I am trying to understand curl and divergences in a more intuitive manner, especially the curl. And is curl a surface phenomenon, if yes then how?
A discussion on the intuitive interpretation of the curl from math SE. And a quote from Wikipedia:
If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid.
For divergence, I'd also point you to Wikipedia:
More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.
I know, it's a lot of quotes and links, but there's nothing new under the sun ;)