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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.
"Overall, this book is a really fun read. The combination of mathematics applied to real physics problems and the historical fabric within which they are woven proved a winner for me. I could write more about this volume, but I think I'll quit here--I want to get to work on some of the challenge problems."--Barry R. Holstein, American Journal of Physics
"This book shows mathematics and physics at their very best, united to explore fascinating phenomena with astonishing results."--Linda Kallam, Mathematics Teacher
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
Overview
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 ...