Mrs. Perkins's Electric Quilt and Other Intriguing Stories of Mathematical Physics

About this book

What does quilting have to do with electric circuit theory? The answer is just one of the fascinating ways that best-selling popular math writer Paul Nahin illustrates the deep interplay of math and physics in the world around us in his latest book of challenging mathematical puzzles, "Mrs. Perkins's Electric Quilt". With his trademark combination of intriguing mathematical problems and the historical anecdotes surrounding them, Nahin invites readers on an exciting and informative exploration of some of the many ways math and physics combine to create something vastly more powerful, useful, and interesting than either is by itself.

In a series of brief and largely self-contained chapters, Nahin discusses a wide range of topics in which math and physics are mutually dependent and mutually illuminating, from Newtonian gravity and Newton's laws of mechanics to ballistics, air drag, and electricity. The mathematical subjects range from algebra, trigonometry, geometry, and calculus to differential equations, Fourier series, and theoretical and Monte Carlo probability. Each chapter includes problems - some three dozen in all - that challenge readers to try their hand at applying what they have learned.

Just as in his other books of mathematical puzzles, Nahin discusses the historical background of each problem, gives many examples, includes MATLAB codes, and provides complete and detailed solutions at the end. "Mrs. Perkins's Electric Quilt" will appeal to students interested in new math and physics applications, teachers looking for unusual examples to use in class - and anyone who enjoys popular math books.

Contents

For the Reader xi Preface xiii Chapter 1: Three Examples of the Mutual Embrace 1 1.1 Unphysical Laws 1 1.2 When Math Goes Wrong 6 1.3 Math from Physics 13 Chapter 2: Measuring Gravity 18 2.1 First, a Little Theory 18 2.2 Out in the Author's Garage 21 Chapter 3: Feynman's Infinite Circuit 24 3.1 An Infinity of Resistors 24 3.2 An Infinity of Reactances, and Recursion 27 3.3 Convergence--or Not? 32 3.4 Three More Infinite, All-Resistor Networks 36 Chapter 4: Air Drag--A Mathematical View 44 4.1 Air Drag Treated Broadly 44 4.2 Air Drag Treated with Some Detail 51 Chapter 5: Air Drag--A Physical View 62 5.1 The Quadratic Force Law 62 5.2 Long Falls through a Real Atmosphere 70 Chapter 6: Really Long Falls 82 6.1 Falling into the Sun 82 6.2 Falling from Heaven to Hell 86 Chapter 7: The Zeta Function--and Physics 94 7.1 A Curious Double Integral 94 7.2 Fourier Series and the Zeta Function 95 7.3 The Zeta Function in Physics 100 Chapter 8: Ballistics--With No Air Drag (Yet) 107 8.1 Shooting a Cannon in a Vacuum 107 8.2 What Makes a Champion Shot-Putter? 112 8.3 Another Cannon Question 116 Chapter 9: Ballistics--With Air Drag 120 9.1 Thin Air Cannot Be Ignored! 120 9.2 Air Drag and Baseball 126 Chapter 10: Gravity and Newton 136 10.1 The Beginnings of Modern Gravity 136 10.2 Newton's Superb Theorems 140 10.3 The Moon Test and Blowing-Up Planets 148 10.4 A Surprising Gravity Calculation 152 10.5 Gravitational Contraction 157 Chapter 11: Gravity Far Above the Earth 170 11.1 Kepler's Laws of Planetary Motion 170 11.2 Weighing the Planets 175 Chapter 12: Gravity Inside the Earth 186 12.1 Newton's Experiment 186 12.2 Gravity Inside the Earth 191 12.3 Pressure at the Center of the Earth 200 12.4 Travel Inside the Earth 203 12.5 Epilogue 209 Chapter 13: Quilts & Electricity 215 13.1 Recreational Mathematics 215 13.2 Electric Quilts 220 13.3 Three Impossibility Proofs 225 Chapter 14: Random Walks 233 14.1 Ronald Ross and the Flight of Mosquitoes 233 14.2 Karl Pearson Formulates a Famous Problem 236 14.3 Gambler's Ruin 241 14.4 The Monte Carlo Method 245 Chapter 15: Two More Random Walks 261 15.1 Brownian Motion 261 15.2 Shrinking Walks 269 Chapter 16: Nearest Neighbors 285 16.1 Cannibals Can Be Fun! 285 16.2 Neighbors Beyond the Nearest 291 16.3 What Happens When We Have Lots of Cannibals 294 16.4 Serious Physics 296 Chapter 17: One Last Random Walk 299 17.1 Resistor Mathematics 299 17.2 Electric Walks 301 17.3 Monte Carlo Circuit Simulation 305 17.4 Symmetry, Superposition, and Resistor Circuits 313 Chapter 18: The Big Noise 321 18.1 An Interesting Textbook Problem 321 18.2 The Polar Equations of the Big-Noise Flight 322 18.3 The Acceleration on a Big-Noise Flight Path 328 SOLUTIONS TO THE CHALLENGE PROBLEMS 333 SPECIAL BONUS DISCUSSION 371 Warning: Do Not Read before Reading Disscussion 17 373 Chapter 19: Electricity in the Fourth Dimension 373 19.1 The Tesseract 373 19.2 Connecting a Tesseract Resistor Cube 376 Acknowledgments 385 Index 387

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Biography

Paul J. Nahin is the author of many best-selling popular math books, including "Digital Dice, Chases and Escapes, Dr. Euler's Fabulous Formula, When Least is Best, Duelling Idiots and Other Probability Puzzlers," and "An Imaginary Tale" (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire.