To truly understand how gauge field is required in the 4d version of the electromagnetism equation,

However when asking what and how covariant derivative is defined and derived, AI hardly can be helpful other reciting dull and dry textbook words.

The best brains of human beings are irreplicable by AI. So here I’d revisit eigenchris’s tensor calculus series to grasp the concept of covariant derivative:

The Abstract Covariant Derivative includes Levi-Civita Connection, and Boring connection. Levi-Civita connection is more interesting and useful in physics.

So the key here is when elevate to 4D to study electromagnetism, we found it’s a U1 symmetry but not as simple as non-curved U1, have to introduce covariant derivative instead of simple normal derivative to account for it, hence the Christopher symbol part as in Tensor calculus, but here it is called “gauge field” to preserve gauge invariant symmetry, now everything works out in the F miu niu tensor field, ideal to describe EM with two properties E and B. All worked out!