Div, grad, and curl are a manifestation of de Rham cohomology that make use of a lot of lucky coincidences. See https://en.wikipedia.org/wiki/De_Rham_cohomology for the formal definition (in which curl becomes the exterior derivative from 1-forms to 2-forms) and https://web.ma.utexas.edu/users/a.debray/lecture_notes/idea_... for a nice exposé.

You use the outer derivative which generalizes the curl as well as a few similar constructs. Technically it's slightly different as it returns a bivector (which is a bit like a plane spanned by 2 vectors), but in 3D both vectors and bivectors form a 3D space and you can freely convert between the two. The difference between the divergence and the curl is basically whether you switch between vectors and bivectors before or after you take the outer derivative.