex]\nabla^i \delta_k^m[/itex]?
     
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