What is the best way to photograph a speeding bullet? How can lost hikers find their way out of a forest? Why does light move through glass in the least amount of time possible? When Least Is Best combines the mathematical history of extrema with contemporary examples to answer these intriguing questions and more. Paul Nahin shows how life often works at the extremes―with values becoming as small (or as large) as possible―and he considers how mathematicians over the centuries, including Descartes, Fermat, and Kepler, have grappled with these problems of minima and maxima. Throughout, Nahin examines entertaining conundrums, such as how to build the shortest bridge possible between two towns, how to vary speed during a race, and how to make the perfect basketball shot. Moving from medieval writings and modern calculus to the field of optimization, the engaging and witty explorations of When Least Is Best will delight math enthusiasts everywhere.
Author(s): Paul J. Nahin
Series: Princeton Science Library 114
Edition: New Paperback Edition
Publisher: Princeton University Press
Year: 2021
Language: English
Pages: 407
Tags: Minima and Maxima
Cover
Title
Copyright
Dedication
Contents
Preface to the 2021 Edition
Preface to the 2007 Paperback Edition
Preface
1. Minimums, Maximums, Derivatives, and Computers
1.1 Introduction
1.2 When Derivatives Don’t Work
1.3 Using Algebra to Find Minimums
1.4 A Civil Engineering Problem
1.5 The AM-GM Inequality
1.6 Derivatives from Physics
1.7 Minimizing with a Computer
2. The First Extremal Problems
2.1 The Ancient Confusion of Length and Area
2.2 Dido’s Problem and the Isoperimetric Quotient
2.3 Steiner’s “Solution” to Dido’s Problem
2.4 How Steiner Stumbled
2.5 A “Hard” Problem with an Easy Solution
2.6 Fagnano’s Problem
3. Medieval Maximization and Some Modern Twists
3.1 The Regiomontanus Problem
3.2 The Saturn Problem
3.3 The Envelope-Folding Problem
3.4 The Pipe-and-Corner Problem
3.5 Regiomontanus Redux
3.6 The Muddy Wheel Problem
4. The Forgotten War of Descartes and Fermat
4.1 Two Very Different Men
4.2 Snell’s Law
4.3 Fermat, Tangent Lines, and Extrema
4.4 The Birth of the Derivative
4.5 Derivatives and Tangents
4.6 Snell’s Law and the Principle of Least Time
4.7 A Popular Textbook Problem
4.8 Snell’s Law and the Rainbow
5. Calculus Steps Forward, Center Stage
5.1 The Derivative: Controversy and Triumph
5.2 Paintings Again, and Kepler’s Wine Barrel
5.3 The Mailable Package Paradox
5.4 Projectile Motion in a Gravitational Field
5.5 The Perfect Basketball Shot
5.6 Halley’s Gunnery Problem
5.7 De L’Hospital and His Pulley Problem, and a New Minimum Principle
5.8 Derivatives and the Rainbow
6. Beyond Calculus
6.1 Galileo’s Problem
6.2 The Brachistochrone Problem
6.3 Comparing Galileo and Bernoulli
6.4 The Euler-Lagrange Equation
6.5 The Straight Line and the Brachistochrone
6.6 Galileo’s Hanging Chain
6.7 The Catenary Again
6.8 The Isoperimetric Problem, Solved (at last!)
6.9 Minimal Area Surfaces, Plateau’s Problem, and Soap Bubbles
6.10 The Human Side of Minimal Area Surfaces
7. The Modern Age Begins
7.1 The Fermat/Steiner Problem
7.2 Digging the Optimal Trench, Paving the Shortest Mail Route, and Least-Cost Paths through Directed Graphs
7.3 The Traveling Salesman Problem
7.4 Minimizing with Inequalities (Linear Programming)
7.5 Minimizing by Working Backwards (Dynamic Programming)
Appendix A. The AM-GM Inequality
Appendix B. The AM-QM Inequality, and Jensen’s Inequality
Appendix C. “The Sagacity of the Bees”
Appendix D. Every Convex Figure Has a Perimeter Bisector
Appendix E. The Gravitational Free-Fall Descent Time along a Circle
Appendix F. The Area Enclosed by a Closed Curve
Appendix G. Beltrami’s Identity
Appendix H. The Last Word on the Lost Fisherman Problem
Appendix I. Solution to the New Challenge Problem
Acknowledgments
Index
