“vanishing” vector fields
The weak version: Let \(\bm{R}\) be a smooth solenoidal vector field and \(f\) be a smooth scalar field on \(\mathbf{R}^3\) that vanish faster than \(1/r^2\) as \(r \to \infty.\) Then there exists a vector field \(\bm{F}\) such that \(\operatorname{div}\bm{F} = f\) and \(\operatorname{curl}\bm{F} = \bm{R}.\) If additionally \(\bm{F}\) vanishes as \(r \to \infty\) then \(\bm{F}\) is unique.
The strong version: For a vector field \(\bm{F}\) with continuous first-order partial derivatives, there exists