Most mathematicians consider the equations for div and curl to be the “dirty equations” while the “perfect mathematical concept” is the exterior derivative of a 1-form.
If the objective is to generalize, then differentiation is simply the adjoint of the boundary operation, which is a homomorphism; extending the notions of closed and exact forms to general differential forms along with that the differential operator is nilpotent allows setting up de Rham cohomology, which also has analogs by multiple authors.
