When Least Is Best : : How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible / / Paul J. Nahin.

• Superior document: Title is part of eBook package: De Gruyter EBOOK PACKAGE COMPLETE 2021 English
• VerfasserIn: Nahin, Paul J.,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2021]; ©2003
• Year of Publication: 2021
• Language: English
• Series: Princeton Science Library ; 114
• Physical Description: 1 online resource (392 p.) :; 99 b/w illus.

A mathematical journey through the most fascinating problems of extremes and how to solve themWhat 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...

Table of Contents

• Frontmatter
• 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” (the preface to Book 5 of Pappus’ Mathematical Collection)
• 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