Not sure, vector calculus isn't very intuitive either if you start with it, with Pauli/gamma matrices it's even worse. Having studied Physics myself, I haven't encountered one lecture where they were able to give a reasonable geometric explanation. (Symplectic Geometry and GA provide it) IMHO if the the same amount of effort was used to force vector calculus into people's heads, it should be doable with these tools as well. Unfortunately there is already a lack of books about the topics