  Aronszajn, Krzywicki and Szarski proved in \cite{AKS62} a strong unique
continuation result for differential forms, satisfying a certain first order
differential inequality, on Riemannian manifolds with empty boundary. The
present paper extends this result to the setting of Riemannian manifold with
non-empty boundary, assuming suitable boundary conditions on the differential
forms. We then present some applications of this extended result. Namely, we
show that the Hausdorff dimension of the zero set of harmonic Neumann and
Dirichlet forms, as well as eigenfields of the curl operator (on
$3$-manifolds), has codimension at least $2$. Again, these bounds were known in
the setting of manifolds without boundary, so that the merit is once more the
inclusion of boundary points.
