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The first bullet would be read "$A$ dot $B$" or "The dot product of $A$ and $B$" The second bullet would be read "$A$ cross $B$" or "The cross product of $A$ and $B$"
So I think $\partial_\mu F$ should also be 4 vectors, each being the directional derivative along a coordinate axis. It's a single covector, not a vector and not a collection of vectors. A gradient of a scalar field is a classic example of how we obtain a covector. In index notation, a free index such as your $\mu$ is interpreted as having the potential ...
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