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Convert tensor from cartesian to cylindrical coordinate

  1. Mar 17, 2012 #1
    1. The problem statement, all variables and given/known data
    Given the tensor
    [tex]
    F_{\mu \nu }=
    \left[ \begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & B_{z} &-B_{y} \\E_{y} & -B_{z} & 0 & B_{x} \\E_{z} & B_{y} & -B_{x} & 0 \end{array} \right]
    [/tex]
    [tex]
    F^{\mu \nu }F_{\mu \nu }=2(B^2-\frac{E^2}{c^2})
    [/tex]
    and metric tensor
    [tex]
    n_{\mu \nu }=
    \left[ \begin{array}{cccc}c^2& 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right]
    [/tex]

    How to convert it into cylindrical coordinates, that is in terms of Eθ,Ez,Er
    More info of this tensor can be viewed at http://en.wikipedia.org/wiki/Electromagnetic_tensor


    2. Relevant equations



    3. The attempt at a solution

    I try to convert it using the transformation matrix and tensor transformation rule but it turns out that
    [tex]
    F^{\mu \nu }F_{\mu \nu }≠2(B^2-\frac{E^2}{c^2})
    [/tex]

    Can anyone give me some idea how to solve this?
    Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 18, 2012 #2
    No one can help?
     
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