Related papers: When Tangent Plane = Limit of Secant Plane

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…

In this article we discuss an important students' misconception about derivatives, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not where the expression is…

Some formal analogies between the Differential Calculus in One Variable and the Differential Calculus in Several Variables are presented. It is studied and introduced the derivability of functions at several variables from the single…

A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…

We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this derivative is invariant under changes of chart and is thus well-defined for functions defined on a…

This article provides several theorems regarding the existence of limit for multivariable function, among which Theorem 1 and Theorem 3 relax the requirement for sequence of Heine's definition of limit. These results address the question of…

A differentiable curve y = y(x) is determined by its tangent lines and is said to be the envelope of its tangent lines. The coefficients of the curve's tangent lines form a curve in another space, called the dual space. There is a…

Given a function f in the exterior of a convex curve in the real plane, we prove that if the restrictions of f to the tangent lines to the curve extend as entire functions, then the function f is an entire function of two variables.

Consider a real valued function defined, but not differentiable at some point. We use sequences approaching the point of interest to define and study sequential concepts of secant and cord derivatives of the function at the point of…

The necessary and sufficient conditions for a function to be totally or partially separable are derived. It is shown that a function is totally separable if and only if each component of the gradient vector of depends only on the…

The necessary and sufficient conditions for differentiability of a function of several real variables stated and proved and its ramifications discussed.

This paper presents an algebraic-geometric construction of the derivative developed initially within the class of polynomial functions without introducing limits at the initial stage. Tangency is characterized by an algebraic condition: the…

Three definitions of a differential form on a tangent structure are considere. It is proved that the (covariant) definition given by Souriau (as a collection of forms indexed by the plaques) is equivalent to a smooth section of the…

Let $\lambda =[d_1,\dots,d_r]$ be a partition of $d$. Consider the variety $\mathbb{X}_{2,\lambda} \subset \mathbb{P}^N$, $N={d+2 \choose 2}-1$, parameterizing forms $F\in k[x_0,x_1,x_2]_d$ which are the product of $r\geq 2$ forms…

The aim of the present article is to establish the connection between the existence of the limit along the normal and an admissible limit at a fixed boundary point for holomorphic functions of several complex variables.

We show that the graph of a bent function is a Salem set in an appropriate sense. We also establish a simple result that quantifies redundancies in the difference operators of a function, which applies to bent functions over fields of odd…

This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…

We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with…

The tangential map is a map on the set of smooth planar curves. It satisfies the 3D-consistency property and is closely related to some well-known integrable equations.

We compute the limit of tangents of an arbitrary surface. We obtain as a byproduct an embedded version of Jung's desingularization theorem for surface singularities with finite limits of tangents.