The divergence of continuously differentiable vector fields

For m≥ 2, we give a complete characterisation of the continuous functions on Rᵐ that are divergences of continuously differentiable vector fields. The characterisation is formulated in terms of distributions and, equivalently, in terms of the continuous dual of a suitably topologised subspace of the Arens--Eells space. The two implications yield contrasting applications. First, there exists a compactly supported continuous function which is the divergence of a bounded Lipschitz vector field but not of any continuously differentiable vector field. Second, every continuous function whose distributional partial derivatives are locally m-integrable (the critical exponent for Sobolev embedding) is the divergence of a continuously differentiable vector field.