Resonant Perturbations and Closed-Form Flux Formulas on Paraboloid-Type Surfaces

This paper establishes two results on a class of planarly perturbed paraboloid surfaces. First, a closed-form boundary formula for the total z-directional flux valid for any differentiable radial profile f: the flux equals π[1−k(f(1)²−f(0)²)]. Second, a necessary and sufficient resonance condition — the Euler-type ODE uf′(u)=kf(u), solved by f(u)=Cu^k — under which the z-component of the normal vector becomes pointwise independent of the azimuthal variable. This pointwise simplification is stronger than integral-level orthogonality and connects to the structure of homogeneous harmonic polynomials. The VMP (vectorial matrix product) framework is introduced as a natural formalism for measurement direction selection on such surfaces.