4) Magnetostatics: We wish to find the magnetic field B(x), x E RS, produced by a (compactly supported) current distribution J(x) in a material with position-dependent per- meability (x). Consider the functional 1 F[A] =3 VX A? - J . A d'x, where A(x) is a candidate vector potential for the field B = V X A. i) Show that FA] is gauge invariant (i.e. it is unchanged in value when A is replaced by A + Vx) provided that V . J = 0. ii) Show that the stationarity condition OF/SA;(x) = 0 leads to A obeying the appropriate Maxwell equation. (You may assume that V x A is zero at infinity. ) iii) By "completing the square," show that FA takes its minimum value when A is a vector potential for the actual magnetic field produced by the current distribution J