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Tensor calculus, dummy indices

  1. Dec 16, 2014 #1
    Hi there. When I have dummy indices in a tensor equation with separate terms, I wanted to know if I can rename the dummies in the separate terms.

    I have, in particular:

    [tex]\displaystyle w_k=-\frac{1}{4}\epsilon_{kpq}\left [ \frac{\partial u_p}{\partial x_q}-\frac{\partial u_q}{\partial x_p} \right ]=-\frac{1}{4}\epsilon_{kpq}\frac{\partial u_p}{\partial x_q}+\frac{1}{4}\epsilon_{kpq}\frac{\partial u_q}{\partial x_p}=\frac{1}{4}\epsilon_{kqp}\frac{\partial u_p}{\partial x_q}+\frac{1}{4}\epsilon_{kpq}\frac{\partial u_q}{\partial x_p} [/tex]

    I've used that [tex]\epsilon_{kpq}=-\epsilon_{kqp}[/tex]

    So, if I can change the dummy indices for the separate terms I can use that:

    [tex]\frac{1}{4}\epsilon_{kpq}\frac{\partial u_q}{\partial x_p}=\frac{1}{4}\epsilon_{kqp}\frac{\partial u_p}{\partial x_q}[/tex]
    to get:
    [tex] w_k= \frac{1}{2} \epsilon_{kqp} \frac{\partial u_p}{\partial x_q} [/tex]

    which is the result I'm looking for, but I wasn't sure if the last step is right.
     
    Last edited: Dec 16, 2014
  2. jcsd
  3. Dec 16, 2014 #2
    mhm, I think that it suffices to just take separate sums for the different terms to see that yes, I just can change the dummies.
     
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