Grad, curl and div are the same operator wearing three hats — and the four big theorems are one theorem.
Grad, curl and div are not three ideas. They are one operator, applied to objects of three different shapes.
Step through the degrees and watch the dictionary line up.
Vector calculus arrives as a pile of separate results: three derivative operators with unrelated definitions, four integral theorems that look nothing alike, and two identities you memorize because they keep coming up. It is a lot to hold, and none of it looks inevitable.
Differential forms collapse the pile. Instead of vector fields there are objects graded by degree — things you integrate over points, over curves, over surfaces, over volumes. There is exactly one derivative, written d, which raises degree by one. Grad, curl and div are what d looks like at degrees 0, 1 and 2, and the reason there are three of them in three dimensions is simply that there are three rungs.
The identities stop being facts to memorize too. curl(grad f) = 0 and div(curl F) = 0 are both the single statement d ∘ d = 0, read at two different degrees. One line, two corollaries.
And the four theorems become one: ∫∂M ω = ∫M dω. What you get integrating a derivative over a region equals what you get integrating the thing itself over that region's boundary. Feed it a curve and you get the fundamental theorem for line integrals; feed it a flat region and you get Green; a surface in space, Stokes; a solid, the divergence theorem. Even the ordinary fundamental theorem of calculus is the degree-0 case, where the "boundary" of an interval is its two endpoints.