Maxwell equations from tensor notation to component notation

[spaghetti3451]

Homework Statement

Verify that $\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}+\partial_{\lambda}F_{\mu\nu}$ is equivalent to $\partial_{[\mu}F_{\nu\lambda]}=0$,

and that they are both equivalent to $\tilde{\epsilon}^{ijk}\partial_{j}E_{k}+\partial_{0}B^{i}=0$ and $\partial_{i}B^{i}=0$.

Homework Equations

The Attempt at a Solution

$\partial_{[\mu}F_{\nu\lambda]}=0$

$\implies \frac{1}{3!}(\partial_{\mu}F_{\nu\lambda}-\partial_{\mu}F_{\lambda\nu}-\partial_{\nu}F_{\mu\lambda}+\partial_{\nu}F_{\lambda\mu}-\partial_{\lambda}F_{\nu\mu}+\partial_{\lambda}F_{\mu\nu})=0$, using the definition of antisymmetrization of a tensor

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

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

$\implies 2(\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}+\partial_{\lambda}F_{\mu\nu})=0$

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

Is my working so far correct?

 

[TSny]

Yes, I believe so.