When least is best : how mathematicians discovered many clever ways to make things as small (or as large) as possible /

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

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Nahin, Paul J. (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Princeton, New Jersey : Princeton University Press, 2021.
Temas:
Maxima and minima.
Mathematics > History.
Maximums et minimums.
Mathématiques > Histoire.
MATHEMATICS / History & Philosophy
Mathematics
Maxima and minima
History
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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.

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