Keisler H. Jerome. Foundations of Infinitesimal Calculus

University of Wisconsin, 2022. — 213 p.The basic concepts of the calculus were originally developed in the seventeenth and eighteenth centuries using the intuitive notion of an infinitesimal, culminating in the work of Gottfried Leibniz (1646-1716) and Isaac Newton (1643-1727). When the calculus was put on a rigorous basis in the nineteenth century, infinitesimals were rejected in favor of the ε, δ approach, because mathematicians had not yet discovered a correct treatment of infinitesimals. Since then generations of students have been taught that infinitesimals do not exist and should be avoided.
In 1960 Abraham Robinson (1918–1974) solved the three hundred year old problem of giving a rigorous development of the calculus based on infinitesimals. Robinson’s achievement was one of the major mathematical advances of the twentieth century. The reason Robinson’s discovery did not come sooner is that the axioms needed to describe the hyperreal numbers are of a kind which were unfamiliar to mathematicians until the mid-twentieth century. Robinson used methods from the branch of mathematical logic called model theory which developed in the 1950’s. Robinson called his method nonstandard analysis because it uses a nonstandard model of analysis. The older name infinitesimal analysis is perhaps more appropriate. The method is surprisingly adaptable and has been applied to many areas of pure and applied mathematics. However, the method is still seen as controversial, and is unfamiliar to most mathematicians.
The purpose of this monograph is to make infinitesimals more readily available to mathematicians and students. Infinitesimals provided the intuition for the original development of the calculus and should help students as they repeat this development. The simple set of axioms for the hyperreal number system given here make it possible to present infinitesimal calculus at the college freshman level, avoiding concepts from mathematical logic. It is shown in Chapter 15 that these axioms are equivalent to Robinson’s approach.Preface.
The Hyperreal Numbers.
Differentiation.
Continuous Functions.
Integration.
Limits.
Applications of the Integral.
Trigonometric Functions.
Exponential Functions.
Infinite Series.
Vectors.
Partial Differentiation.
Multiple Integration.
Vector Calculus.
Differential Equations.
Logic and Superstructures.
References.
Index.A5 format