(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 Ax, Ay. and Az such that: (i) ∂Az/∂y - ∂Ay/∂z = Fx; (ii) ∂Ax/∂z – ∂Az/∂x = Fy; and (iii) ∂Ay/∂x – ∂Ax/∂y = Fz. Here's one way to do it: Pick Ax = 0, and solve (ii) and (iii) for Ay and Az. Note that the "constants of integration" here 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 ∆ x 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 F = yx + zy + xz. Calculate A, and confirm that ∆ x A = F.