Computer algebra recipes An introductory guide to the mathematical models of science

Computer algebra systems are revolutionizing the teaching, the learning, and the exploration of science. Not only can students and researchers work through mathematical models more efficiently and with fewer errors than with pencil and paper, they can also easily explore, both analytically and numerically, more complex and computationally intensive models.

Aimed at science and engineering undergraduates at the sophomore/junior level, this introductory guide to the mathematical models of science is filled with examples from a wide variety of disciplines, including biology, economics, medicine, engineering, game theory, mathematics, physics, and chemistry. The topics are organized into the Appetizers dealing with graphical aspects, the Entrees concentrating on symbolic computation, and the Desserts illustrating numerical simulation.

The heart of the text is a large number of computer algebra recipes based on the Maple 10 software system. These have been designed not only to provide tools for problem solving, but also to stimulate the reader's imagination. Associated with each recipe is a scientific model or method and an interesting or amusing story (accompanied with a thought-provoking quote) that leads the reader through the various steps of the recipe. Each section of recipes is followed by a set of problems that readers can use to check their understanding or to develop the topic further.

This text is the first of two volumes, the advanced guide, aimed at junior/senior/graduate level students, dealing with more advanced differential equation models.

Preface	p. v
Introduction	p. 1
A Computer Algebra Systems	p. 1
B Computer Algebra Recipes	p. 3
C Introductory Recipe: Bridge Design 101	p. 5
D Maple Help	p. 9
E How to Use This Text	p. 10
I The Appetizers	p. 11
1 The Pictures of Science	p. 13
1.1 Data and Function Plots	p. 13
1.1.1 Correcting for Inflation	p. 15
1.1.2 The Plummeting Badminton Bird	p. 22
1.1.3 Minimizing the Travel Time	p. 31
1.2 Log-Log (Power Law) Plots	p. 36
1.2.1 Chimpanzee Brain Size	p. 36
1.2.2 Scaling Arguments and Gulliver's Travels	p. 40
1.3 Contour and Gradient Plots	p. 46
1.3.1 The Secret Message	p. 46
1.3.2 Designing a Ski Hill	p. 49
1.4 Animated Plots	p. 56
1.4.1 Waves Are Dynamic	p. 56
1.4.2 The Sands of Time	p. 59
1.4.3 These Arrows Are Useful	p. 61
2 Deriving Model Equations	p. 65
2.1 Linear Correlation	p. 66
2.1.1 The Corn Palace	p. 67
2.2 Least Squares Derivations	p. 69
2.2.1 Will You Be Better Off Than Your Parents?	p. 71
2.2.2 What Was the Heart Rate of a Brachiosaur?	p. 76
2.2.3 Senate Renewal	p. 84
2.2.4 Bikini Sales and the Logistic Curve	p. 87
2.2.5 Following the Dow Jones Index	p. 91
2.2.6 Variation of "g" with Latitude	p. 98
2.2.7 Finding Romeo a Juliet	p. 103
2.3 Multiple Regression Equations	p. 106
2.3.1 Real Estate Appraisals	p. 107
2.3.2 And the Winner Is?	p. 113
II The Entrees	p. 119
3 Algebraic Models. Part I	p. 121
3.1 Scalar Models	p. 121
3.1.1 Bombs Versus Schools	p. 122
3.1.2 Kirchhoff Rules the Electrical World	p. 129
3.1.3 The Window Washer's Secret	p. 136
3.1.4 The Science Student's Summer Job Interview	p. 142
3.1.5 Envelope of Safety	p. 148
3.1.6 Rainbow County	p. 152
3.2 Integral Examples	p. 156
3.2.1 The Great Pyramid of Cheops	p. 156
3.2.2 Noah's Ark	p. 162
4 Algebraic Models. Part II	p. 173
4.1 Vector Models	p. 173
4.1.1 Vectoria's Mathematical Heritage	p. 174
4.1.2 Vectoria and Fowles's Fly	p. 179
4.1.3 Ain't She Sweet	p. 183
4.1.4 Born Curl-Free	p. 188
4.1.5 Of Coordinates and Circulation Too	p. 194
4.1.6 All Is Flux	p. 199
4.2 Matrix Models	p. 202
4.2.1 Secret Message Revisited	p. 202
4.2.2 A Fishy Tale	p. 205
4.2.3 Population Waves	p. 208
5 Linear ODE Models	p. 213
5.1 Phase-Plane Portraits	p. 214
5.1.1 Tenure Policy at Erehwon University	p. 216
5.1.2 Vectoria Investigates the RLC Circuit	p. 221
5.2 First-Order ODE Models	p. 229
5.2.1 There Goes Louie's Alibi	p. 229
5.2.2 The Water Skier	p. 238
5.2.3 Ready to Charge	p. 242
5.3 Second-Order ODE Models	p. 245
5.3.1 Shrinking the Safety Envelope	p. 245
5.3.2 Frank N. Stein Is Not Heartless	p. 251
5.3.3 Halley's Comet	p. 255
5.3.4 Wheel of misFortune	p. 260
5.3.5 The Weedeater	p. 266
5.3.6 Can an Unstable Spring Find Stability?	p. 269
6 Difference Equation Models	p. 271
6.1 Linear Models	p. 272
6.1.1 Those Dratted Gnats	p. 272
6.1.2 Gone Fishing	p. 276
6.1.3 Fibonacci's Adam and Eve Rabbit	p. 279
6.1.4 How Red Is Your Blood?	p. 283
6.1.5 Fermi-Pasta-Ulam Is Not a Spaghetti Western	p. 285
6.2 Nonlinear Models	p. 292
6.2.1 Competition for Available Resources	p. 292
6.2.2 The Logistic Map and Cobweb Diagrams	p. 299
6.2.3 The Bouncing Ball Art Gallery	p. 306
6.2.4 Onset of Chaos: A Model for the Outbreak of War	p. 310
III The Desserts	p. 317
7 Monte Carlo Methods	p. 319
7.1 Random Walks	p. 321
7.1.1 The Soccer Fan's Drunken Walk	p. 324
7.1.2 Blowin' in the Wind	p. 329
7.1.3 Flight of Penelope Jitter Bug	p. 333
7.1.4 That Meandering Perfume Molecule	p. 335
7.2 Monte Carlo Integration	p. 338
7.2.1 Numerical Integration Methods	p. 339
7.2.2 Wait and Buy Later!	p. 344
7.2.3 Wait and Buy Later! The Sequel	p. 348
7.2.4 Estimating [pi]	p. 353
7.2.5 Chariot of Fire and Destruction	p. 355
7.3 Probability Distributions	p. 361
7.3.1 Of Nuts and Bolts and Hospital Beds Too	p. 361
7.3.2 The Ice Wines of Rainbow County	p. 367
7.4 Monte Carlo Statistical Distributions	p. 372
7.4.1 Estimating e	p. 372
7.4.2 Vapor Deposition	p. 376
8 Fractal Patterns	p. 381
8.1 Difference Equations	p. 384
8.1.1 Wallpaper for the Mind	p. 384
8.1.2 Sierpinski's Fractal Gasket	p. 386
8.1.3 Barnsley's Fern	p. 391
8.1.4 Douady's Rabbit and Other Fauna and Flora	p. 396
8.1.5 The Rings of Saturn	p. 400
8.2 Cellular Automata	p. 408
8.2.1 A Navaho Rug Design	p. 408
8.2.2 The One out of Eight Rule	p. 411
8.3 Strange Attractors	p. 414
8.3.1 Lorenz's Butterfly	p. 414
Epilogue	p. 416
Bibliography	p. 417
Index	p. 421

Available:*

On Order

Summary

Summary

Table of Contents