I realize there's probably an answer to this question somewhere on this site, but it would seem I'm having trouble picking the right search terms.
In my multivariable calculus class, the professor just spent some time going over the relationships between the integral theorems of calculus by essentially saying that they all are examples of how integrating some function over the boundary of a domain (path, surface, solid) is the same as integrating some form of its "derivative" over the interior of that domain. The issue is, while the boundary operator seems to be consistent across all of the integral theorems presented (line integral theorem, Stokes' theorem, and Gauss' theorem), the notion of the derivative is different in each one: for Stokes', it's the curl; for Gauss', it's the divergence; and for the line integral theorem, it's the gradient.
I was wondering: is there any natural way of generalizing the curl, divergence, and gradient operators, so that they become special cases of some more broad notion of "taking the derivative"? Or perhaps to other dimensions? (Or is there something special about $\mathbb R^{3}$ that makes it unique for vector calculus?)
(Sorry if this question is too open-ended or ill-defined for this site; regardless, if anybody has any suggestions on topics to study or other resources to consult, I would very much appreciate those as well.)