Localization through a vector potential

Core idea

Let $A$ be a smooth vector potential on $U\subseteq\mathbb R^3$ , with $u=\nabla\times A$ , and let $\chi$ be a smooth cutoff with compact support in $U$ . Define

v=\nabla\times(\chi A) =\chi u+\nabla\chi\times A,

extending $\chi A$ by zero outside $U$ . Then $v$ is smooth, compactly supported, and divergence free on $\mathbb R^3$ .

Plateau and correction

Where $\chi=1$ on a neighborhood, $v=u$ . The term $\nabla\chi\times A$ is the cutoff correction and is supported where the cutoff varies. The identity follows by the componentwise product rule for curl, while $\nabla\cdot v=0$ follows from the divergence-of-curl identity.

Why the potential is needed

Multiplication alone gives $\nabla\cdot(\chi u)=\nabla\chi\cdot u$ , which is not usually zero. The construction assumes a potential on the region being localized; existence of a global potential can depend on the domain.

Infinite asymptotic correction families

Solenoidal asymptotic summation applies the same potential construction term by term with shrinking cutoffs and quantitative derivative tails.