Tensor calculus, dummy indices

[Telemachus]

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:

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

I've used that $$\epsilon_{kpq}=-\epsilon_{kqp}$$

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

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

which is the result I'm looking for, but I wasn't sure if the last step is right.

 

[Telemachus]

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.