An engineering problem concerned with relating the coordinates of the centre of a spherical cutting tool to the actual cutting surface leads to a potentially rich example of problem-solving techniques. Basic calculus, Lagrange multipliers and vector calculus techniques are employed to produce solutions that may be compared to better understand…

The aim of this note is to give some insight into the formal unity of a very applicable area of mathematics by showing an interesting analogy between the weak part of the Rouche-Frobenius theorem and the existence result for the initial value problem for the general first-order linear two-dimensional PDE.

This paper introduces a concept of differentiability for functions in several variables which has a very natural definition and shares all the properties one expects of a notion of differentiability, but is shown to be weaker than the usual concept of differentiability defined as local linearizability. The exact connection between the two…

There has been a lot of material written about logarithmic spirals of golden proportion but this author states that he has never come across an article that states the exact equation of the spiral which ultimately spirals tangentially to the sides of the rectangles. In this article, the author intends to develop such an equation. (Contains 5…

In this article, the author explores the history and the mathematics used by Newton and Leibniz in their invention of calculus. The exploration of this topic is intended to show students that mathematics is a human invention. Suggestions are made to help teachers incorporate the mathematics and the history into their own lessons. (Contains 3…

A formal approach to the differentiation of vectors from an axiomatic point of view is introduced, and applications to dynamics are presented. (DT)

Several examples of apparent contradictions which can arise from failure to state vigorous conditions under which a result holds. (MM)

This article shows how ideas of maximum or minimum values of fundamental quantities have developed through the ages. (Author/JG)

This essay investigates the rhetoric surrounding the appearance of the concept of the infinitesimal in the seventeenth-century Calculus of Sir Isaac Newton and Gottfried Wilhelm Leibniz. Although historians often have positioned rhetoric as a supplemental discipline, this essay shows that rhetoric is the "material" out of which a new and powerful…

Recent events have led to an increased interest in emerging infectious diseases. This article applies various deterministic models to the SARS epidemic of 2003 and a measles outbreak in the Netherlands in 1999-2000. We take a historical approach beginning with the well-known logistic curve and a lesser-known extension popularized by Pearl and Reed…

Recent developments are explained which use the concepts and methods of mathematical logic to provide a suitable framework for the development of the differential and integral calculus. (DT)

Notation and definition of the derivative function are discussed. (DT)