I am a little bit confused over the exact conceptual meaning of the curl vector. So I am familiar with the paddle wheel interpretation, but I don't think I am satisfied with that analogy because it could give different meanings. There are two ways which I am imagining the curl. To make this explanation easier, I will talk about a fluid on a two dimensional plane where the vector field is the velocity vector field given by F= M(x,y)i+ N(x,y)j, as usual.
So when we talk about some fluid, say water, and we say there is a curl vector field on the plane, are we saying that it is the water molecules themselves which are rotating about their axis due to this field? (suppose that these water molecules are perfect spheres and it's axis is parallel to the z-axis). In other words, is it the water molecule itself that is located at (x_o,y_o) rotating about it's axis which is parallel to the z-axis?
Or, are we saying that these water molecules (perhaps some bunch) are "rotating" in the sense that they follow a circular path which is centered at (x_o, y_o) and so it is not actually the molecules itself rotating about their own axis.
I am thinking it is the first one. Thanks in advance!
P.S. if anyone would like the edit my question using Latex, please do so! Thank you.