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Product of dyadic and a vector

  1. May 1, 2014 #1
    I have:

    dVμ = (∂Vμ/∂xη)dxη where Vμ is a contravariant vector field

    I believe the () term on the RHS is a covariant tensor. Is the dot product of () and dxη a scalar and how do I write this is compact form. I know how this works for scalars but am not clear when tensors are involved.
  2. jcsd
  3. May 1, 2014 #2

    Ben Niehoff

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    Science Advisor
    Gold Member

    You haven't been very specific. What kind of space are you in? Is it flat or curved?

    Generically speaking, ##\partial_\nu V^\mu## is not a tensor at all, because in curved space, partial derivatives do not transform nicely under changes of coordinates. Using a covariant derivative,

    [tex]\nabla_\nu V^\mu[/tex]
    is a tensor of mixed type.

    In curved space, "##\mathrm{d} V^\mu##" is not really a sensible thing to do, because it is not covariant under coordinate changes.
  4. May 1, 2014 #3
    Flat space. I am trying to show that the gradient of a contravariant vector is a covariant vector. I understand how to show this for a scalar, but not sure how to extend this to vectors/tensors.
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