Newton and Leibniz are commonly credited with discovering the fundamental theorem of calculus, and therefore, for the invention/discovery of the subject itself. It's been discussed many times on this site that there were lots of other mathematicians making contributions to the development of calculus from antiquity through Fermat. It seems obvious that people must have known about partial derivatives once they knew about derivatives at all, because there's no shortage of functions of several variables running around in physics. It also seems obvious that they must have known about vectors since Newton because Newtonian physics is filled with vector quantities they want to differentiate or integrate in order to calculate e.g. work. Maybe they didn't call them vectors yet, but that doesn't seem important.
What was the state of knowledge of calculus like after the fundamental theorem and before Green's Theorem or Stokes? Was it considered "open" to generalize the fundamental theorem of calculus? Did nobody think about generalizations until electro-magnetism was of interest? Or maybe these results were already known to Newton in some form or another. It would amaze me if it took us all the way to e.g. Gauss.
My university education gave me this narrative about calculus that's focused on the 1-dimensional case: we had the pre-rigorous stage, the crisis, and the rigorous stage, culminating in what we now call analysis. But this narrative spans over a century, and if I recall correctly, Stokes' theorem was published in the same ballpark as Cauchy's Course d'Analyse. Where does it fit into this story? I'm especially interested in earlier spans of time, for example, if there were any noteworthy pre-rigorous "proofs" of vector formulations of the fundamental theorem of calculus or if anyone ever speculated about higher dimensional analogues before they were proven.