Chapter 16.5, Problem 39E

We have seen that all vector fields of the form F = ∇g satisfy the equation curt F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appropriate partial derivatives). This suggests the question: are there any equations that all functions of the form f = div G must satisfy? Shoes that the answer to this question is “No” by proving that every continuous function f on ℝ³ is the divergence of sonic vector field.

[Hint: Let G(x, y, z) = ⟨g(x, y, z), 0, 0⟩, where $g(x,y,z)\text{ =}{\displaystyle {\int }_0^{x}f(t,y,z)dt}$.]