In the text, we observed that although the inverse-square radial vector field F = er r2

Chapter 17, Problem 42

(choose chapter or problem)

In the text, we observed that although the inverse-square radial vector field F = er r2 satisfies div(F) = 0, F cannot have a vector potential on its domain {(x, y, z) = (0, 0, 0)} because the flux of F through a sphere containing the origin is nonzero. (a) Show that the method of Exercise 39 produces a vector potential A such that F = curl(A) on the restricted domain D consisting of R3 with the y-axis removed. (b) Show that F also has a vector potential on the domains obtained by removing either the x-axis or the z-axis from R3. (c) Does the existence of a vector potential on these restricted domains contradict the fact that the flux of F through a sphere containing the origin is nonzero?