I have to show, how the contravariant components of the electromagnetic field tensor behave under Lorentz transformation.
I guess the answer should look something like this $$F'^{\mu\nu}=\frac{\partial x'^\alpha}{\partial x^\gamma}\frac{\partial x'^\beta}{\partial x^\delta}=\Lambda^\alpha_\gamma\Lambda^\beta_\delta F^{\gamma\delta}$$$$F'^{\mu\nu}=\frac{\partial x'^\alpha}{\partial x^\gamma}\frac{\partial x'^\beta}{\partial x^\delta}=\Lambda^\alpha{}_\gamma\Lambda^\beta{}_\delta F^{\gamma\delta}$$ That's not a big deal, one can see this immediately (although I don't really get the message of this exercise, since $F^{\mu\nu}$ just transforms as a 2nd rank tensor has to by definition)
However, the exercise wants me to derive this from the inhomogeneous Maxwell equations in covariant formulation:
$$\partial_\mu F^{\mu\nu}=\mu_0j^\nu$$
I know how $\partial_\mu$ transforms: $$\partial'_\mu=\frac{\partial x^\nu}{\partial x'^\nu}\frac{\partial}{\partial x^\nu}=\Lambda_\mu^\nu \partial_nu=(\Lambda^{-1})^\nu_\mu \partial_\nu$$$$\partial'_\mu=\frac{\partial x^\nu}{\partial x'^\nu}\frac{\partial}{\partial x^\nu}=\Lambda_\mu{}^\nu \partial_\nu=(\Lambda^{-1})^\nu{}_\mu \partial_\nu$$
as well as $j^\nu$:
$$j'^\mu=\frac{\partial x'^\mu}{\partial x^\nu}j^\nu=\Lambda^\mu_\nu j^\nu$$$$j'^\mu=\frac{\partial x'^\mu}{\partial x^\nu}j^\nu=\Lambda^\mu{}_\nu j^\nu$$
But how do I link these equations to the 3d one and arrive at the 1st ?
Any hints?
EDIT:How do I insert horizontal spacings in front of my subscripts/superscritps?
