Rank 3 tensor created by taking the derivative of electromagnetic field tensor

[mjordan2nd]

Homework Statement

Show that the rank 3 tensor $$S_{\alpha \beta \gamma}=F_{\alpha \beta , \gamma} + F_{\beta \gamma , \alpha} + F_{\gamma \alpha , \beta}$$ is completely antisymmetric.

I just don't feel comfortable doing this stuff at all. Each of the three terms seems like they should be exactly the same to me. Could someone show me how I would start doing something like this please? Furthermore, if this is a rank 3 tensor, what would it mean if this tensor equals 0?

Thanks. :-\

 

[gabbagabbahey]

If $S_{\alpha\beta\gamma}$ is completely antisymmetric, then

$$S_{\alpha\beta\gamma}=-S_{\alpha\gamma\beta}$$
$$S_{\alpha\beta\gamma}=-S_{\beta\alpha\gamma}$$

and

$$S_{\alpha\beta\gamma}=-S_{\gamma\beta\alpha}$$

That is, $S_{\alpha\beta\gamma}$ is antisymmetric on all pairs of indices...

So, start by comparing $S_{\alpha\beta\gamma}$ to $S_{\alpha\gamma\beta}$ using the definition you posted...