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Sum of tensors

  1. Jun 2, 2005 #1
    show that [tex]B_{ij}[/tex] can be written as the sum of a symmetric tensor
    [tex]B^S_{ij}[/tex] and an antisymmetric tensor [tex]B^A_{ij}[/tex]

    i dont know how to do this one.
    for a symmetric tensor we have
    [tex]B^S_{ij} = B^S_{ji}[/tex]

    and for an antisymmetric tensor we have
    [tex]B^A_{ij} = -B^A_{ji}[/tex]

    the only thing my book says is that the sum should be a tensor of the same type.
     
  2. jcsd
  3. Jun 2, 2005 #2

    robphy

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    Hint:
    define the "symmetric part of B" to be
    [tex](B_{ij})^S = \frac{1}{2}\left(B_{ij}+ B_{ji}\right)[/tex]
    ... quite analogous to defining the "real part of a complex number z" as (z+z*)/2. [Check for yourself that this "symmetric part" is truly symmetric.]

    I'm sure you can finish the rest.
     
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