The surface tangent paradox and the difference vector quotient of a secant plane

TL;DR

This paper explores a paradox in multivariable calculus where secant planes do not necessarily approach tangent planes, and introduces a new approach using Clifford's geometric product to analyze this phenomenon.

Contribution

The paper reveals the surface tangent paradox in multivariable functions and proposes a novel method involving Clifford's geometric product to understand secant plane limits.

Findings

Secant planes of certain functions do not always approach tangent planes.

Clifford's geometric product helps analyze secant plane behavior.

Some classical analogies extend to multivariable functions with this approach.

Abstract

If a one-variable function is sufficiently smooth, then the limit position of secant lines its graph is a tangent line. By analogy, one would expect that the limit position of secant planes of a two-variable smooth function is a plane tangent to its graph. Amazingly, this is not necessarily true, even when the function is a simple polynomial. Despite this paradox, we show that some analogies with the one-variable case still hold in the multi-variable context, provided we use a particular vector product: the Clifford's geometric one.