Computer algebra recipes : a gourmet's guide to mathematical models of science

Computer algebra systems have the potential to revolutionize the teaching of and learning of science. Not only can students work thorough mathematical models much more efficiently and with fewer errors than with pencil and paper, they can also work with much more complex and computationally intensive models. Thus, for example, in studying the flight of a golf ball, students can begin with the simple parabolic trajectory, but then add the effects of lift and drag, of winds, and of spin. Not only can the program provide analytic solutions in some cases, it can also produce numerical solutions and graphic displays.
Aimed at undergraduates in their second or third year, this book is filled with examples from a wide variety of disciplines, including biology, economics, medicine, engineering, game theory, physics, chemistry. The text is organized along a spiral, revisiting general topics such as graphics, symbolic computation, and numerical simulation in greater detail and more depth at each turn of the spiral.
The heart of the text is a large number of computer algebra recipes. 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 a story that leads the reader through 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.

Reviews 1

Choice Review

Enns (physics, Simon Fraser Univ.) and McGuire (physics, University College of the Fraser Valley) previously published a book on nonlinear physics using the Maple computer algebra system, Nonlinear Physics with Maple for Scientists and Engineers (1997). The book under review and accompanying CD-ROM are a collection of examples of the application of Maple to scientific problems. Each section consists of a short description of a physical problem with a corresponding mathematical model and Maple worksheet. Exercises ask the student to modify or extend the Maple worksheet. The examples are organized into three themes of graphical, numeric, and symbolic computation. The mathematics used ranges from algebra to calculus, ordinary differential equations, partial differential equations, and probability theory. Some of the sections are classic fare, such as the analysis of fox and rabbit populations, while other examples are new and interesting. This book will serve as a useful source of examples and exercises for both teachers and students. Undergraduates. B. Borchers New Mexico Institute of Mining and Technology

Preface	p. vii
Introduction	p. 1
A. Computer Algebra Systems	p. 1
B. The Spiral Staircase to Learning	p. 3
C. How to Climb the Spiral Staircase	p. 7
I The Appetizers	p. 9
1 The Pictures of Science	p. 11
1.1 Introduction	p. 11
1.2 Data and Function Plots	p. 13
1.2.1 Correcting for Inflation	p. 13
1.2.2 The Plummeting Badminton Bird	p. 20
1.2.3 Minimizing the Travel Time	p. 29
1.3 Log-Log (Power Law) Plots	p. 33
1.3.1 Chimpanzee Brain Size	p. 33
1.3.2 Scaling Arguments and Gulliver's Travels	p. 38
1.4 Contour and Gradient Plots	p. 43
1.4.1 The Secret Message	p. 43
1.4.2 Designing a Ski Hill	p. 47
1.5 Animated Plots	p. 53
1.5.1 Waves Are Dynamic	p. 53
1.5.2 The Sands of Time	p. 56
2 Deriving Model Equations	p. 59
2.1 Introduction	p. 59
2.2 Linear Correlation	p. 60
2.2.1 What Is Linear Correlation?	p. 60
2.2.2 The Corn Palace	p. 61
2.3 Least Squares Derivations	p. 63
2.3.1 Regression Analysis	p. 63
2.3.2 Will You Be Better Off Than Your Parents?	p. 65
2.3.3 What Was the Heart Rate of a Brachiosaurus?	p. 71
2.3.4 Senate Renewal	p. 78
2.3.5 Bikini Sales and the Logistic Curve	p. 81
2.3.6 Following the Dow Jones Index	p. 86
2.3.7 Variation of "g" with Latitude	p. 93
2.3.8 Finding Romeo a Juliet	p. 97
2.4 Multiple Regression Equations	p. 101
2.4.1 Real Estate Appraisals	p. 102
2.4.2 And the Winner Is?	p. 107
3 Algebraic Models	p. 113
3.1 Introduction	p. 113
3.2 Algebraic Examples	p. 114
3.2.1 Bombs Versus Schools	p. 114
3.2.2 Kirchhoff Rules the Electrical World	p. 121
3.2.3 The Window Washer's Secret	p. 128
3.2.4 The Science Student's Summer Job Interview	p. 134
3.2.5 Envelope of Safety	p. 140
3.2.6 Rainbow County	p. 144
3.3 Integral Examples	p. 150
3.3.1 The Great Pyramid of Cheops	p. 150
3.3.2 Noah's Ark	p. 156
3.4 Vector Examples	p. 166
3.4.1 Vectoria's Mathematical Heritage	p. 166
3.4.2 Ain't She Sweet	p. 173
3.4.3 Born Curl Free	p. 181
3.4.4 Of Flux and Circulation and Coordinates Too	p. 186
4 Monte Carlo Methods	p. 195
4.1 Introduction	p. 195
4.2 Random Walks	p. 197
4.2.1 The Concept	p. 197
4.2.2 The Soccer Fan's Drunken Walk	p. 200
4.2.3 Blowin' in the Wind	p. 205
4.2.4 Flight of Penelope Jitter Bug	p. 209
4.2.5 That Meandering Perfume Molecule	p. 212
4.3 Monte Carlo Integration	p. 215
4.3.1 Standard Numerical Integration Algorithms	p. 215
4.3.2 Monte Carlo Integration	p. 219
4.3.3 Wait and Buy Later!	p. 220
4.3.4 Wait and Buy Later! The Sequel	p. 224
4.3.5 Estimating [pi]	p. 229
4.3.6 Chariot of Fire and Destruction	p. 231
4.4 Probability Distributions	p. 236
4.4.1 Of Nuts and Bolts and Hospital Beds Too	p. 236
4.4.2 The Ice Wines of Rainbow County	p. 243
4.5 Monte Carlo Statistical Distributions	p. 250
4.5.1 Estimating e	p. 250
4.5.2 Vapor Deposition	p. 255
II The Entrees	p. 261
5 Phase-Plane Portraits	p. 263
5.1 Introduction	p. 263
5.2 Phase-Plane Portraits	p. 263
5.2.1 Stationary or Singular Points	p. 265
5.3 Linear ODE Models	p. 268
5.3.1 Tenure Policy at Erehwon University	p. 268
5.3.2 Vectoria Investigates the RLC Circuit	p. 273
5.4 Nonlinear ODE Models	p. 280
5.4.1 Classification of Stationary Points	p. 280
5.4.2 Rabbits and Foxes	p. 284
5.4.3 The Mona Lisa of Nonlinear Science	p. 292
5.4.4 Mike Creates a Higher-Order Singular Point	p. 301
5.4.5 The Gnus and Sung of Erehwon	p. 308
5.5 Nonautonomous ODEs	p. 314
5.5.1 Can an Unstable Spring Find Stability?	p. 314
5.5.2 The Period Doubling Route to Chaos	p. 317
6 Linear ODE Models	p. 325
6.1 Introduction	p. 325
6.2 Solving Linear ODEs with Maple	p. 326
6.3 First-Order ODE Models	p. 332
6.3.1 There Goes Louie's Alibi	p. 332
6.3.2 The Water Skier	p. 341
6.4 Second-Order ODE Models	p. 345
6.4.1 Shrinking the Safety Envelope	p. 345
6.4.2 Halley's Comet	p. 350
6.4.3 Frank N. Stein Is Not Heartless	p. 358
6.4.4 Vectoria Feels the Force and Hits the Bottle	p. 362
6.5 Bessel and Legendre ODE Models	p. 369
6.5.1 Introduction to Special Functions	p. 369
6.5.2 The Vibrating Bungee Cord	p. 376
6.5.3 Wheel of misFortune	p. 382
6.5.4 The Weedeater	p. 391
7 Nonlinear ODE Models	p. 397
7.1 Introduction	p. 397
7.2 First-Order Models	p. 398
7.2.1 The Nonlinear Diode	p. 398
7.2.2 The Bad Bird Equation	p. 402
7.2.3 The Struggle for Existence	p. 407
7.3 Second-Order Models	p. 414
7.3.1 Pirates of the Caribbean	p. 414
7.3.2 Oh What Sounds We Hear!	p. 418
7.3.3 Those Lennard-Jones Vibrational Blues	p. 424
7.3.4 Golf Is Such an "Uplifting" Experience	p. 432
7.3.5 This Would Be a Great Amusement Park Ride	p. 438
7.4 Limit Cycles	p. 445
7.4.1 The Bizarre World of the Tunnel Diode Oscillator	p. 445
7.4.2 Follow That Rabbit	p. 452
8 Difference Equation Models	p. 459
8.1 Introduction	p. 459
8.2 Linear Difference Equation Models	p. 460
8.3 First-Order Linear Models	p. 461
8.3.1 Those Dratted Gnats	p. 461
8.3.2 Gone Fishing	p. 464
8.4 Second-Order Linear Models	p. 467
8.4.1 Fibonacci's Adam and Eve Rabbit	p. 467
8.4.2 How Red Is Your Blood?	p. 471
8.4.3 Fermi-Pasta-Ulam Is Not a Spaghetti Western	p. 473
8.5 Nonlinear Difference Equation Models	p. 484
8.6 First-Order Nonlinear Models	p. 484
8.6.1 Competition for Available Resources	p. 484
8.6.2 The Logistic Map and Cobweb Diagrams	p. 492
8.7 Second-Order Nonlinear Models	p. 499
8.7.1 The Bouncing Ball Art Gallery	p. 499
8.7.2 Onset of Chaos: A Model for the Outbreak of War	p. 503
8.8 Numerically Solving ODEs	p. 513
8.8.1 Finite Difference Approximations to Derivatives	p. 513
8.8.2 Rabbits and Foxes: The Sequel	p. 516
8.8.3 Glycolytic Oscillator	p. 521
9 Some Analytic Approaches	p. 527
9.1 Introduction	p. 527
9.2 Checking Solutions	p. 527
9.2.1 The Palace of the Governors	p. 527
9.2.2 Play It, Sam	p. 532
9.2.3 The Three-Piece String	p. 536
9.3 Calculus of Variations	p. 541
9.3.1 Dress Design, The Erehwonese Way	p. 541
9.3.2 Queen Dido's Problem	p. 548
9.3.3 The Human Fly Plans His Escape Route	p. 552
9.4 Fourier Series	p. 559
9.4.1 Hi C Is Not Always a Drink	p. 562
9.4.2 Play It, Sam: A New Perspective	p. 565
9.4.3 Vectoria Sums a Series	p. 569
10 Fractal Patterns	p. 573
10.1 Introduction	p. 573
10.2 Difference Equation Patterns	p. 574
10.2.1 Wallpaper for the Mind	p. 574
10.2.2 Sierpinski's Fractal Gasket	p. 576
10.2.3 Barnsley's Fern	p. 583
10.2.4 Douady's Rabbit and Other Fauna and Flora	p. 588
10.2.5 The Rings of Saturn	p. 592
10.3 ODE Patterns	p. 601
10.3.1 The Butterfly Attractor	p. 601
10.3.2 Rossler's Strange Attractor	p. 606
10.4 Cellular Automata Patterns	p. 608
10.4.1 A Navaho Rug Design	p. 608
10.4.2 The One Out of Eight Rule	p. 611
III The Desserts	p. 615
11 Diagnostic Tools for Nonlinear Dynamics	p. 617
11.1 Introduction	p. 617
11.2 The Poincare Section	p. 617
11.2.1 The Concept	p. 617
11.2.2 A Rattler Signals Chaos	p. 618
11.3 The Power Spectrum	p. 622
11.3.1 The Concept	p. 622
11.3.2 The Rattler Returns	p. 624
11.4 The Bifurcation Diagram	p. 628
11.4.1 The Concept	p. 628
11.4.2 Pitchforks and Other Bifurcations	p. 629
11.5 The Lyapunov Exponent	p. 632
11.5.1 The Concept	p. 632
11.5.2 Mr. Lyapunov Agrees	p. 633
11.6 Reconstructing an Attractor	p. 635
11.6.1 The Concept	p. 635
11.6.2 Chaos Versus Noise	p. 636
12 Linear PDE Models	p. 641
12.1 Introduction	p. 641
12.1.1 The Linear PDEs of Mathematical Physics	p. 641
12.1.2 Separation of Variables	p. 643
12.2 Diffusion and Laplace's Equation Models	p. 647
12.2.1 Freeing Excalibur	p. 647
12.2.2 Aussie Barbecue	p. 651
12.2.3 Erehwon Institute of Technology	p. 655
12.2.4 Hugo and the Atomic Bomb	p. 659
12.2.5 Hugo Prepares for his Job Interview	p. 666
12.3 Wave Equation Models	p. 672
12.3.1 Vectoria Encounters Simon Legree	p. 672
12.3.2 Homer's Jiggle Test	p. 676
12.3.3 Vectoria's Second Problem	p. 681
12.4 Semi-Infinite and Infinite Domains	p. 685
12.4.1 Vectoria's Third Problem	p. 686
12.4.2 Assignment Complete!	p. 688
12.4.3 Radioactive Contamination	p. 691
12.4.4 "Play It, Sam" Revisited	p. 696
13 Nonlinear PDE Models: Solition Solutions	p. 701
13.1 Introduction	p. 701
13.2 Solitary Waves	p. 702
13.3 The Graphical Hunt for Solitons	p. 704
13.3.1 Of Kinks and Antikinks	p. 704
13.3.2 In Search of Bright Solitons	p. 708
13.3.3 Can Three Solitary Waves Live Together?	p. 712
13.4 Analytic Soliton Solutions	p. 715
13.4.1 Follow That Wave!	p. 715
13.4.2 Looking for a Kinky Solution	p. 719
14 Simulating PDE Models	p. 723
14.1 Introduction	p. 723
14.2 Diffusion and Wave Equation Models	p. 724
14.2.1 Freeing Excalibur the Numerical Way	p. 724
14.2.2 Vectoria Secret	p. 728
14.2.3 Enjoy the Klein-Gordon Vibes	p. 730
14.3 Soliton Collisions	p. 734
14.3.1 To Be or Not to Be a Soliton	p. 734
14.3.2 Are Diamonds a Kink's Best Friend?	p. 738
Epilogue	p. 745
Bibliography	p. 747
Index	p. 753

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