I know both Stokes' and divergence theorem are a part of more generalised (albeit beyond my scope) theorem. And I also know a similar question to this exists, but I want to clarify that that question doesn't talk about divergence theorem in 2 dimensions. I have started engineering physics quite recently and am introduced to the prerequisites like divergence, curl etc. (vector calculus) Having studied from Khan academy, I have the following intuition for stokes theorem (and it's not a generalized form of it but how I think of it intuitively):
In traversing a directed closed path (henceforth, contour), the sum of values of a vector function in a direction of the contour at all points (viz. dot product of small length element $dl$ and vector at that point) is proportional to how much curl the surface defined by that contour encloses i.e. sum of local curls times small area, $da$ over the entire enclosed surface.
And the following for divergence theorem in two dimensions again from Khan Academy:
The sum of values of a vector function in a direction normal to the path at all points multiplied by the little length $dl$ at which it is being calculated is proportional to how much divergence is enclosed in small areas of that surface. (and it makes sense because first we're calculating how much the vector is "outwards" at a point and then multiplying by a small portion of the path, integrating over the entire path, essentially giving us how much of something is flowing out of the loop. Divergence, in essence, is also the measure of how much in or out something is flowing.)
My questions are:
Is what I'm saying correct? if not, please suggest the part where I am not making sense.
Aren't the two theorems suspiciously similiar and so different at the same time, just because in one we're taking a dot along the path, and other perpendicular to the path, one corresponds to curl and other to divergence.