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H. Jerome Keisler : Foundations of Infinitesimal Calculus FOUNDATIONS OF INFINITESIMAL CALCULUS H. JEROME KEISLER Department of Mathematics University of Wisconsin, Madison, Wisconsin, USA [email protected] June 4, 2011 ii This work is licensed under the Creative Commons Attribution-Noncommercial- Share Alike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ Copyright c 2007 by H. Jerome Keisler CONTENTS Preface................................................................ vii Chapter 1. The Hyperreal Numbers.............................. 1 1A. Structure of the Hyperreal Numbers (x1.4, x1.5) . 1 1B. Standard Parts (x1.6)........................................ 5 1C. Axioms for the Hyperreal Numbers (xEpilogue) . 7 1D. Consequences of the Transfer Axiom . 9 1E. Natural Extensions of Sets . 14 1F. Appendix. Algebra of the Real Numbers . 19 1G. Building the Hyperreal Numbers . 23 Chapter 2. Differentiation........................................ 33 2A. Derivatives (x2.1, x2.2) . 33 2B. Infinitesimal Microscopes and Infinite Telescopes . 35 2C. Properties of Derivatives (x2.3, x2.4) . 38 2D. Chain Rule (x2.6, x2.7). 41 Chapter 3. Continuous Functions ................................ 43 3A. Limits and Continuity (x3.3, x3.4) . 43 3B. Hyperintegers (x3.8) . 47 3C. Properties of Continuous Functions (x3.5{x3.8) . 49 Chapter 4. Integration ............................................ 59 4A. The Definite Integral (x4.1) . 59 4B. Fundamental Theorem of Calculus (x4.2) . 64 4C. Second Fundamental Theorem of Calculus (x4.2) . 67 Chapter 5. Limits ................................................... 71 5A. "; δ Conditions for Limits (x5.8, x5.1) . 71 5B. L'Hospital's Rule (x5.2) . 74 Chapter 6. Applications of the Integral........................ 77 6A. Infinite Sum Theorem (x6.1, x6.2, x6.6) . 77 6B. Lengths of Curves (x6.3, x6.4) . 82 6C. Improper Integrals (x6.7). 87 iii iv CONTENTS Chapter 7. Trigonometric Functions ............................ 91 7A. Inverse Function Theorem (x7.3) . 91 7B. Derivatives of Trigonometric Functions (x7.1, x7.2) . 94 7C. Area in Polar Coordinates (x7.9) . 95 Chapter 8. Exponential Functions ............................... 99 8A. Extending Continuous Functions . 99 x 8B. The Functions a and logb x (x8.1, x8.2) . 100 8C. Derivatives of Exponential Functions (x8.3) . 102 Chapter 9. Infinite Series ......................................... 105 9A. Sequences (x9.1) . 105 9B. Series (x9.2 { x9.6) . 108 9C. Taylor's Formula and Higher Differentials (x9.10). 110 Chapter 10. Vectors ...............................................115 10A. Hyperreal Vectors (x10.8) . 115 10B. Vector Functions (x10.6) . 118 Chapter 11. Partial Differentiation .............................121 11A. Continuity in Two Variables (x11.1, x11.2) . 121 11B. Partial Derivatives (x11.3, x11.4) . 122 11C. Chain Rule and Implicit Functions (x11.5, x11.6) . 125 11D. Maxima and Minima (x11.7) . 128 11E. Second Partial Derivatives (x11.8) . 133 Chapter 12. Multiple Integration................................137 12A. Double Integrals (x12.1, x12.2) . 137 12B. Infinite Sum Theorem for Two Variables (x12.3) . 140 12C. Change of Variables in Double Integrals (x12.5) . 144 Chapter 13. Vector Calculus.....................................151 13A. Line Integrals (x13.2) . 151 13B. Green's Theorem (x13.3, x13.4). 154 Chapter 14. Differential Equations ............................. 161 14A. Existence of Solutions (x14.4) . 162 14B. Uniqueness of Solutions (x14.4). 167 14C. An Example where Uniqueness Fails (x14.3) . 171 Chapter 15. Logic and Superstructures . 175 15A. The Elementary Extension Principle . 175 15B. Superstructures . 180 15C. Standard, Internal, and External Sets . 184 15D. Bounded Ultrapowers . 189 15E. Saturation and Uniqueness . 194 CONTENTS v References............................................................199 Index. 201 PREFACE In 1960 Abraham Robinson (1918{1974) solved the three hundred year old problem of giving a rigorous development of the calculus based on infinitesi- mals. Robinson's achievement was one of the major mathematical advances of the twentieth century. This is an exposition of Robinson's infinitesimal cal- culus at the advanced undergraduate level. It is entirely self-contained but is keyed to the 2000 digital edition of my first year college text Elementary Cal- culus: An Infinitesimal Approach [Keisler 2000] and the second printed edition [Keisler 1986]. Elementary Calculus: An Infinitesimal Approach is available free online at www.math.wisc.edu/∼Keisler/calc. This monograph can be used as a quick introduction to the subject for mathematicians, as background ma- terial for instructors using the book Elementary Calculus, or as a text for an undergraduate seminar. This is a major revision of the first edition of Foundations of Infinitesimal Calculus [Keisler 1976], which was published as a companion to the first (1976) edition of Elementary Calculus, and has been out of print for over twenty years. A companion to the second (1986) edition of Elementary Calculus was never written. The biggest changes are: (1) A new chapter on differential equations, keyed to the corresponding new chapter in Elementary Calculus. (2) The ax- ioms for the hyperreal number system are changed to match those in the later editions of Elementary Calculus. (3) An account of the discovery of Kanovei and Shelah [KS 2004] that the hyperreal number system, like the real number system, can be built as an explicitly definable mathematical structure. Ear- lier constructions of the hyperreal number system depended on an arbitrarily chosen parameter such as an ultrafilter. The basic concepts of the calculus were originally developed in the seven- teenth 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 "; 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. vii viii Preface 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. 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 nonstan- dard 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. It is also used in such fields as econom- ics and physics as a source of mathematical models. (See, for example, the books [AFHL 1986] and [ACH 1997]). However, the method is still seen as controversial, and is unfamiliar to most mathematicians. The purpose of this monograph, and of the book Elementary Calculus, is to make infinitesimals more readily available to mathematicians and students. In- finitesimals provided the intuition for the original development of the calculus and should help students as they repeat this development. The book Ele- mentary Calculus treats infinitesimal calculus at the simplest possible level, and gives plausibility arguments instead of proofs of theorems whenever it is appropriate. This monograph presents the subject from a more advanced view- point and includes proofs of almost all of the theorems stated in Elementary Calculus. Chapters 1{14 in this monograph match the chapters in Elementary Calcu- lus, and after each section heading the corresponding sections of Elementary Calculus are indicated in parentheses. In Chapter 1 the hyperreal numbers are first introduced with a set of axioms and their algebraic structure is studied. Then in Section 1G the hyperreal numbers are built from the real numbers. This is an optional section which is more advanced than the rest of the chapter and is not used later. It is included for the reader who wants to see where the hyperreal numbers come from. Chapters 2 through 14 contain a rigorous development of infinitesimal cal- culus based on the axioms in Chapter 1. The only prerequisites are the tra- ditional three semesters of calculus and a certain amount of mathematical maturity. In particular, the material is presented without using notions from mathematical logic. We will use some elementary set-theoretic notation famil- iar to all mathematicians, for example the function concept and the symbols ;;A [ B; fx 2 A: P (x)g. Frequently, standard results are given alternate proofs using infinitesimals. In some cases a standard result which is beyond the scope of beginning calculus is rephrased as a simpler infinitesimal result and used effectively in Elementary Calculus; some examples are the Infinite Sum Theorem, and the two-variable criterion for a global maximum. Preface ix The last chapter of this monograph, Chapter 15, is a bridge between the simple treatment of infinitesimal calculus given here and the more advanced subject of infinitesimal analysis found in the research literature. To go be- yond infinitesimal calculus one should at least be familiar with some basic notions
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