When Tangent Plane = Limit of Secant Plane

TL;DR

This paper establishes a new fundamental equivalence in multivariable calculus, showing that differentiability corresponds to the tangent plane being the limit of secant planes, extending a core concept from single-variable calculus.

Contribution

It provides the first rigorous proof that for multivariable functions, the tangent plane can be characterized as the limit of secant planes, generalizing a key differentiability criterion.

Findings

Tangent plane equals the limit of secant planes for differentiable multivariable functions.

First rigorous proof of this equivalence in several variables.

Extends the fundamental concept of differentiability from single-variable to multivariable calculus.

Abstract

For function of one variable, differentiability is equivalent to the existence of tangent line as the limit of secant line. The genuine counterpart of this equivalence for function of several variables is obtained for the first time.