Let F be a smooth vector field on a simply connected open subset U C Rº. Show that F is both irrotational and incompressible if and only if it can be written as F = Vf for a smooth function f : U → R...

Extracted text: Let F be a smooth vector field on a simply connected open subset U C Rº. Show that F is both irrotational and incompressible if and only if it can be written as F = Vf for a smooth function f : U → R satisfying V² f = 0, where V2 is the Laplacian operator. Note that a function f that is a solution of the equation V2 f = 0 is called a harmonic function.