Deriving Maxwell's equations and stress tensor from action in Peskin & Schroeder 2.1

[Matterwave]

Hello, so I was asked a question in two parts (Peskin & Schroeder problem 2.1). The first part asked me to derive the source-free Maxwell's equations from the action:
$$S=\int{d^4 x \frac{-1}{4}F_{\mu\nu}F^{\mu\nu}}$$

Given that the vector potential itself is the dynamical variable.

I derived (source-free) Gauss's law and Ampere's law from that action by setting the variation to zero. Is it possible to derive Faraday's law and Gauss's law for magnetism from that action? I was under the impression that those two laws just came from our definition of the E and B field. Since I carried through the whole process and got the 4 equations of motion:
$$\partial_\nu F^{\mu\nu}=0$$

I don't see how I could extract the other two Maxwell's equations from this action. Certainly I can't extract them from this equation of motion.

The second part of the problem asked me to find the energy-momentum-stress tensor of E&M. I started by using Noether's theorem, using the translational (in 4 directions) invariance of the action. I think I was going somewhere until I hit a roadblock.

In a nutshell, I used the transformations:
$$x^\mu \rightarrow x^\mu+\epsilon^\mu$$

As my symmetry, and then as per the usual formulation I made $\epsilon$ a function of the space-time coordinates instead of a constant vector. I reached a point where:
$$F_{\mu\nu} \rightarrow F_{\mu\nu} + \epsilon^\rho(\partial_\rho F_{\mu\nu})+(\partial_\rho A_\nu)(\partial_\mu \epsilon^\rho)-(\partial_\rho A_\mu)(\partial_\nu \epsilon^\rho)$$

I think if I can just take care of the last 2 terms, it should be ok, but I can't think of a way to manipulate them in such a way to combine them into one term, or factor it or something like that. Any ideas? Thanks!

EDIT
If I'm not providing enough info, just tell me, but some help would be appreciated, thanks!

 

[dextercioby]

I don't understand. dF=0 is, because F=dA and A is a genuine 1-form. You don't need the action for that.

As for the E_M tensor part, why do you take the parameters as functions of x ?

 

[vela]

I don't see how I could extract the other two Maxwell's equations from this action. Certainly I can't extract them from this equation of motion.

You should take a look in Jackson or some other text on electromagnetism. If I recall correctly, the other two equations are the result of a dual formulation of $\partial_\mu F^{\mu\nu}=0$.

 

[Matterwave]

I don't understand. dF=0 is, because F=dA and A is a genuine 1-form. You don't need the action for that.

As for the E_M tensor part, why do you take the parameters as functions of x ?

I'm not sure what you mean by the top part. Are you saying the 2 Maxwell equations I derived need not come from the action?

I make the parameter a function of x because that's how I was taught to find the conserved current.

 

[Matterwave]

I figured it out!