a) Complete the proof of Theorem 2, Sect. 1.6.2. That is, show that any divergenceless v...

a) Complete the proof of Theorem 2, Sect. 1.6.2. That is, show that any divergenceless vector field F can be written as the curl of a vector potential A. What you have to do is find A_x , A_y, and A_z such that (i) ∂ A_z/∂y − ∂ A_y/∂_z = Fx ; (ii) ∂ A_x/∂_z − ∂ A_z/∂_x = F_y ; and (iii) ∂ A_y/∂_x − ∂ A_x/∂_y = F_z . Here’s one way to do it: Pick A_x = 0, and solve (ii) and (iii) for A_y and A_z . Note that the “constants of integration” are themselves functions of y and z—they’re constant only with respect to x. Now plug these expressions into (i), and use the fact that ∇ · F = 0 to obtain

(b) By direct differentiation, check that the A you obtained in part (a) satisfies ∇ × A = F. Is A divergenceless? [This was a very asymmetrical construction, and it would be surprising if it were—although we know that there exists a vector whose curl is F and whose divergence is zero.]

(c) As an example, let Calculate A, and confirm that ∇ × A = F. (For further discussion, see Prob. 5.53.)

Reference prob 5.53

Reference Theorem 2

Divergence-less (or “solenoidal”) fields. The following conditions are equivalent: