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Covariant differentiation twice

  1. Jul 9, 2014 #1
    Doing some problems in D'INVERNO GR textbook and I am stuck on taking the covariant derivation of a tensor twice. Please see the attached picture and please do inform me if something is not clear :smile:
     

    Attached Files:

  2. jcsd
  3. Jul 11, 2014 #2
    Hello.
    Where do e comes from??

    Take ##A^i=B^i_jC^j##.
    j is a dummy index, there is a summation over j.
    i can be 1, 2 or 3. This equation tells that the equality is true for all three values of i.
    ##A^k=B^k_jC^j## is exactly the same as ##A^i=B^i_jC^j##, you must not keep track of indices from one equation to another.
     
  4. Jul 11, 2014 #3
    Heya bloby,

    For e, I just chose a new tensor to represent the covariant derivative of the original tensor. Then what should I have named it? T[itex]^{a}_{b}[/itex] ?

    Thanks
     
  5. Jul 11, 2014 #4
    Rather ##T^a_d## the same indices than LHS. The indices are related to basis element. They must be consistent within an equation, like ##v^i=\frac{dx^i}{dt}##, not ##v^i=\frac{dx^j}{dt}##. The 3rd and 4th line of the thumbnail are the same(after corrections) with renamed indices .
     
  6. Jul 11, 2014 #5
    Much Appreciated bloby :smile:

    Thanks
     
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