Dual Reciprocal Scalar Potentials Paired via Differential Operators in Frenet Frames Make the Operators to Act Simultaneously in Each of Two Paired 3D Reciprocal Spaces

Extending the operating domain of 3D geometric differential operator expands also the range of its operations onto paired 3D dual reciprocal spaces as well as the scope of their validity into paired 3D multispatial structures. Intraspatial duality principle for paired 3D dual reciprocal spaces is inferred from differential operations performed on the dual reciprocal 3D spaces. The new scalar differential operator SCovar as multiplicative inverse of the scalar gradient differential operator SGrad is proposed here to deliver scalar components of covariant differentials in order to accommodate both the operational and structural legitimacy of differential and integral operations performed in 3D dual reciprocal spaces. From preliminarily formulated abstract intraspatial duality principle a generalized interspatial duality principle is deduced and the connection of paired multispatial structures established. It is shown that the finegrained geometric differential operator GDiff acts simultaneously in each of the paired dual reciprocal spaces, which is its formerly unknown operational feature.