From what I understand, curl is the "spinning" at a small point in the vector field (and not the overall vector field's appearance) So although both these vector fields look the same (apart from the origin), and a particle placed in the field will move in a circular path around the origin, to compute curl, I need to do $\nabla \times \pmb{F}$
For the first one, I get $-2\pmb{k}$ and for the second one I get $\pmb{0}$ but how do I actually interpret this?
For the second one, I would understand the particle doesn't spin as it goes in that circular path, but for the first one, would it spin?
I just find it counter-intuitive why having the vector field defined or undefined at the origin would make such a big difference. I guess I'm asking, is there an intuitive reason why there is such a difference in curl even though the vector fields look so similar?