How do Maxwells equations result from the field tensor?

[Azelketh]

Hi,
I've been trying to solve problem 2.1 a in Peskin and schroeder, an introduction to QFT.
The problem is to derive Maxwells equations for free space, which I have almost managed to do,
using the Euler- lagrange euqation And the definition of the field tensor as
$$F_{μv} = d_μ A_v - d_v A_μ$$
So I have managed to get to;
$$0=d_μ F^{μv}$$
But I am unable to see how this shows Maxwells equations.
Any points would be appreciated.
Thanks.

 

[Vanadium 50]

In what variables are Maxwell's equations usually expressed?

 

[Azelketh]

My apologies for a less than comprehensive post at first;

Well, usually in the vector form of E and B or the 4 vector A,
where;
$$A= ( \phi, \vec{A} )$$

where$$\vec{A}$$is the magnetic vector
$$\phi$$ is the scalar electric potential.

$$E = -∇\phi - \frac{\partial\vec{A}}{\partial t}$$
And,
$$B= ∇ X \vec{A}$$
It's clear to me that 2 of maxwells equations result directly from this definition;
$$∇.B = ∇. ( ∇ X \vec{A} ) = 0$$
and
$$∇ X E = ∇ X ∇\phi - ∇ X \frac{\partial\vec{A}}{\partial t}$$$$∇ X E = ∇(∇ X \phi) - \frac{\partial (∇ X\vec{A} )}{\partial t}$$$$∇ X E = 0 - \frac{\partial ( B )}{\partial t}$$$$∇ X E = - \frac{\partial B}{\partial t}<br />$$

Which leaves
$$∇.E = 0$$
and
$$∇ X B = \frac{\partial E}{\partial t}$$
to be found from
$$0=\partial_μ F^{μv}$$

So I have tried putting in
$$F_{μv} = \partial_μ A_v - \partial_v A_μ$$
To give;
$$0=\partial_μ ( \partial_μ A_v - \partial_v A_μ )$$

$$0=\partial_μ \partial_μ A_v - \partial_μ \partial_v A_μ$$
so I then tried expanding this out, hoping that some terms would cancel and that I would recognize others and perhaps then they would be close to the E and B field formulation that I am more familier with.
This yielded;
$$\partial_μ \partial_μ A_v= \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \vec(A)}{\partial t^2} + ∇^2 \phi + ∇^2 \vec{A}$$
And
$$\partial_μ \partial_v A_μ = \partial_v \partial_μ A_μ = \frac{\partial^2 \phi}{\partial t^2} - \frac{\partial \vec(∇A)}{\partial t} - \frac{\partial ∇\phi}{\partial t} + ∇^2 \vec{A}$$
which when combined gives;
$$0= \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \vec(A)}{\partial t^2} + ∇^2 \phi + ∇^2 \vec{A} - ( \frac{\partial^2 \phi}{\partial t^2} - \frac{\partial \vec(∇A)}{\partial t} - \frac{\partial ∇\phi}{\partial t} + ∇^2 \vec{A} )$$
$$0= \frac{\partial^2 \vec(A)}{\partial t^2} + ∇^2 \phi + \frac{\partial \vec(∇A)}{\partial t} + \frac{\partial ∇\phi}{\partial t}$$
which is where I am scratching my head...

EDIT:replaced d's with $$\partial$$ as per dextercioby's suggestion.

 

[dextercioby]

I suggest one tiny bit of LaTex: $\partial$, i.e. \partial.

 

[Vanadium 50]

Wouldn't it make more sense to work with E's and B's if you want equations giving you E's and B's?

 

[jfy4]

at the end of post #3, like in the last 4 lines, you have vector things added to scalar things, which is no good. And on the left hand side of those lines there is a free index $\nu$ I think... Those are vectors. You can write it in a vector format, but the index is not summed over.

 

[Azelketh]

I see how I've gone wrong on the last few lines.

Thanks for your help vanadium 50 , jfy4 and dextercioby!