- #1
Telemachus
- 820
- 30
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.
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.
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