A vector field \(\mathbf{F}\) is incompressible if \(\operatorname{div}(\mathbf{F})=0\) and is irrotational if \(\operatorname{curl}(\mathbf{F})=\mathbf{0}\).

Prove that the cross product of two irrotational vector fields is incompressible, and explain why this implies that the cross product of two conservative vector fields is incompressible.