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.