Calculus for Cranks

by Nets Hawk Katz

Overview

A new approach to the foundations of single variable calculus, based on the introductory course taught at Caltech

In mathematics, “cranks” are people who insist they understand something new about math even when the world tells them they are doing it wrong. This introduction to calculus is written with those cranks in mind, based on the foundational course that Nets Katz teaches at Caltech. It emphasizes the practical purposes of the foundations—for example, for tracking errors in calculations. In addition to covering the basics of single variable calculus, the book outlines the mathematical method—the ability to express oneself with absolute precision and then to use logical proofs to establish that certain statements are universally true. Katz emphasizes conceptual clarity as well as testing hypotheses and writing complete proofs. The result is a rigorous calculus book of use not only to future mathematicians but also to scientists and engineers.

Product Details

ISBN-13:	9780300242799
Publisher:	Yale University Press
Publication date:	01/26/2021
Pages:	264
Product dimensions:	6.12(w) x 9.25(h) x 0.44(d)

About the Author

Nets Hawk Katz is International Business Machines Professor of Mathematics at the California Institute of Technology. He was a Guggenheim fellow in 2012 and the winner of the Clay Research Award in 2016.

Table of Contents

Preface vii

1 Induction and the Real Numbers 1

1.2 Induction 1

1.2 Induction for derivatives of polynomials 13

1.3 The real numbers 23

1.4 Limits 36

2 Sequences and Series 46

2.1 Cauchy sequences and the Bolzano-Weierstrass and Squeeze theorems 46

2.2 Infinite series 57

2.3 Power series 68

3 Functions and Derivatives 80

3.1 Continuity and limits 80

3.2 Derivatives 91

3.3 Mean Value Theorem 100

3.4 Applications of the Mean Value Theorem 108

3.5 Exponentiation 121

3.6 Smoothness and series 129

3.7 Inverse function theorem 137

4 Integration 144

4.1 Definition of the Riemann integral 144

4.2 Integration and uniform continuity 155

4.3 The fundamental theorem 161

4.4 Taylor's theorem with remainder 170

4.5 Numerical integration 178

5 Convexity 185

5.1 Convexity and optimization 185

5.2 Inequalities 191

5.3 Economics 199

6 Trigonometry, Complex Numbers, and Power Series 205

6.1 Trigonometric functions by arclength 205

6.2 Complex numbers 211

6.3 Power series as functions 217

7 Complex Analysis 222

7.1 Roots of polynomials 222

7.2 Analytic functions 226

7.3 Cauchy's theorem 231

7.4 Taylor series of analytic functions 236

A Appendix 241

A.1 How to Use This Book 241

A.2 Logic, Axioms, and Problem Solving 243

A.3 Further Reading 247

Index 249