# Derivative of covariant EM tensor

I cannot seem to prove that the derivative of the duel tensor = 0.

$$\frac{1}{2}\partial_{\alpha}\epsilon^{\alpha \beta \gamma \delta} F_{\gamma \delta} = 0.$$

Writing this out I get (for some fixed $\alpha$ and $\beta$),

$$\partial_{\alpha} (\partial_{\gamma}A_{\delta} - \partial_{\delta}A_{\gamma}).$$

From here I get stuck.

Any ideas?

$\epsilon^{xyab}\partial_a\partial_b=(-\epsilon^{xyba})\partial_a\partial_b=(-\epsilon^{xyba})\partial_b\partial_a=(-\epsilon^{xycd})\partial_c\partial_d=-\epsilon^{xyab}\partial_a\partial_b$
$\Longrightarrow\ \ \epsilon^{xyab}\partial_a\partial_b=0$
More abstractly, if $A^{ab}=-A^{ba}$ and $S_{ab}=S_{ba}$, then $A^{ab}S_{ab}=0$.