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

mjordan2nd

1. The problem statement, all variables and given/known data

Show that the rank 3 tensor [tex]S_{\alpha \beta \gamma}=F_{\alpha \beta , \gamma} + F_{\beta \gamma , \alpha} + F_{\gamma \alpha , \beta}[/tex] 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 [itex]S_{\alpha\beta\gamma}[/itex] is completely antisymmetric, then

[tex]S_{\alpha\beta\gamma}=-S_{\alpha\gamma\beta}[/tex]
[tex]S_{\alpha\beta\gamma}=-S_{\beta\alpha\gamma}[/tex]

and

[tex]S_{\alpha\beta\gamma}=-S_{\gamma\beta\alpha}[/tex]

That is, [itex]S_{\alpha\beta\gamma}[/itex] is antisymmetric on all pairs of indices....

So, start by comparing [itex]S_{\alpha\beta\gamma}[/itex] to [itex]S_{\alpha\gamma\beta}[/itex] using the definition you posted...