Let v : R 3 → → R 3 be a solenoidal vector field which is twice continuously differentiable. Assume that v ( x ) decreases at least as fast as 1 / ‖ ‖ x ‖ ‖ for ‖ ‖ x ‖ ‖ → → ∞ ∞ . Define A ( x ) = 1 4 π π ∫ ∫ R 3 ∇ ∇ y × × v ( y ) ‖ x − − y ‖ d 3 y where ∇ ∇ y × × denotes curl with respect to variable y . Then A is a vector potential for v . That is, ∇ ∇ × × A = v .