General Directional Mapping: Admissibility and Tangent Generation (Interdisciplinary Concept)

In classical analysis the tangent is introduced as a primitive object. We propose adifferent starting point, the general directional mapping D(x, u), which assigns to eachpair of points a displacement-like quantity. A direction is called admissible at u when itis generated by D through a limiting process, and the collection of admissible directionsis the generated tangent cone TD(u).Two theorems are proved:• The first shows that, when D has an invertible linear part A, the generated coneis the classical tangent cone transported by A, that lower-order corrections donot change it, and that polar cones transform by A−⊤.• The second applies this to equilibrium: the equilibrium condition of a potentialon a constrained set is unchanged by the choice of mapping, provided the forcetransforms covariantly.The relations among classical, linearizable, invex, Zulfia and non-linearizable mappingsare summarized, and connections to statistics and spacetime geometry are listed asdirections to investigate, not as established consequences.