Computer Algebra Recipes : A Gourmet's Guide to Mathematical Models of Science

by Enns, Richard H.; McGuire, George C.

• ISBN13: 9780387951485
• ISBN10: 0387951482
• Format: Hardcover
• Copyright: 2001-06-01
• Publisher: Springer Verlag

Summary

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. The recipes are also included on the CD-ROM enclosed with the book. 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.

Table of Contents

Preface	vii
Introduction	1	(1)
Computer Algebra Systems	1	(2)
The Spiral Staircase to Learning	3	(4)
How to Climb the Spiral Staircase	7	(2)
I THE APPETIZERS	9	(252)
The Pictures of Science	11	(48)
Introduction	11	(2)
Data and Function Plots	13	(20)
Correcting for Inflation	13	(7)
The Plummeting Badminton Bird	20	(9)
Minimizing the Travel Time	29	(4)
Log-Log (Power Law) Plots	33	(10)
Chimpanzee Brain Size	33	(5)
Scaling Arguments and Gulliver's Travels	38	(5)
Contour and Gradient Plots	43	(10)
The Secret Message	43	(4)
Designing a Ski Hill	47	(6)
Animated Plots	53	(6)
Waves Are Dynamic	53	(3)
The Sands of Time	56	(3)
Deriving Model Equations	59	(54)
Introduction	59	(1)
Linear Correlation	60	(3)
What Is Linear Correlation?	60	(1)
The Corn Palace	61	(2)
Least Squares Derivations	63	(38)
Regression Analysis	63	(2)
Will You Be Better Off Than Your Parents?	65	(6)
What Was the Heart Rate of a Brachiosaurus?	71	(7)
Senate Renewal	78	(3)
Bikini Sales and the Logistic Curve	81	(5)
Following the Dow Jones Index	86	(7)
Variation of ``g'' with Latitude	93	(4)
Finding Romeo a Juliet	97	(4)
Multiple Regression Equations	101	(12)
Real Estate Appraisals	102	(5)
And the Winner Is?	107	(6)
Algebraic Models	113	(82)
Introduction	113	(1)
Algebraic Examples	114	(36)
Bombs Versus Schools	114	(7)
Kirchhoff Rules the Electrical World	121	(7)
The Window Washer's Secret	128	(6)
The Science Student's Summer Job Interview	134	(6)
Envelope of Safety	140	(4)
Rainbow County	144	(6)
Integral Examples	150	(16)
The Great Pyramid of Cheops	150	(6)
Noah's Ark	156	(10)
Vector Examples	166	(29)
Vectoria's Mathematical Heritage	166	(7)
Ain't She Sweet	173	(8)
Born Curl Free	181	(5)
Of Flux and Circulation and Coordinates Too	186	(9)
Monte Carlo Methods	195	(66)
Introduction	195	(2)
Random Walks	197	(18)
The Concept	197	(3)
The Soccer Fan's Drunken Walk	200	(5)
Blowin' in the Wind	205	(4)
Flight of Penelope Jitter Bug	209	(3)
That Meandering Perfume Molecule	212	(3)
Monte Carlo Integration	215	(21)
Standard Numerical Integration Algorithms	215	(4)
Monte Carlo Integration	219	(1)
Wait and Buy Later!	220	(4)
Wait and Buy Later! The Sequel	224	(5)
Estimating π	229	(2)
Chariot of Fire and Destruction	231	(5)
Probability Distributions	236	(14)
Of Nuts and Bolts and Hospital Beds Too	236	(7)
The Ice Wines of Rainbow County	243	(7)
Monte Carlo Statistical Distributions	250	(11)
Estimating e	250	(5)
Vapor Deposition	255	(6)
II THE ENTREES	261	(354)
Phase-Plane Portraits	263	(62)
Introduction	263	(1)
Phase-Plane Portraits	263	(5)
Stationary or Singular Points	265	(3)
Linear ODE Models	268	(12)
Tenure Policy at Erehwon University	268	(5)
Vectoria Investigates the RLC Circuit	273	(7)
Nonlinear ODE Models	280	(34)
Classification of Stationary Points	280	(4)
Rabbits and Foxes	284	(8)
The Mona Lisa of Nonlinear Science	292	(9)
Mike Creates a Higher-Order Singular Point	301	(7)
The Guns and Sung of Erehwon	308	(6)
Nonautonomous ODEs	314	(11)
Can an Unstable Spring Find Stability?	314	(3)
The Period Doubling Route to Chaos	317	(8)
Linear ODE Models	325	(72)
Introduction	325	(1)
Solving Linear ODEs with Maple	326	(6)
First-Order ODE Models	332	(13)
There Goes Louie's Alibi	332	(9)
The Water Skier	341	(4)
Second-Order ODE Models	345	(24)
Shrinking the Safety Envelope	345	(5)
Halley's Comet	350	(8)
Frank N. Stein Is Not Heartless	358	(4)
Vectoria Feels the Force and Hits the Bottle	362	(7)
Bessel and Legendre ODE Models	369	(28)
Introduction to Special Functions	369	(7)
The Vibrating Bungee Cord	376	(6)
Wheel of misFortune	382	(9)
The Weedeater	391	(6)
Nonlinear ODE Models	397	(62)
Introduction	397	(1)
First-Order Models	398	(16)
The Nonlinear Diode	398	(4)
The Bad Bird Equation	402	(5)
The Struggle for Existence	407	(7)
Second-Order Models	414	(31)
Pirates of the Caribbean	414	(4)
Oh What Sounds We Hear!	418	(6)
Those Lennard-Jones Vibrational Blues	424	(8)
Golf Is Such an ``Uplifting'' Experience	432	(6)
This Would Be a Great Amusement Park Ride	438	(7)
Limit Cycles	445	(14)
The Bizarre World of the Tunnel Diode Oscillator	445	(7)
Follow That Rabbit	452	(7)
Difference Equation Models	459	(68)
Introduction	459	(1)
Linear Difference Equation Models	460	(1)
First-Order Linear Models	461	(6)
Those Dratted Gnats	461	(3)
Gone Fishing	464	(3)
Second-Order Linear Models	467	(17)
Fibonacci's Adam and Eve Rabbit	467	(4)
How Red Is Your Blood?	471	(2)
Fermi-Pasta-Ulam Is Not a Spaghetti Western	473	(11)
Nonlinear Difference Equation Models	484	(1)
First-Order Nonlinear Models	484	(15)
Competition for Available Resources	484	(8)
The Logistic Map and Cobweb Diagrams	492	(7)
Second-Order Nonlinear Models	499	(14)
The Bouncing Ball Art Gallery	499	(4)
Onset of Chaos: A Model for the Outbreak of War	503	(10)
Numerically Solving ODEs	513	(14)
Finite Difference Approximations to Derivatives	513	(3)
Rabbits and Foxes: The Sequel	516	(5)
Glycolytic Oscillator	521	(6)
Some Analytic Approaches	527	(46)
Introduction	527	(1)
Checking Solutions	527	(14)
The Palace of the Governors	527	(5)
Play It, Sam	532	(4)
The Three-Piece String	536	(5)
Calculus of Variations	541	(18)
Dress Design, The Erehwonese Way	541	(7)
Queen Dido's Problem	548	(4)
The Human Fly Plans His Escape Route	552	(7)
Fourier Series	559	(14)
Hi C Is Not Always a Drink	562	(3)
Play It, Sam: A New Perspective	565	(4)
Vectoria Sums a Series	569	(4)
Fractal Patterns	573	(42)
Introduction	573	(1)
Difference Equation Patterns	574	(27)
Wallpaper for the Mind	574	(2)
Sierpinski's Fractal Gasket	576	(7)
Barnsley's Fern	583	(5)
Douady's Rabbit and Other Fauna and Flora	588	(4)
The Rings of Saturn	592	(9)
ODE Patterns	601	(7)
The Butterfly Attractor	601	(5)
Rossler's Strange Attractor	606	(2)
Cellular Automata Patterns	608	(7)
A Navaho Rug Design	608	(3)
The One Out of Eight Rule	611	(4)
III THE DESSERTS	615	(130)
Diagnostic Tools for Nonlinear Dynamics	617	(24)
Introduction	617	(1)
The Poincare Section	617	(5)
The Concept	617	(1)
A Rattler Signals Chaos	618	(4)
The Power Spectrum	622	(6)
The Concept	622	(2)
The Rattler Returns	624	(4)
The Bifurcation Diagram	628	(4)
The Concept	628	(1)
Pitchforks and Other Bifurcations	629	(3)
The Lyapunov Exponent	632	(3)
The Concept	632	(1)
Mr. Lyapunov Agrees	633	(2)
Reconstructing an Attractor	635	(6)
The Concept	635	(1)
Chaos Versus Noise	636	(5)
Linear PDE Models	641	(60)
Introduction	641	(6)
The Linear PDEs of Mathematical Physics	641	(2)
Separation of Variables	643	(4)
Diffusion and Laplace's Equation Models	647	(25)
Freeing Excalibur	647	(4)
Aussie Barbecue	651	(4)
Erehwon Institute of Technology	655	(4)
Hugo and the Atomic Bomb	659	(7)
Hugo Prepares for his Job Interview	666	(6)
Wave Equation Models	672	(13)
Vectoria Encounters Simon Legree	672	(4)
Homer's Jiggle Test	676	(5)
Vectoria's Second Problem	681	(4)
Semi-Infinite and Infinite Domains	685	(16)
Vectoria's Third Problem	686	(2)
Assignment Complete!	688	(3)
Radioactive Contamination	691	(5)
``Play It, Sam'' Revisited	696	(5)
Nonlinear PDE Models: Soliton Solutions	701	(22)
Introduction	701	(1)
Solitary Waves	702	(2)
The Graphical Hunt for Solitons	704	(11)
Of Kinks and Antikinks	704	(4)
In Search of Bright Solitons	708	(4)
Can Three Solitary Waves Live Together?	712	(3)
Analytic Soliton Solutions	715	(8)
Follow That Wave!	715	(4)
Looking for a Kinky Solution	719	(4)
Simulating PDE Models	723	(22)
Introduction	723	(1)
Diffusion and Wave Equation Models	724	(10)
Freeing Excalibur the Numerical Way	724	(4)
Vectoria Secret	728	(2)
Enjoy the Klein-Gordon Vibes	730	(4)
Solition Collisions	734	(11)
To Be or Not to Be a Soliton	734	(4)
Are Diamonds a Kink's Best Friend?	738	(7)
Epilogue	745	(2)
Bibliography	747	(6)
Index	753