Almost invariant manifolds for divergence-free fields

It is shown that toroidal surfaces that extremize a properly weighted surface integral of the squared normal component of a solenoidal three-vector field capture the local invariant dynamics, in that a field line that is anywhere tangential to the surface must be confined to the surface everywhere. In addition to an elementary three-vector calculus derivation, which relies on a curvilinear toroidal coordinate system, a coordinate-free geometric approach applicable to hypersurfaces (codimension-one submanifolds) of manifolds of arbitrary dimension is sketched.