[Koshkin]

> vector calculus isn't very intuitive either

That's, by the way, why we have the calculus of differential forms which, unlike vectors with all their flavors (free; polar; axial/pseudo), have a clear geometric meaning, and with which many statements about fields acquire an especially simple form. There are many excellent guides; for the motivation, see, for example, https://www.jpier.org/PIER/pier148/09.14063009.pdf

[onedognight]

Differential forms are a sub-algebra of the geometric algebra, so you don’t give up any of the beautiful things you mentioned.

[fango]

yes ... V.I.Arnold - important symplectic geometry author tells us: "Hamilton mechanics cannot be understood without differential forms ...". Thanks for your link.