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How can we know that the electromagnetic tensor $F^munu$ can be written in terms of a four-vector potential $A^mu$ as $F^mu nu = partial^mu A^nu - partial^nu A^mu$? In the vector calculus approach, this is not really hard to see (under reasonable 'smoothness' conditions on the fields), but I would like to know how one would see this in the four-vector approach.

More specifically, how can we prove (mathematically) that given the electromagnetic tensor, there exists a four-vector such that $F^munu = partial^mu A^nu - partial^nu A^mu$.

The Bianchi identity $mathrmdF~=~0$ together with Poincare lemma guarantee the local existence of $A$ in contractible regions of spacetime. See also this related Phys.SE post.

One way to write the homogenous Maxwell's equations is
with the Levi-Civita symbol $epsilon$:
$$epsilon^alphabetamunu partial_beta F_munu = 0$$

Solution to this is obviously (with arbitrary potential $A$):
$$F_munu = partial_mu A_nu - partial_nu A_mu$$

It is easy to verify using the antisymmetry of $epsilon^alphabetamunu$
upon swapping any 2 indexes, together with $partial_mupartial_nu = partial_nupartial_mu$.

You are asking "how we know...". This may not be a fair question. We created this formalism. You could also ask how do we know that we can write Maxwell's equations using vectors. Long ago they were not, they were written as a large set of coupled scalar (scalar type) equations. The formalism of vector notation was still evolving and being accepted and one has to PROVE that a set of quantities actually behaves like a vector under coordinate transformations.

This is a key to understanding the 4-vector approach. There is a scalar E field potential, Phi, and a vector potential A, in classical electrodynamics.

Once Einstein presented Lorentz invariance in physics (I'm not going to write extensively about that history here) we started on the path of putting all equations in a covariant format. It is the nature of light that governs this invariance, and the postulate that the speed of light is the same for all relatively interital observers. Even Newton's laws of mechanics were elevated to a covariant 4-vector form, as was momentum (E, p) where Energy, E, was thought to be a scalar.

We know that electromagnetism needs to be Lorentz invariant and this motivates elevating Phi to the time component of a 4-vector, just like E is the time component of 4-momentum. This is also indicated by seeing how the equations transform under Lorentz. The 4-potential (Phi, A) must be as is to obey this symmetry. Then the rest follows.