Infinite Powers

Front Cover......Page 1
Front Flap......Page 2
 Half Title......Page 3
 Title......Page 5
 Copyright......Page 6
 Contents......Page 7
 Introduction......Page 9
  Calculus for Everyone......Page 10
  The World According to Calculus......Page 12
  Calculus Is More than a Language......Page 14
  Unreasonably Effective......Page 15
  The Infinity Principle......Page 16
  The Golem of Infinity......Page 18
  Curves, Motion, and Change......Page 19
1. Infinity......Page 27
 Infinity as a Bridge Builder......Page 29
 A Pizza Proof......Page 30
 Limits and the Riddle of the Wall......Page 34
 The Parable of .333 . . .......Page 35
 The Parable of the Infinite Polygon......Page 37
 The Allure and Peril of Infinity......Page 39
 The Sin of Dividing by Zero......Page 40
 The Sin of Completed Infinity......Page 41
 Zeno’s Paradoxes......Page 42
 Zeno Goes Digital......Page 45
 Zeno Meets the Quantum......Page 47
2. The Man Who Harnessed Infinity......Page 53
 Squeezing Pi......Page 55
 The Tao of Pi......Page 59
 Cubism Meets Calculus......Page 61
 A Cheesy Argument......Page 65
 The Method......Page 68
 Archimedes Today: From Computer Animation to Facial Surgery......Page 76
 Onward to the Mystery of Motion......Page 82
3. Discovering the Laws of Motion......Page 85
 The World According to Aristotle......Page 86
 Enter Galileo......Page 90
 Falling, Rolling, and the Law of Odd Numbers......Page 92
 The Art of Scientific Minimalism......Page 95
 From a Swinging Chandelier to the Global Positioning System......Page 97
 Kepler and the Mystery of Planetary Motion......Page 104
 Kepler’s First Law: Elliptical Orbits......Page 107
 Kepler’s Second Law: Equal Areas in Equal Times......Page 108
 Kepler’s Third Law and the Sacred Frenzy......Page 110
 Kepler and Galileo, the Same and Not the Same......Page 111
 Storm Clouds Gathering......Page 112
4. The Dawn of Differential Calculus......Page 115
 The Rise of Algebra in the East......Page 116
 How Algebra Waxed While Geometry Waned......Page 117
 Algebra Meets Geometry......Page 119
 Equations as Curves......Page 122
 Fermat Versus Descartes......Page 124
 The Search for Analysis, the Long-Lost Method of Discovery......Page 127
 Optimizing for the Overhead Bin......Page 129
 How Fermat Helped the FBI......Page 133
 The Principle of Least Time......Page 139
 The Tussle over Tangents......Page 144
 Within Sight of the Promised Land......Page 146
5. The Crossroads......Page 149
 The Function of Functions......Page 151
 Power Functions......Page 152
 Exponential Functions......Page 153
 Powers of Ten......Page 154
 Logarithms......Page 157
 The Natural Logarithm and Its Exponential Function......Page 160
 The Mechanism Behind Exponential Growth and Decay......Page 163
6. The Vocabulary of Change......Page 167
 The Three Central Problems of Calculus......Page 170
 Linear Functions and Their Constant Rates......Page 172
 A Nonlinear Function and Its Changing Rate......Page 175
 Derivatives as Rates of Change of Day Length......Page 180
 Derivatives as Instantaneous Speeds......Page 185
7. The Secret Fountain......Page 193
 Areas, Integrals, and the Fundamental Theorem......Page 194
 Visualizing the Fundamental Theorem with Motion......Page 195
 Constant Acceleration......Page 198
 A Paint-Roller Proof of the Fundamental Theorem......Page 201
 The Meaning of the Fundamental Theorem......Page 205
 The Holy Grail of Integral Calculus......Page 206
 Local Versus Global......Page 211
 A Lonesome Boy......Page 212
 At Play with Power Series......Page 214
 Newton as Mash-Up Artist......Page 219
 A Private Calculus......Page 221
8. Fictions of the Mind......Page 225
 In the Twinkling of an Eyelid......Page 227
 Infinitesimals......Page 228
 The Cube of Numbers near 2......Page 230
 Differentials......Page 231
 Derivatives via Differentials......Page 232
 The Fundamental Theorem via Differentials......Page 235
 What Led Leibniz to Differentials and the Fundamental Theorem?......Page 237
 Fighting HIV with an Assist from Calculus......Page 244
9. The Logical Universe......Page 253
 The Logic of Nature......Page 255
 The Two-Body Problem......Page 260
 Newton Meets Hidden Figures......Page 263
 Calculus and the Enlightenment......Page 264
 From Discrete to Continuous Systems......Page 267
 Ordinary Versus Partial Differential Equations......Page 268
 Partial Differential Equations and the Boeing 787......Page 270
 The Ubiquity of Partial Differential Equations......Page 273
10. Making Waves......Page 275
 String Theory......Page 278
 Why Sine Waves?......Page 282
 Visualizing Modes of Vibration: Chladni Patterns......Page 285
 The Noblest Courage......Page 287
 Microwave Ovens......Page 288
 Why Microwave Ovens Used to Be Called Radar Ranges......Page 289
 Computerized Tomography and Brain Imaging......Page 291
11. The Future of Calculus......Page 297
 The Writhing Number of DNA......Page 299
 Determinism and Its Limits......Page 303
 Nonlinearity......Page 305
 Chaos......Page 307
 Poincaré’s Visual Approach......Page 308
 Nonlinearity Goes to War......Page 310
 The Alliance Between Calculus and Computers......Page 311
 Complex Systems and the Curse of High Dimensions......Page 313
 Computers, Artificial Intelligence, and the Mystery of Insight......Page 317
 Eight Decimal Places......Page 321
 Summoning the Positron......Page 323
 The Mystery of a Comprehensible Universe......Page 325
 Acknowledgments......Page 329
 Illustration Credits......Page 333
 Notes......Page 335
 Bibliography......Page 359
 Index......Page 371
Back Flap......Page 387
Back Cover......Page 388
Spine......Page 0