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

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

by Paul J. Nahin
     
 

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ISBN-10: 0691070784

ISBN-13: 9780691070780

Pub. Date: 11/24/2003

Publisher: Princeton University Press

What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area? By combining the mathematical history of extrema with contemporary examples, Paul J

Overview

What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area? By combining the mathematical history of extrema with contemporary examples, Paul J. Nahin answers these intriguing questions and more in this engaging and witty volume. He shows how life often works at the extremes -- with values becoming as small (or as large) as possible -- and how mathematicians over the centuries have struggled to calculate these problems of minima and maxima. From medieval writings to the development of modern calculus to the current field of optimization, Nahin tells the story of Dido's problem, Fermat and Descartes, Torricelli, Bishop Berkeley, Goldschmidt, and more. Along the way, he explores how to build the shortest bridge possible between two towns, how to shop for garbage bags, how to vary speed during a race, and how to make the perfect basketball shot. Written in a conversational tone and requiring only an early undergraduate level of mathematical knowledge, When Least Is Best is full of fascinating examples and ready-to-try-at-home experiments. This is the first book on optimization written for a wide audience, and math enthusiasts of all backgrounds will delight in its lively topics.

Product Details

ISBN-13:
9780691070780
Publisher:
Princeton University Press
Publication date:
11/24/2003
Edition description:
With a New preface by the author
Pages:
400
Product dimensions:
6.20(w) x 9.46(h) x 1.13(d)

Related Subjects

Table of Contents

Prefacexiii
1.Minimums, Maximums, Derivatives, and Computers1
1.1Introduction1
1.2When Derivatives Don't Work4
1.3Using Algebra to Find Minimums5
1.4A Civil Engineering Problem9
1.5The AM-GM Inequality13
1.6Derivatives from Physics20
1.7Minimizing with a Computer24
2.The First Extremal Problems37
2.1The Ancient Confusion of Length and Area37
2.2Dido's Problem and the Isoperimetric Quotient45
2.3Steiner's "Solution" to Dido's Problem56
2.4How Steiner Stumbled59
2.5A "Hard" Problem with an Easy Solution62
2.6Fagnano's Problem65
3.Medieval Maximization and Some Modern Twists71
3.1The Regiomontanus Problem71
3.2The Saturn Problem77
3.3The Envelope-Folding Problem79
3.4The Pipe-and-Corner Problem85
3.5Regiomontanus Redux89
3.6The Muddy Wheel Problem94
4.The Forgotten War of Descartes and Fermat99
4.1Two Very Different Men99
4.2Snell's Law101
4.3Fermat, Tangent Lines, and Extrema109
4.4The Birth of the Derivative114
4.5Derivatives and Tangents120
4.6Snell's Law and the Principle of Least Time127
4.7A Popular Textbook Problem134
4.8Snell's Law and the Rainbow137
5.Calculus Steps Forward, Center Stage140
5.1The Derivative: Controversy and Triumph140
5.2Paintings Again, and Kepler's Wine Barrel147
5.3The Mailable Package Paradox149
5.4Projectile Motion in a Gravitational Field152
5.5The Perfect Basketball Shot158
5.6Halley's Gunnery Problem165
5.7De L'Hospital and His Pulley Problem, and a New Minimum Principle171
5.8Derivatives and the Rainbow179
6.Beyond Calculus200
6.1Galileo's Problem200
6.2The Brachistochrone Problem210
6.3Comparing Galileo and Bernoulli221
6.4The Euler-Lagrange Equation231
6.5The Straight Line and the Brachistochrone238
6.6Galileo's Hanging Chain240
6.7The Catenary Again247
6.8The Isoperimetric Problem, Solved (at last!)251
6.9Minimal Area Surfaces, Plateau's Problem, and Soap Bubbles259
6.10The Human Side of Minimal Area Surfaces271
7.The Modern Age Begins279
7.1The Fermat/Steiner Problem279
7.2Digging the Optimal Trench, Paving the Shortest Mail Route, and Least-Cost Paths through Directed Graphs286
7.3The Traveling Salesman Problem293
7.4Minimizing with Inequalities (Linear Programming)295
7.5Minimizing by Working Backwards (Dynamic Programming)312
Appendix A.The AM-GM Inequality331
Appendix B.The AM-QM Inequality, and Jensen's Inequality334
Appendix C."The Sagacity of the Bees"342
Appendix D.Every Convex Figure Has a Perimeter Bisector345
Appendix E.The Gravitational Free-Fall Descent Time along a Circle347
Appendix F.The Area Enclosed by a Closed Curve352
Appendix G.Beltrami's Identity359
Appendix H.The Last Word on the Lost Fisherman Problem361
Acknowledgments365
Index367

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