The local Schwarz paradox and the gradient of strongly differentiable functions

Abstract

We show that the gradient of a multi-variable strongly differentiable function at a point is the limit of a single coordinate-free Clifford quotient between a multi-difference pseudo-vector and a pseudo-scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both built on a same non degenerate simplex contracting to that point. Then, we provide some conjectures implied by the foregoing result.