A man in Tennessee has written to me, asking for a reference about the method in my answer to https://math.stackexchange.com/questions/81405/anti-curl-operator He is writing an article on finite element methods for the unit cube. His main reference is Ciarlet, The Finite Element Method for Elliptic Problems
The version I gave there is called the Poincare Lemma by Spivak (Calculus on Manifolds). The integration is along a line segment beginning at the origin.
My friend Dmitry says that most texts use "partial integration," which is how he describes three integrals, along segments parallel to the axes.
A very different approach, and likely the earliest, is due to Helmholtz, who wrote an integral over all of $\mathbb R^3.$
Anyway, I am still uncertain what to tell him. He wants a reference as part of an article he is submitting soon. He said he does not recognize the Poincare version, so he is more familiar with the other two methods. He plans to include his own short proof in any case.
from my answer: Given your divergence-free vector field $(F_1(x,y,z), \; F_2(x,y,z), \; F_3(x,y,z)),$ the $x$-coordinate of the new vector field $G$ is
$$ G_1(x,y,z) = \int_0^1 \; \left( \; t z F_2(tx, ty,tz) - t y F_3(tx, ty,tz) \; \right) \; dt, $$
the $y$-coordinate is
$$ G_2(x,y,z) = \int_0^1 \; \left( \; t x F_3(tx, ty,tz) - t z F_1(tx, ty,tz) \; \right) \; dt, $$
with $z$-coordinate $$ G_3(x,y,z) = \int_0^1 \; \left( \; t y F_1(tx, ty,tz) - t x F_2(tx, ty,tz) \; \right) \; dt. $$