Chapter 1.6, Problem 1.50P

(a) Let ${\text{F}}_1=x^2\text{<mover accent="true"> <mo> z </mo> <mo> ^ </mo> </mover>}$ and ${\text{F}}_2=x\text{<mover accent="true"> <mo> x </mo> <mo> ^ </mo> </mover>}+y\text{<mover accent="true"> <mo> y </mo> <mo> ^ </mo> </mover>}+z\text{<mover accent="true"> <mo> z </mo> <mo> ^ </mo> </mover>}$ . Calculate the divergence and curl of F₁ and F₂. Which one can be written as the gradient of a scalar? Find a scalar potential that does the job. Which one can be written as the curl of a vector? Find a suitable vector potential.
(b) Show that ${\text{F}}_3=yz\text{<mover accent="true"> <mo> x </mo> <mo> ^ </mo> </mover>}+zx\text{<mover accent="true"> <mo> y </mo> <mo> ^ </mo> </mover>}+xy\text{<mover accent="true"> <mo> z </mo> <mo> ^ </mo> </mover>}$ can be written both as the gradient of a scalar and as the curl of a vector. Find scalar and vector potentials for this function.