A vector field with a vector potential has zero flux through every closed surface in its domain. it is observed that although the inverse-square radial vector field F = (e^r)/(p^2) satisfies div(F) = 0, F cannot have a vector potential on its domain {(x,y,z)does not equal (0,0,0)} because the flux of F through a sphere containing the origin is nonzero.

Question

A vector field with a vector potential has zero flux through every closed surface in its domain. it is observed that although the inverse-square radial vector field F = (e^r)/(p^2) satisfies div(F) = 0, F cannot have a vector potential on its domain {(x,y,z)does not equal (0,0,0)} because the flux of F through a sphere containing the origin is nonzero.
(a) show that F also has a vector potential on the domains obtained by removing either the x-axis or the z-axis from R^3.
(b) Does the existence of a vector on these restricted domains contradict the fact that the flux of F through a sphere containing the origin is nonzero?