Computer Algebra Recipes for Mathematical Physics

by Enns, Richard H.

• ISBN13: 9780817632236
• ISBN10: 0817632239
• Edition: CD
• Format: Paperback
• Copyright: 2004-12-01
• Publisher: Birkhauser

Summary

Over two hundred novel and innovative computer algebra worksheets or "recipes" will enable readers in engineering, physics, and mathematics to easily and rapidly solve and explore most problems they encounter in their mathematical physics studies. While the aim of this text is to illustrate applications, a brief synopsis of the fundamentals for each topic is presented, the topics being organized to correlate with those found in traditional mathematical physics texts. The recipes are presented in the form of stories and anecdotes, a pedagogical approach that makes a mathematically challenging subject easier and more fun to learn. Key features: * Uses the MAPLE computer algebra system to allow the reader to easily and quickly change the mathematical models and the parameters and then generate new answers * No prior knowledge of MAPLE is assumed; the relevant MAPLE commands are introduced on a need-to-know basis * All recipes are contained on a CD-ROM provided with the text * All MAPLE commands are indexed for easy reference * A classroom-tested story/anecdote format is used, accompanied with amusing or thought-provoking quotations * Study problems, which are presented as Supplementary Recipes, are fully solved and annotated and also provided on the CD-ROM This is a self-contained and standalone text, similar in style and format to Computer Algebra Recipes: A Gourmet's Guide to Mathematical Models of Science (ISBN 0-387-95148-2), Springer New York 2001 and Computer Algebra Recipes for Classical Mechanics (ISBN 0-8176-4291-9), Birkhäuser 2003. Computer Algebra Recipes for Mathematical Physics may be used in the classroom, for self-study, as a reference, or as a text for an online course.

Table of Contents

Preface	v
Introduction	1	(1)
Computer Algebra Systems	1	(1)
Computer Algebra Recipes	2	(1)
Maple Help	3	(1)
Introductory Recipes	4	(6)
A Dangerous Ride?	4	(3)
The Patrol Route of Bertie Bumblebee	7	(3)
How to Use this Text	10	(1)
I THE APPETIZERS	11	(114)
Linear ODEs of Physics	13	(46)
Linear ODEs with Constant Coefficients	13	(23)
Dazzling Dsolve Debuts	14	(4)
The Tale of the Turbulent Tail	18	(3)
This Bar Doesn't Serve Drinks	21	(4)
Shake, Rattle, and Roll	25	(2)
``Resonances'', A Recipe by I. M. Curious	27	(5)
Mr. Dirac's Famous Function	32	(4)
Linear ODEs with Variable Coefficients	36	(18)
Introducing the Sturm-Liouville Family	36	(6)
Onset of Bending in a Vertical Antenna	42	(2)
The Quantum Oscillator	44	(4)
Going Green, the Mathematician's Way	48	(3)
In Search of a More Stable Existence	51	(3)
Supplementary Recipes	54	(5)
Newton's Law of Cooling	54	(1)
Charging a Capacitor	55	(1)
Radioactive Chain	55	(1)
Newton and Stokes Join Forces	55	(1)
Exploring the RLC Series Circuit	55	(1)
The Whirling Bar Revisited	56	(1)
Driven Coupled Oscillators	56	(1)
Some Properties of the Delta Function	56	(1)
A Green Function	56	(1)
A Potpourri of General Solutions	57	(1)
Chebyshev Polynomials	57	(1)
The Growing Pendulum	57	(1)
Another Green Function	58	(1)
Going Green, Once Again	58	(1)
Applications of Series	59	(28)
Taylor Series	59	(6)
Polynomial Approximations	59	(2)
Finite Difference Approximations	61	(4)
Series Solutions of LODEs	65	(7)
Jennifer Renews an Old Acquaintance	65	(3)
Another Old Acquaintance	68	(4)
Fourier Series	72	(8)
Madeiran Levadas and the Gibb's Phenomenon	74	(2)
Sine or Cosine Series?	76	(2)
How Sweet This Is!	78	(2)
Summing Series	80	(4)
I. M. Curious Sums a Series	80	(2)
Spiegel's Series Problem	82	(2)
Supplementary Recipes	84	(3)
Euler and Bernoulli Numbers	84	(1)
Ms. Curious Approximates an Integral	84	(1)
More Finite Difference Approximations	84	(1)
Series Solution	85	(1)
Chebyshev Polynomials Revisited	85	(1)
A Fourier Series	85	(1)
Fourier Sine Series	85	(1)
Fourier Cosine Series	85	(1)
Legendre Series	86	(1)
Directly Evaluating Series Sums	86	(1)
Another Cosine Series	86	(1)
The Complex Series Trick Again	86	(1)
Vectors and Matrices	87	(38)
Vectors: Cartesian Coordinates	87	(11)
Bobby Blowfly	88	(2)
Hiking in the Southern Chilkotin	90	(4)
Establishing These Identities is Easy	94	(1)
This Task is Not a Chore	95	(3)
Vectors: Curvilinear Coordinates	98	(10)
From Scale Factors to Vector Operators	98	(2)
Vector Operators the Easy Way	100	(2)
These Operators Do Not Have an Identity Crisis	102	(1)
Is This Vector Field Conservative?	103	(2)
The Divergence Theorem	105	(3)
Matrices	108	(14)
Some Matrix Basics	109	(2)
Eigenvalues and Eigenvectors	111	(4)
Diagonalizing a Matrix	115	(2)
Orthogonal and Unitary Matrices	117	(2)
Introducing the Euler Angles	119	(3)
Supplementary Recipes	122	(3)
Bobby Blowfly Seeks a Warmer Clime	122	(1)
Jennifer's Vector Assignment	122	(1)
Another Vector Operator Identity	123	(1)
Another Maple Approach	123	(1)
Conservative or Non-conservative?	123	(1)
Basic Matrix Operations	123	(1)
The Cayley-Hamilton Theorem	123	(1)
Simultaneous Diagonalization	124	(1)
Orthonormal Vectors	124	(1)
Stokes's Theorem	124	(1)
Solving Linear Equation Systems	124	(1)
II THE ENTREES	125	(164)
Linear PDEs of Physics	127	(58)
Three Cheers for the String	128	(13)
Jennifer Finds the General Solution	128	(2)
Daniel Separates Strings: I Separate Variables	130	(3)
Daniel Strikes Again: Mr. Fourier Reappears	133	(2)
The 3-Piece String	135	(2)
Encore?	137	(4)
Beyond the String	141	(17)
Heaviside's Telegraph Equation	141	(2)
Spiegel's Diffusion Problems	143	(3)
Introducing Laplace's Equation	146	(2)
Grandpa's ``Trampoline''	148	(2)
Irma Insect's Isotherm	150	(2)
Daniel Hits Middle C	152	(3)
A Poisson Recipe	155	(3)
Beyond Cartesian Coordinates	158	(20)
Is It Separable?	158	(4)
A Shell Problem, Not a Shell Game	162	(4)
The Little Drummer Boy	166	(3)
The Cannon Ball	169	(2)
Variation on a Split-sphere Potential	171	(3)
Another Poisson Recipe	174	(4)
Supplementary Recipes	178	(7)
General Solutions	178	(1)
Balalaika Blues	179	(1)
Damped Oscillations	179	(1)
Kids Will Be Kids	179	(1)
Energy of a Vibrating String	179	(1)
Vibrations of a Tapered String	180	(1)
Green Function for Forced Vibrations	180	(1)
Plane-wave Propagation in a 5-Piece String	180	(1)
Transverse Vibrations of a Whirling String	180	(1)
Newton Would Think That This Recipe Is Cool	181	(1)
Locomotive on a Bridge	181	(1)
The Temperature Switch	181	(1)
Telegraph Equation Revisited	181	(1)
Another ``Trampoline'' Example	182	(1)
An Electrostatic Poisson Problem	182	(1)
SHE Does Not Want to Separate	182	(1)
WE Can Separate	182	(1)
The Stark Effect	183	(1)
Annular Temperature Distribution	183	(1)
Split-boundary Temperature Problem	183	(1)
Fluid Flow Around a Sphere	183	(1)
Sound of Music?	183	(2)
Complex Variables	185	(36)
Introduction	185	(5)
Jennifer Tests Basics	186	(2)
The Stream Function	188	(2)
Contour Integrals	190	(6)
Jennifer Tests Cauchy's Theorem	191	(2)
Cauchy's Residue Theorem	193	(3)
Definite Integrals	196	(9)
Infinite Limits	196	(2)
Poles on the Contour	198	(3)
An Angular Integral	201	(1)
A Branch Cut	202	(3)
Laurent Expansion	205	(3)
Ms. Curious Meets Mr. Laurent	205	(2)
Converge or Diverge?	207	(1)
Conformal Mapping	208	(10)
Field Around a Semi-infinite Plate	209	(3)
A Clever Transformation	212	(4)
Schwarz-Christoffel Transformation	216	(2)
Supplementary Recipes	218	(3)
Roots	218	(1)
Fluid Flow Around a Cylinder	218	(1)
Constructing f(z)	219	(1)
Analytic or Non-analytic?	219	(1)
A Contour Integral	219	(1)
A Higher-order Pole	219	(1)
Another Angular Integral	219	(1)
A Removable Singularity	219	(1)
Another Contour Integral	219	(1)
Fluid Flow & Electric Field Around a Plate	219	(1)
Another Branch Cut	220	(1)
Laurent Expansion	220	(1)
Capacitor Edge Effects	220	(1)
Integral Transforms	221	(36)
Fourier Transforms	221	(13)
Some Fourier Transform Shapes	223	(2)
A Northern Weenie Roast	225	(2)
Turn Off the Boob Tube and Concentrate	227	(3)
Diffusive Heat Flow	230	(2)
Deja Vu	232	(2)
Laplace Transforms	234	(9)
Jennifer Consults Mr. Spiegel	235	(1)
Jennifer's Heat Diffusion Problem	236	(2)
Daniel Strikes Yet Again: Mr. Laplace Appears	238	(1)
Infinite-medium Green's Function	239	(2)
Our Field of Dreams	241	(2)
Bromwich Integral and Contour Integration	243	(9)
Spiegel's Transform Problem Revisited	245	(1)
Ms. Curious's Branch Point	246	(3)
Cooling That Weenie Rod	249	(3)
Other Transforms	252	(3)
Meet the Hankel Transform	253	(2)
Supplementary Recipes	255	(2)
Verifying the Convolution Theorem	255	(1)
Bandwidth Theorem	255	(1)
Solving an Integral Equation	255	(1)
Verifying Parseval's Theorem	255	(1)
Heat Diffusion in a Copper Rod	255	(1)
Solving Another Integral Equation	255	(1)
Free Vibrations of an Infinite Beam	256	(1)
A Potential Problem	256	(1)
Solving an ODE	256	(1)
Impulsive Force	256	(1)
Bromwich Integral	256	(1)
Branch Point	256	(1)
Calculus of Variations	257	(32)
Euler-Lagrange Equation	257	(10)
Betsy's In A Hurry	257	(5)
Fermat's Principle	262	(2)
Betsy's Other Path	264	(3)
Subsidiary Conditions	267	(5)
Ground State Energy	267	(2)
Erehwon Hydro Line	269	(3)
Lagrange's Equations	272	(7)
Daniel's Chaotic Pendulum	273	(2)
Van Allen Belts	275	(4)
Rayleigh-Ritz Method	279	(7)
I. M. Estimates a Bessel Zero	280	(3)
I. M. Estimates the Ground State Energy	283	(3)
Supplementary Recipes	286	(3)
Geodesic	286	(1)
Laws of Geometrical Optics	286	(1)
Bending of Starlight	286	(1)
Another Refractive Index	286	(1)
Mirage	286	(1)
A Constrained Extremum	287	(1)
Maximum Volume	287	(1)
Eigenvalue Estimate	287	(1)
Surface of Revolution	287	(1)
Dido Wasn't a Dodo	287	(1)
Another Approach to the String Equation	287	(1)
Betsy Bug's Ride	288	(1)
III THE DESSERTS	289	(86)
NLODEs & PDEs of Physics	291	(46)
Nonlinear ODEs: Exact Methods	291	(9)
Jacob Bernoulli and the Nonlinear Diode	291	(3)
The Chase	294	(3)
Not As Hard As It Seems	297	(3)
Nonlinear ODEs: Graphical Methods	300	(9)
Joe and the Van der Pol Scroll	300	(3)
Squid Munch (Slurp?) Herring	303	(6)
Nonlinear ODEs: Approximate Methods	309	(12)
Poisson's Method Isn't Fishy	309	(3)
Lindstedt Saves the Day	312	(4)
Krylov-Bogoliubov Have A Say	316	(2)
A Ritzy Approach	318	(3)
Nonlinear PDEs	321	(12)
John Scott Russell's Chance Interview	321	(3)
There is a Similarity	324	(4)
Creating Something Out Of Nothing	328	(2)
Portrait of a Nerve Impulse	330	(3)
Supplementary Recipes	333	(4)
A Bunch of Bernoulli equations	333	(1)
Introducing the Riccati Equation	333	(1)
Period of the Plane Pendulum	334	(1)
The Child-Langmuir Law	334	(1)
Soft Spring	334	(1)
Gnits vs. Gnots	334	(1)
The Vibrating Eardrum	335	(1)
Van der Pol Transient Growth	335	(1)
Another Ritzy Solution	335	(1)
Portrait of a Dark Solution	335	(1)
Bright Soliton Solution	336	(1)
Numerical Methods	337	(38)
Ordinary Differential Equations	337	(16)
Joe's Problem Revisited	338	(2)
Survival of the Fittest	340	(3)
A Chemical Reaction	343	(3)
Parametric Excitation	346	(2)
A Stiff System	348	(3)
A Strange Attractor	351	(2)
Partial Differential Equations	353	(17)
Steady-State Temperature Distribution	354	(2)
1-Dimensional Heat Flow	356	(3)
Von Neumann Stability Analysis	359	(2)
Sometimes It Pays to be Backwards	361	(2)
Daniel Still Separates, I Now Iterate	363	(2)
Interacting Laser Beams	365	(3)
KdV Solitons	368	(2)
Supplementary Recipes	370	(5)
White Dwarf Equation	370	(1)
Spruce Budworm Infestation	371	(1)
A Math Example	371	(1)
Hermione Hippo	371	(1)
The Oregonator	371	(1)
Lorenz's Butterfly	372	(1)
A Stiff Harmonic Oscillator	372	(1)
Courant Stability Condition	372	(1)
Poisson's Equation	372	(1)
Crank-Nicolson Method	372	(1)
Klein-Gordon Equation	373	(2)
Bibliography	375	(4)
Index	379