What is the use of divergence and curl in CG?

What is the use of divergence and curl in CG?

I believe it was mentioned that the modeling of water currents in graphics uses a divergence-free vector field, because water is incompressible, as an example

Is curl signed towards clockwise or anti-clockwise direction?

counter-clockwise I believe

Are there any counterparts for divergence and curl in dimensions higher than 3?

counter-clockwise