The local Schwarz paradox and the gradient of strongly differentiable functions

TL;DR

This paper introduces a novel Clifford quotient approach to characterize the gradient of strongly differentiable functions, linking geometric algebra with differential calculus and proposing new conjectures.

Contribution

It presents a coordinate-free Clifford quotient method to compute gradients, offering a new geometric perspective and formulating related conjectures.

Findings

Gradient expressed as a Clifford quotient limit

Coordinate-free geometric interpretation of derivatives

Proposed conjectures based on Clifford algebra approach

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.