Applications of Clifford ratios unaffected by the local Schwarz paradox

We show that the gradient of a strongly differentiable function at a point is the limit of a single coordinate-free Clifford quotient between a multidifference pseudo-vector and a pseudo-scalar, or of a sum of Clifford quotients between scalars (as numerators) and vectors (as denominators), both evaluated at the vertices of a same non-degenerate simplex contracting to that point. Such result allows to fix an issue with a defective definition of pseudo-scalar field in Sobczyck's simplicial calculus. Then, we provide some consequences and conjectures implied by the foregoing results.