I think that the explanation of circulation, which is pretty straightforward, detracts a little from the explanation of curl. Firstly, the whirlpool example is little confusing, because with a boat in water, as soon as you match the water’s speed, the force is zero. So it’s not analogous to circulation in curling vector fields (i.e. the magnetic field) and confuses the definition of conservative. In fact, a boat twisting in a whirlpool IS conservative. The water does work on the boat which is stored in rotational kinetic energy which could then be extracted from the boat. It’s not analogous to energy gain due to a curling field because in such a field the force is perpendicular to the field and not in the direction of the path.

A much better example is when you first talked about a boat moving AROUND the whirlpool, i.e. anti-centripetal force that only changes direction and whose work can never be re-extracted from the system because the force is always perpendicular to the path of motion (inward), just like a magnetic field creating a cyclotron. I can’t say that I have a good enough intuition to know for sure, but the river going in different speeds right next to each other seemed to make sense too. In any case, when I then try to “shrink down” the concept to an infinitely small area to get an intuition of curl, I can’t wrap my head around the role of area in circulation.

I understand path integrals, and I get calculus, but I still don’t understand the physical meaning of circulation per unit area. What area is it? The area inside the path? Since area drops off with the square of the size, how can you shrink it the same as you can path length?

What I can’t see is how the dotting of a vector along an infinitely small path (i.e. the circulation) in a continuous field can not produce a zero result in an infinitely small path where the field, assuming a smooth field gradient, becomes essentially constant. The whole reason that circulation in a field need not be conservative is because, like you said, the field does not need to be symmetric, and can follow the path of circulation to an extent. But it seems like shrinking down to an infinitely small path would void this possibility.

Finally, how do you relate the concept of curl to the concepts of divergence and/or cross-product as in the latter del-cross-F notation? Is this in another discussion? The relationship is not making sense to me without the math.

Thanks so much and sorry for such a long post!