1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: A Tensor Problem: A skew-symmetric tensor and another tensor

  1. Oct 3, 2012 #1
    1. The problem statement, all variables and given/known data

    If [itex]A_{ij}[/itex] is a skew-symmetric tensor, and [itex]B_{ij}[/itex] is a second-order tensor, evaluate the expression

    [tex](B_{ij} B_{kl} + B_{il}B_{kj})A_{ik}[/tex]

    and express the final answer in its simplest form.

    2. Relevant equations

    For a skew-symmetric tensor, [itex]A_{ik}=-A_{ki}[/itex]

    3. The attempt at a solution

    I'm stuck and unsure what's the first step. I notice that the expression in the bracket looks similar to what happens when two Levi-Civita symbols come together to form an expression of two pairs of the Kronecker delta. Other than that I'm quite lost. Can I get a tip please?
  2. jcsd
  3. Oct 3, 2012 #2
    You are probably summing over all the repeated indices, right? Remember then that the dummy indices are arbitrary, and you can for example swap k and i if you feel like it. Using this, maybe you can write the expression into a form where you take BijBkl as a common factor, multiplying some expression containing the tensor A.
  4. Oct 3, 2012 #3
    Here's what I have so far:

    (B_{ij} B_{kl} + B_{il}B_{kj})A_{ik}\\
    =B_{ij} B_{1l}A_{i1}+B_{ij}B_{2l}A_{i2}+B_{il} B_{1l}A_{i1}+B_{il}B_{2j}A_{i2}\\
    =B_{1j} B_{1l}A_{11}+B_{2j} B_{1l}A_{21}+B_{1j}B_{2l}A_{12}+B_{2j}B_{2l}A_{22}+B_{1l} B_{1l}A_{11}+B_{2l} B_{1l}A_{21}+B_{1l}B_{2j}A_{12}+B_{2l}B_{2j}A_{22}\\
    =2B_{1j} B_{1l}A_{11}+2B_{2l}B_{2j}A_{22}[/tex]

    So the final answer that I can give is... [tex]2\sum_i B_{ij} B_{il} A_{ii}[/tex] or in the Einstein summation, [tex]2B_{mj} B_{ml} A_{nn}[/tex] with nn no sum.

    If this is correct, is there any other way to write this without the no sum?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Similar Threads for Tensor Problem skew Date
Vector Calculus - Tensor Identity Problem Dec 7, 2016
Tensor problems please help Apr 27, 2012
Simple problem - contracting tensors Aug 16, 2011
Fluids/tensors problem Feb 7, 2011
Tensor product of vector space problems Jun 25, 2010