Show that $\nabla\cdot\ast F = 0$ (divergence of the dual of the EMF tensor) is a geometric frame independent version of $F_{ab,c} + F_{bc,a} + F_{ca,b} = 0$, where $F$ is the electromagnetic field tensor
I don't know how to answer this question.
We know that $\ast F_{ab} = \frac12 F^{\mu \nu} \epsilon _{\mu \nu a b}$, but i am having problem to define the divergence here, i mean, if we denote $\nabla = \sum_{i=0}^3 \partial_{i}$, we will need to compute a lot of things, i mean, F already has 16 components (i am not counting the symmetries here), plus this derivatives, it will be really a problematic computation. Summarizing, i would demonstrate at first that this divergence is zero, and so demonstrate that it is invariant under Lorentz transformation, but i think this is not the best way.
So i gave up this alternative, and i would appreciate another way to solve it.