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Incidentally, with antisymmetric covectors (differential forms) one can define another type of derivative called 'd' that doesn't seem to have any direction (not a directional derivative).

If you add a metric tensor, then you can define another derivative for a vector field called (at least in physics) the covariant derivative, which is a partial derivative plus a connection term.

My question is which derivative is really the directional derivative of a vector field: the Lie derivative, or the covariant derivative?

Also, probably related, is the definition of a directional derivative unique?

And how can there be a derivative 'd' that has no direction? That would have to imply, roughly speaking, that 'd' is a derivative averaged around a circle at a point?