Maxwell equations in Special Relativity I'm currently studying special relativity, and with it, tensor algebra. I have some difficulties in deriving a tensor differential relation involving the field tensor $F_{\mu\nu}$.
I have the Maxwell equations:
$$\partial_{\mu}F^{\mu\nu}=J^{\nu} \tag{1}$$
$$\epsilon^{\mu\nu\sigma\rho}\partial_{\nu}F_{\sigma\rho}=0 \tag{2}$$
where $F^{\mu\nu}=-F^{\nu\mu}$ and I want to derive:
$$
\partial_{\mu}F_{\nu\sigma}+\partial_{\nu}F_{\sigma\mu}+\partial_{\sigma}F_{\mu\nu}=0 \tag{4}$$
Can you give me some advice on how to do it?
 A: Note that $ \epsilon^{\mu\nu\sigma\rho} $ is the totally anti-symmetric tensor which is defined as:
$$ \epsilon^{\mu\nu\sigma\rho}=
    \begin{cases}
      1, & \text{for cyclic permutations of } (\mu,\nu,\sigma,\rho) \\
      0, & \text{otherwise}
    \end{cases} $$
Start by checking that $\epsilon^{\mu\nu\sigma\rho} \partial_{(\nu}F_{\sigma\rho)} = \epsilon^{\mu\nu\sigma\rho} \partial_{\nu}F_{\sigma\rho}$. After that, I am sure you understand how everything follows.
A: I will not give a proof but I will give instructions on how to prove it.  
You must start from the second equation that you gave. Then, you should write down the whole sum with arbitrary choice of the index μ(because what you choose μ to be will only affect the sign in front of each term in the sum and because the left side of the equation is equal to zero this choice does not make a difference), so you might as well choose μ=0.
After expanding the sum, use the antisymmetry of the electromagnetic field tensor(for example this means that the diagonal components of the tensor are zero) along with the definition of the rank-4 Levi-Civita symbol(for example this leaves you only with the choice of 1, 2 or 3 for the indices ν, σ, ρ and none should be the same as another) to obtain the last formula.
