Infinitesimals to derive chain rule12/27/2023 The purpose of this monograph, and of the book Elementary Calculus, is to make infinitesimals more readily available to mathematicians and students. However, the method is still seen as controversial, and is unfamiliar to most mathematicians. It is also used in such fields as economics and physics as a source of mathematical models. The method is surprisingly adaptable and has been applied to many areas of pure and applied mathematics. The older name infinitesimal analysis is perhaps more appropriate. Robinson called his method nonstandard analysis because it uses a nonstandard model of analysis. Robinson used methods from the branch of mathematical logic called model theory which developed in the 1950’s. ![]() 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. The actual situation, as suggested by Leibniz and carried out by Robinson, is that one can form the hyperreal number system by adding infinitesimals to the real number system, and obtain a powerful new tool in analysis. Since then generations of students have been taught that infinitesimals do not exist and should be avoided. 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. 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). Earlier constructions of the hyperreal number system depended on an arbitrarily chosen parameter such as an ultrafilter. (3) An account of the discovery of Kanovei and Shelah that the hyperreal number system, like the real number system, can be built as an explicitly definable mathematical structure. (2) The axioms for the hyperreal number system are changed to match those in the later editions of Elementary Calculus. The biggest changes are: (1) A new chapter on differential equations, keyed to the corresponding new chapter in Elementary Calculus. A companion to the second (1986) edition of Elementary Calculus was never written. This is a major revision of the first edition of Foundations of Infinitesimal Calculus, which was published as a companion to the first (1976) edition of Elementary Calculus, and has been out of print for over twenty years. Elementary Calculus: An Infinitesimal Approach is available free online at This monograph can be used as a quick introduction to the subject for mathematicians, as background material for instructors using the book Elementary Calculus, or as a text for an undergraduate seminar. It is entirely self-contained but is keyed to the 2000 digital edition of my first year college text Elementary Calculus: An Infinitesimal Approach and the second printed edition. This is an exposition of Robinson’s infinitesimal calculus at the advanced undergraduate level. Robinson’s achievement was one of the major mathematical advances of the twentieth century. In 1960 Abraham Robinson (1918–1974) solved the three hundred year old problem of giving a rigorous development of the calculus based on infinitesimals. An Example where Uniqueness Fails (§14.3). Change of Variables in Double Integrals (§12.5). Infinite Sum Theorem for Two Variables (§12.3). Chain Rule and Implicit Functions (§11.5, §11.6). Taylor’s Formula and Higher Differentials (§9.10). Derivatives of Exponential Functions (§8.3). The Functions ax and logb x (§8.1, §8.2). ![]() Derivatives of Trigonometric Functions (§7.1, §7.2). Second Fundamental Theorem of Calculus (§4.2). ![]() Properties of Continuous Functions (§3.5–§3.8). Infinitesimal Microscopes and Infinite Telescopes. Axioms for the Hyperreal Numbers (§Epilogue). Structure of the Hyperreal Numbers (§1.4, §1.5). This work is licensed under the Creative Commons Attribution-NoncommercialShare Alike 3.0 Unported License. JEROME KEISLER Department of Mathematics University of Wisconsin, Madison, Wisconsin, USA 4, 2011
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