Why does $epsilon_{munurhosigma}partial^{rho}partial^mu A^nu-epsilon_{munurhosigma}partial^{rho}partial^nu A^mu=0$?

Question

The tensor $F^{munu}$ is defined as $partial^mu A^nu-partial^nu A^mu$. Why is the equation
$$epsilon_{munurhosigma}partial^{rho} F^{munu} = 0$$
identically satisfied by $F^{munu}=partial^mu A^nu-partial^nu A^mu$?
We have
$$epsilon_{munurhosigma}partial^{rho} (partial^mu A^nu-partial^nu A^mu)=epsilon_{munurhosigma}partial^{rho}partial^mu A^nu-epsilon_{munurhosigma}partial^{rho}partial^nu A^mu$$
I am told that since $epsilon$ is antisymmetric and $partialpartial$ is symmetric (no doubts about it), the product $(text{antisymmetric})(text{symmetric})=0$. Here is my attempt at understanding this last statement:
$$epsilon_{munurhosigma}partial^{rho}partial^mu A^nu=epsilon_{munurhosigma}partial^{mu}partial^rho A^nu=-epsilon_{rhonumusigma}partial^{mu}partial^rho A^nu=-epsilon_{munurhosigma}partial^{rho}partial^mu A^nu$$

Step 1: symmetry of $partial^rhopartial^mu$
Step 2: antisymmetry of $epsilon$
Step 3: I call $mu$ $rho$ and viceversa, since they are to be summed over

Then I got $epsilon_{munurhosigma}partial^{rho}partial^mu A^nu=-epsilon_{munurhosigma}partial^{rho}partial^mu A^nu=0$. Are these steps right?

Emmy · Accepted Answer

You got it a little bit wrong, but the main ideas are here. Starting from $epsilon_{mu nurhosigma}partial^rhopartial^mu A^nu$, you commute $partial^rho$ with $partial^mu$ without changing anything. Then, you use anti symmetry of $epsilon$ to exchange the two indices $mu$ and $rho$. At this point: $$epsilon_{mu nurhosigma}partial^rhopartial^mu A^nu=-epsilon_{rho numusigma}partial^mupartial^rho A^nu$$
And since $mu$ and $rho$ are dummy indices, you can exchange them in the right hand side:$$epsilon_{mu nurhosigma}partial^rhopartial^mu A^nu=-epsilon_{munurhosigma}partial^rhopartial^mu A^nu$$
Since that thing is equal to its opposite, it should be zero indeed.

Why does $epsilon_{munurhosigma}partial^{rho}partial^mu A^nu-epsilon_{munurhosigma}partial^{rho}partial^nu A^mu=0$?

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