Mathematical Background: Foundations of Innitesimal Calculus second edition

by

K. D. Stroyan

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1997 by Academic Press, Inc. - All rights reserved.

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Preface to the Mathematical Background

We want you to reason with mathematics. We are not trying to get everyone to give formalized proofs in the sense of contemporary mathematics; ‘proof’ in this course means ‘convincing argument.’ We expect you to use correct reasoning and to give careful explanations. The projects bring out these issues in the way we nd best for most students, but the pure mathematical questions also interest some students. This book of mathematical “background” shows how to ll in the mathematical details of the main topics from the course. These proofs are completely rigorous in the sense of modern mathematics – technically bulletproof. We wrote this book of foundations in part to provide a convenient reference for a student who might like to see the “theorem - proof” approach to calculus. We alsowroteit for the interestedinstructor. In re-thinking the presentationof beginning calculus, we found that a simpler basis for the theory was both possible and desirable. The pointwise approach most books give to the theory of derivatives spoils the subject. Clear simple argumentslike the proofof the FundamentalTheoremat the startof Chapter 5 below are not possible in that approach. The result of the pointwise approach is that instructors feeltheyhaveto either bedishonestwithstudents ordisclaimgoodintuitiveapproximations. This is sad because it makes a clear subject seem obscure. It is also unnecessary – by and large, the intuitive ideas work provided your notion of derivative is strong enough. This book shows how to bridge the gap between intuition and technical rigor. A function with a positive derivative ought to be increasing. After all, the slope is positive and the graph is supposed to look like an increasing straight line. How could the function NOT be increasing? Pointwise derivatives make this bizarre thing possible - a positive “derivative” of a non-increasing function. Our conclusion is simple. That denition is WRONG in the sense that it does NOT support the intended idea. You might agree that the counterintuitive consequences of pointwise derivatives are unfortunate, but are concerned that the traditional approach is more “general.” Part of the point of this book is to show students and instructors that nothing of interest is lost and a great deal is gained in the straightforward nature of the proofs based on “uniform” derivatives. It actually is not possible to give a formula that is pointwise dierentiable and not uniformly dierentiable. The pieced together pointwise counterexamples seem contrived and out-of-place in a course where students are learning valuable new rules. It is a theorem that derivatives computed by rules are automatically continuous where dened. We want the course development to emphasize good intuition and positive results. This background shows that the approach is sound. This book also shows how the pathologies arise in the traditional approach – we left pointwise pathology out of the main text, but present it here for the curious and for comparison. Perhaps only math majors ever need to know about these sorts of examples, but they are fun in a negative sort of way. This book also has several theoretical topics that are hard to nd in the literature. It includes a complete self-contained treatment of Robinson’s modern theory of innitesimals, rst discovered in 1961. Our simple treatment is due to H. Jerome Keisler from the 1970’s. Keisler’s elementary calculus using innitesimals is sadly out of print. It used pointwise derivatives, but had many novel ideas, including the rst modern use of a microscope to describe the derivative. (The l’Hospital/Bernoulli calculus text of 1696 said curves consist of innitesimal straight segments, but I do not know if that was associated with a magnifying transformation.) Innitesimals give us a very simple way to understand the uniform

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derivatives, although this can also be clearly understood using function limits as in the text by Lax, et al, from the 1970s. Modern graphical computing can also help us “see” graphs convergeas stressed in our main materials and in the interesting Uhl, Porta, Davis, Calculus & Mathematica text. Almost all the theorems in this book are well-known old results of a carefully studied subject. The well-known ones are more important than the few novel aspects of the book. However, some details like the converseof Taylor’s theorem – both continuous and discrete – arenotsoeasytondintraditionalcalculussources. Themicroscopetheoremfordierential equations does not appear in the literature as far as we know, though it is similar to research work of Francine and Marc Diener from the 1980s. We conclude the book with convergence results for Fourier series. While there is nothing novel in our approach, these results have been lost from contemporary calculus and deserve to be part of it. Our development follows Courant’s calculus of the 1930s giving wonderful results of Dirichlet’s era in the 1830s that clearly settle some of the convergence mysteries of Euler from the 1730s. This theory and our development throughout is usually easy to apply. “Clean” theory should be the servant of intuition – building on it and making it stronger and clearer. There is more that is novel about this “book.” It is free and it is not a “book” since it is not printed. Thanks to small marginal cost, our publisher agreed to include this electronic text on CD at no extra cost. We also plan to distribute it over the world wide web. We hope our fresh look at the foundations of calculus will stimulate your interest. Decide for yourself what’s the best way to understand this wonderful subject. Give your own proofs.

Contents

Part 1 Numbers and Functions Chapter 1. Numbers 3 1.1 Field Axioms 3 1.2 Order Axioms 6 1.3 The Completeness Axiom 7 1.4 Small, Medium and Large Numbers 9 Chapter 2. Functional Identities 17 2.1 Specic Functional Identities 17 2.2 General Functional Identities 18 2.3 The Function Extension Axiom 21 2.4 Additive Functions 24 2.5 The Motion of a Pendulum 26 Part 2 Limits Chapter 3. The Theory of Limits 31 3.1 Plain Limits 32 3.2 Function Limits 34 3.3 Computation of Limits 37 Chapter 4. Continuous Functions 43 4.1 Uniform Continuity 43 4.2 The Extreme Value Theorem 44

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4.3 Bolzano’s Intermediate Value Theorem 46 Part 3 1 Variable Dierentiation Chapter 5. The Theory of Derivatives 49 5.1 The Fundamental Theorem: Part 1 49 5.1.1 Rigorous Innitesimal Justication 52 5.1.2 Rigorous Limit Justication 53 5.2 Derivatives, Epsilons and Deltas 53 5.3 Smoothness ⇒ Continuity of Function and Derivative 54 5.4 Rules ⇒ Smoothness 56 5.5 The Increment and Increasing 57 5.6 Inverse Functions and Derivatives 58 Chapter 6. Pointwise Derivatives 69 6.1 Pointwise Limits 69 6.2 Pointwise Derivatives 72 6.3 Pointwise Derivatives Aren’t Enough for Inverses 76 Chapter 7. The Mean Value Theorem 79 7.1 The Mean Value Theorem 79 7.2 Darboux’s Theorem 83 7.3 Continuous Pointwise Derivatives are Uniform 85 Chapter 8. Higher Order Derivatives 87 8.1 Taylor’s Formula and Bending 87 8.2 Symmetric Dierences and Taylor’s Formula 89 8.3 Approximation of Second Derivatives 91 8.4 The General Taylor Small Oh Formula 92 8.4.1 The Converse of Taylor’s Theorem 95 8.5 Direct Interpretation of Higher Order Derivatives 98 8.5.1 Basic Theory of Interpolation 99 8.5.2 Interpolation where f is Smooth 101 8.5.3 Smoothness From Dierences 102 Part 4 Integration Chapter 9. Basic Theory of the Denite Integral 109 9.1 Existence of the Integral 110

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9.2 You Can’t Always Integrate Discontinuous Functions 114 9.3 Fundamental Theorem: Part 2 116 9.4 Improper Integrals 119 9.4.1 Comparison of Improper Integrals 121 9.4.2 A Finite Funnel with Innite Area? 123 Part 5 Multivariable Dierentiation Chapter 10. Derivatives of Multivariable Functions 127 Part 6 Dierential Equations Chapter 11. Theory of Initial Value Problems 131 11.1 Existence and Uniqueness of Solutions 131 11.2 Local Linearization of Dynamical Systems 135 11.3 Attraction and Repulsion 141 11.4 Stable Limit Cycles 143 Part 7 Innite Series Chapter 12. The Theory of Power Series 147 12.1 Uniformly Convergent Series 149 12.2 Robinson’s Sequential Lemma 151 12.3 Integration of Series 152 12.4 Radius of Convergence 154 12.5 Calculus of Power Series 156 Chapter 13. The Theory of Fourier Series 159 13.1 Computation of Fourier Series 160 13.2 Convergence for Piecewise Smooth Functions 167 13.3 Uniform Convergence for Continuous Piecewise Smooth Functions 173 13.4 Integration of Fourier Series 175

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Part 1

Numbers and Functions

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CHAPTER 1 Numbers This chapter gives the algebraic laws of the number systems used in calculus.

Numbers represent various idealized measurements. Positive integers may count items, fractions may represent a part of an item or a distance that is part of a xed unit. Distance measurements go beyond rational numbers as soon as we consider the hypotenuse of a right triangle or the circumference of a circle. This extension is already in the realm of imagined “perfect” measurements because it corresponds to a perfectly straight-sided triangle with perfect rightangle, or a perfectly round circle. Actual realmeasurements arealwaysrational and have some error or uncertainty. The various “imaginary” aspects of numbers are very useful ctions. The rules of computation with perfect numbers are much simpler than with the error-containing real measurements. This simplicity makes fundamental ideas clearer. Hyperreal numbers have ‘teeny tiny numbers’ that will simplify approximationestimates. Direct computations with the ideal numbers produce symbolic approximations equivalent to the function limits needed in dierentiation theory (that the rules of Theorem 1.12 give a direct way to compute.) Limit theory does not give the answer, but only a way to justify it once you have found it.

1.1 Field Axioms

The laws of algebra follow from the eld axioms. This means that algebra is the same with Dedekind’s “real” numbers, the complex numbers, and Robinson’s “hyperreal” numbers.

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4 1. Numbers

Axiom 1.1. Field Axioms A “eld” of numbers is any set of objects together with two operations, addition and multiplication where the operations satisfy: • The commutative laws of addition and multiplication, a1 + a2 = a2 + a1 & a1 ·a2 = a2 ·a1 • The associative laws of addition and multiplication, a1 +(a2+a3)=(a1+a2)+a3 & a1·(a2·a3)=(a1·a2)·a3 •The distributive law of multiplication over addition, a1 ·(a2 + a3)=a1·a2+a1·a3 •There is an additive identity, 0, with0+a=afor every number a. • There is an multiplicative identity, 1, with1·a=afor every number a 6=0. •Each number a has an additive inverse, −a, witha+(−a)=0. •Each nonzero number a has a multiplicative inverse, 1 a, witha·1 a=1. A computation needed in calculus is Example 1.1. The Cube of a Binomial

(x +x)3 =x3+3x2x+3xx2+x3 =x3+3x2x+(x(3x +x))x

We analyze the term ε = (x(3x +x)) in dierentiation. Thereadercouldlaboriouslydemonstratethatonlytheeldaxiomsareneededtoperform the computation. This means it holds for rational, real, complex, or hyperreal numbers. Here is a start. Associativity is needed so that the cube is well dened, or does not depend on the order we multiply. We use this in the next computation, then use the distributive property, the commutativity and the distributive property again, and so on.

(x+x)3 =(x+x)(x +x)(x +x) =(x+x)((x +x)(x +x)) =(x+x)((x +x)x+(x+x)x) =(x+x)((x2 + xx)+(xx+x2)) =(x+x)(x2 + xx + xx +x2) =(x+x)(x2 +2xx+x2) =(x+x)x2+(x+x)2xx +(x+x)x2) . . .

The natural counting numbers 1,2,3,...have operations of addition and multiplication, but do not satisfy all the properties needed to be a eld. Addition and multiplication do satisfy the commutative, associative, and distributive laws, but there is no additive inverse

Field Axioms 5

0 in the counting numbers. In ancient times, it was controversial to add this element that could stand for counting nothing, but it is a useful ction in many kinds of computations. The negative integers −1,−2,−3,...are another idealization added to the natural numbers that make additive inverses possible - they are just new numbers with the needed property. Negative integers have perfectly concrete interpretations such as measurements to the left, rather than the right, or amounts owed rather than earned. The set of all integers; positive, negative, and zero, still do not form a eld because there are no multiplicative inverses. Fractions,±1/2,±1/3, ...are the needed additional inverses. When they are combined with the integers through addition, we have the set of all rational numbers of the form ±p/q for natural numbers p and q 6= 0. The rational numbers are a eld, that is, they satisfy all the axioms above. In ancient times, rationals were sometimes considered only “operators” on “actual” numbers like 1,2,3,.... The point of the previous paragraphs is simply that we often extend one kind of number system in order to have a new system with useful properties. The complex numbers extend the eld axioms above beyond the “real” numbers by adding a number i that solves the equation x2 = −1. (See the CD Chapter 29 of the main text.) Hundreds of years ago this number was controversial and is still called “imaginary.” In fact, all numbers are useful constructs of our imagination and some aspects of Dedekind’s “real” numbers are much more abstract than i2 = −1. (For example, since the reals are “uncountable,” “most” real numbers have no description what-so-ever.) The rationals are not “complete” in the sense that the linear measurement of the side of an equilateral right triangle (√2) cannot be expressed as p/q for p and q integers. In Section 1.3 we “complete” the rationals to form Dedekind’s “real” numbers. These numbers correspond to perfect measurements along an ideal line with no gaps. The complex numbers cannot be ordered with a notion of “smaller than” that is compatible with the eld operations. Adding an “ideal” number to serve as the square root of−1 is not compatible with the square of every number being positive. When we make extensions beyond the real number system we need to make choices of the kind of extension depending on the properties we want to preserve. Hyperreal numbers allow us to compute estimates or limits directly, rather than making inverse proofs with inequalities. Like the complex extension, hyperreal extension of the reals loses a property; in this case completeness. Hyperreal numbers are explained beginning in Section 1.4 below and then are used extensively in this background book to show how many intuitive estimates lead to simple direct proofs of important ideas in calculus. The hyperreal numbers (discovered by Abraham Robinson in 1961) are still controversial because they contain innitesimals. However, they are just another extended modern number system with a desirable new property. Hyperreal numbers can help you understand limits of real numbers and many aspects of calculus. Results of calculus could be proved without innitesimals, just as they could be proved without real numbers by using only rationals. Many professors still prefer the former, but few prefer the latter. We believe that is only because Dedekind’s “real” numbers are more familiar than Robinson’s, but we will make it clear how both approaches work as a theoretical background for calculus. There is no controversy concerning the logical soundness of hyperreal numbers. The use ofinnitesimals inthe earlydevelopmentofcalculus beginning with Leibniz, continuing with Euler, and persisting to the time of Gauss was problematic. The founders knew that their use of innitesimals was logically incomplete and could lead to incorrect results. Hyperreal numbers are a correct treatment of innitesimals that took nearly 300 years to discover.

6 1. Numbers

With hindsight, they also have a simple description. The Function Extension Axiom 2.1 explained in detail in Chapter 2 was the missing key.

Exercise set 1.1 1. Show that the identity numbers 0 and 1 are unique. (HINT: Suppose 00 + a = a. Add −ato both sides.) 2. Show that 0·a =0. (HINT: Expand0+b a·awith the distributive law and show that0 ·a+b = b. Then use the previous exercise.) 3. The inverses −a and 1 a are unique. (HINT: Suppose not, 0=a−a=a+b. Add−a to both sides and use the associative property.) 4. Show that −1·a =−a. (HINT: Use the distributive property on 0 = (1−1)·a and use the uniqueness of the inverse.) 5. Show that (−1)·(−1) = 1. 6. Other familiar properties of algebra follow from the axioms, for example, if a3 6=0and a4 6=0, then a 1+a 2 a 3 =a 1 a 3 +a 2 a 3 , a 1·a 2 a 3·a 4 =a 1 a 3· a 2 a 4 & a 3·a 46=0

1.2 Order Axioms Estimation is based on the inequality ≤ of the real numbers.

Oneimportantrepresentationofrationalandrealnumbersisasmeasurementsofdistance along a line. The additive identity 0 is located as a starting point and the multiplicative identity 1 is marked o (usually to the right on a horizontal line). Distances to the right correspond to positive numbers and distances to the left to negative ones. The inequality < indicates which numbers are to the left of others. The abstract properties are as follows. Axiom 1.2. Ordered Field Axioms A a number system is an ordered eld if it satises the eld Axioms 1.1 and has a relation < that satises: • Every pair of numbers a and b satises exactly one of the relations a = b, a