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

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...
Ausführliche Beschreibung

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 Ausführliche Beschreibung

Autor*in:	Nahin, Paul J [verfasserIn]

Format:	E-Book
Sprache:	Englisch

Erschienen:	Princeton, NJ: Princeton University Press ; 2021

Rechteinformationen:	Restricted Access

Schlagwörter:	Mathematics, History Maxima and minima MATHEMATICS / History & Philosophy

Umfang:	1 Online-Ressource (392 p) ; 99 b/w illus
Beschreibung:	Mode of access: Internet via World Wide Web.

Reproduktion:	[Online-Ausgabe]

Reihe:	Princeton Science Library ; 118

Links:	Link aufrufen Link aufrufen Link aufrufen Cover Cover
ISBN:	978-0-691-22038-3

DOI / URN:	10.1515/9780691220383

Katalog-ID:	1761838199