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I am studying Ex.1 of chapter $2$ in Quantum Field Theory's book by Peskin and Schroeder, whose solution is available.

Note we're working with Maxwell's equations in vacuum.

In section a) we are asked to derive all Maxwell's equations given the the following Lagrangian and the definition of the antysymmetric field tensor $F_{\mu \nu}$

$$\mathscr{L} = -\frac 1 4 F_{\mu \nu} F^{\mu \nu}, \ \ \ \ \ \ F_{\mu \nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$$

I understand how to use the Euler-Lagrange equation to get $\partial_{\mu} F^{\mu \nu}=0$ (which represents two of Maxwell's equations).

I also know how to get the remaining two out of an equation satified by the antysymmetric field tensor $F_{\mu \nu}$. Such an equation is known as Bianchi's identity:

$$\partial_{\lambda} F_{\mu \nu} + \partial_{\mu} F_{\nu \lambda} + \partial_{\nu} F_{\mu \lambda} = 0$$

My issue is that I do not understand how to write Maxwell's equations $\partial_{\mu} F^{\mu \nu}=0$ and $\partial_{\lambda} F_{\mu \nu} + \partial_{\mu} F_{\nu \lambda} + \partial_{\nu} F_{\mu \lambda} = 0$ in standard form.

$$\partial^i E^i = 0, \ \ \epsilon^{i j k} \partial^j B^k - \partial^0 E^i = 0, \ \ \epsilon^{i j k} \partial^j E^k = 0, \ \ \partial^i B^i = 0$$

We are given that $E^i = - F^{0 i}$ and $\epsilon^{i j k} B^k =-F^{i j}$

The reason why I am not getting it is because I am not used to work with the Levi-Civita symbol.

In Peskin and Schroeder's book they use the convention $\epsilon^{0 1 2 3} := 1$

Any help is appreciated.

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Starting with the $F_{\mu,\nu}$ form of the Maxwell equations, rewrite them in terms of $E$ and $B$, by just plugging in the definitions of $E$ and $B$. That may look scary, because in the Bianchi identities you have lots of choices for $\lambda, \mu,\nu$, but it's not so bad because many of those equations are redundant, in view of the cyclic symmetry of the equations with respect to $\lambda, \mu,\nu$ and the antisymmetry of $F$. Finally, observe that the equations you got, in terms of $E$ and $B$, are exactly the (components of) the Maxwell equations in standard form.

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