What are Irrotational and Solenoidal Vector Fields?

An irrotational vector field is a vector field where the curl of the field is equal to zero at every point in space. Geometrically, this means that the field lines of an irrotational vector field form closed loops, without any net circulation around any closed path in the field. Examples of irrotational vector fields include gravitational fields and electrostatic fields.

On the other hand, a solenoidal vector field is a vector field where the divergence of the field is equal to zero at every point in space. Geometrically, this means that the field lines of a solenoidal vector field are always either closed loops or extend to infinity, and there is no source or sink of the field at any point. Examples of solenoidal vector fields include magnetic fields and fluid flow fields.