Mathematics for Physical Science and Engineering: Symbolic Computing Applications in Maple and Mathematica

780 Pages·2014·8.8 MB·English

by  Frank E. Harris

Mathematics for Physical Science
and Engineering
Mathematics for Physical Science
and Engineering
Symbolic Computing Applications in Maple

and Mathematica
Frank E. Harris
University of Utah, Salt Lake City, UT
and University of Florida, Gainesville, FL
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PREFACE
This text is designed to provide an understanding of the mathematical concepts that
form a basis for our understanding of physics and engineering, and to introduce stu-
dents to the symbolic computation tools that have become essential for the use of that
mathematics. The book is at a level of sophistication and rigor appropriate to students
in the latter part of a standard undergraduate curriculum in physics or engineering. It
should be an easy read for well-prepared graduate students or professionals whose
main need is for a good knowledge of symbolic computing.
There are many books that contain the mathematical topics that are covered in the pres-
ent text, but this author knows of no others that tie that material in an effective and com-
prehensive way to symbolic computation. Such a connection is of importance, in part
because understanding is enhanced by using symbolic techniques to apply the math-
ematics and visualize its results. In addition, many important mathematical methods
become cumbersome when applied to real-world problems, and symbolic computa-
tion methods are often superior to the purely numerical approaches that until recently
dominated in practical applications.
Today’s physical science and engineering students are adept in using computers for
information retrieval and some are skilled in digital computer programming, but too
many have little or no experience in symbolic computing. This lack of experience is par-
ticularly unfortunate because symbolic computing can aid greatly in the visualization
of concepts, can eliminate drudgery (and errors) from otherwise complicated computa-
tions, and can even facilitate studies of types that were previously completely impracti-
cal. But despite the rapid advances in symbolic computing and the increasing extent of
its use in the physical and engineering sciences, very little of the instructional material
in this area is designed for students of advanced mathematical methods in science and
engineering.
In determining how best to incorporate symbolic computation into a presenta-
tion of the mathematics, the issue of the choice of symbolic language immediately
xii PREFACE
arises. One could take the view that a noncommercial open-access language should
be chosen, but no widely accepted such language presently exists. The alterna-
tive of discussing symbolic computation in an artificial “pseudocode” (a device
sometimes used to present numerical computational procedures) does not provide
students with the necessary practice or experience. It seemed inevitable that one
must choose among the two symbolic languages that presently dominate, maple
and mathematica. Each of these languages has its passionate defenders and equally
vocal critics, and each has its individual strengths and weaknesses. Moreover, as
a practical matter the choice of language depends upon the language that is actu-
ally available to the student, and that is often determined by the instructor or the
educational institution.
This book approaches the language dilemma by developing both languages in paral-
lel, thereby giving both the students and their instructor the opportunity to make lan-
guage choices that fit their current instructional situation. This approach therefore has
features that extend its value to a large population of practicing physicists and engi-
neers: it enables those fluent in one of the two languages rapidly to gain proficiency
in the other. In addition, the comparative discussion of the two languages helps to
identify areas where one has advantages over the other.
It is important to realize that skill in symbolic computing cannot be a sub-
stitute for an understanding of the related mathematics, and the exposition
of this text starts each topic with a discussion of the mathematics involved.
To reinforce this point, the Exercises include many that focus on the analytic proper-
ties of the quantities involved, and not just on using computers to provide answers.
We have in addition placed some emphasis on the use of symbolic computation to
explore the content hidden in the underlying mathematics, and have also encouraged
students to use symbolic methods to check results obtained by other means. In some
areas (e.g., matrix eigenvalue problems, evaluation of inverse Laplace transforms)
symbolic computation permits the exploration of a much wider collection of practi-
cal examples than would otherwise be possible.
Because this is not a reference book or an advanced treatise, we have omitted proofs
of some theorems when we believe they are not essential to a basic understanding of
the issues involved. This decision enables us to create a text that falls within the time
constraints of instructional programs while still including a wide range of useful top-
ics. Readers desiring treatment of topics at greater depth may find what they seek in
the more comprehensive work by Arfken, Weber, and Harris, Mathematical Methods for
Physicists, 7th edition (New York: Academic Press, 2013).
PREFACE xiii
While a different choice might have been more mathematically elegant, the text
has been constructed in a manner that does not require the use of complex vari-
able theory (in particular, the residue theorem and contour integration) in the
discussion of special functions, especially the gamma function. We do, however,
present early in the book and use throughout it the algebra of complex functions,
including Euler’s formula for eiθ, de Moivre’s theorem, multiple-valued functions,
and formulas for inverse trigonometric functions.
There is one technical issue which arises so often that it should be addressed here:
The coding that illustrates symbolic computing methods has been written assuming
that the user has not defined quantities or procedures that can interfere with the prob-
lem under discussion. The presence of potentially interfering definitions is probably
the most frequent source of puzzling errors in symbolic computation, and both the
instructor and the student need to be cognizant of the need to make computations
in a “clean” symbolic environment, obtained either by opening a new computing
session or by being vigilant in undefining quantities that are no longer needed or
relevant.
In general, this text has been prepared in a way such that the background needed for
each chapter has been presented in earlier chapters. However, one can omit or delay
the study of Chapter 8 (Tensor Analysis), Chapter 12 (Fourier Series), Chapter 13
(Integral Transforms), Chapter 16 (Calculus of Variations), or Chapter 17 (Complex
Variable Theory) without much impact on the remainder of the book. On the other
hand, it is recommended not to be dismissive of the material in the Appendices. Much
of that material is there, not because it is parenthetical, unimportant, or too special-
ized, but because it also applies in contexts far removed from that which represented
its first use. That material deserves detailed discussion at appropriate points in an
instructional program.
An examination of the introductory chapter will reveal that the discussion of sym-
bolic computing assumes that the student has already gotten the symbolic system
running on the computer to be used. An instructor may need to supplement this text
by providing instructions as to how to access the symbolic system available to his/her
students, how to obtain computer output, and how to preserve and reuse workspaces.
The author has benefited from the advice and help of a number of people. Professor
Nelson H. F. Beebe of the University of Utah provided invaluable counsel regarding
symbolic computing languages, in which he is a recognized expert. At Elsevier, sub-
stantial assistance was provided at all stages of the publication process by Editorial
Project Manager Jessica Vaughan. The author also gratefully acknowledges the support
xiv PREFACE
and encouragement of his friend and partner Sharon Carlson. Without her, he might
not have had the energy and sense of purpose needed to bring this project to a timely
fruition.
Additional Information
Qualified instructors: See www.textbooks.elsevier.com for the in-depth Instructor’s
Guide
Chapter 1
COMPUTERS, SCIENCE,
AND ENGINEERING
Digital computers are revolutionizing the ways in which scientists and engineers
can solve numerical problems of practical interest. This fact is well appreciated
both in academic institutions and in governmental and industrial laboratories, and
over the past half-century research and development organizations have invested
tremendous amounts of money and computer personnel into ever larger and faster
digital computing environments. These resources permit the solution of problems for
whichtherelevantscienceisknown,butwhicharetoocomplicatedtosolveeitherby
formal analysis or by hand computation. Typical examples of problems whose study
has been made possible by digital computation include:
• Engineering problems such as the design of aircraft (requiring the application
of compressible fluid mechanics [air] to complicated geometric shapes),
• Physics instrumentation problems such as the design of nuclear reactors (re-
quiring analysis of heat flow and neutron transport through diverse materials
in difficult geometries as well as modeling the nuclear reactions involved),
• Modeling of chemical reactions (involving solution of the Schro¨dinger equation
for the quantum states of the relevant species and study of the time-dependent
dynamics of the reactants and products),
• Modeling of the physical properties of materials (involving the simulation of
processes such as cracking or fragmentation, time evolution of defects or grain
boundaries, and other changes in morphology),
• Prediction of electronic or optical properties in complex systems (involving
studiesoftheeffectsofbasiccompositionandimpurities,nonlinearresponseto
incoming signals, and other effects),
• Analysis of the time evolution of the spatial configuration of a large molecular
system (examples include biological systems such as the folding processes in
proteins, but also the formation or destruction of inanimate structures such as
carbon cages or nanotubes).
What these processes have in common is that they involve numerical computations
thataresolengthyorcomplicatedthattheycannotbedonewithoutusingmachinery
to perform the computations and store their results.
MathematicsforPhysicalScienceandEngineering.
http://dx.doi.org/10.1016/B978-0-12-801000-6.00001-8
1
©2014ElsevierInc. Allrightsreserved.
2 CHAPTER 1. COMPUTERS, SCIENCE, AND ENGINEERING
More recently, accelerating into prominence within the past 25 years, computer
software has emerged for carrying out mathematical analyses that are not basically
numerical.Softwareforthispurpose,sometimesreferredtoascomputer algebraor
symbolic computing, can assist human investigators in exploring the application
of more advanced mathematics. Again, we illustrate with a short list:
• Conversion of the form of algebraic expressions. These conversions include ex-
pansions of products of polynomials, solution of algebraic equations or systems
of such equations, and insertion of the explicit form of defined quantities,
• Expansions into power series, trigonometric series, asymptotic series, or other
types of expansions,
• Integration or differentiation of algebraic forms,
• Solution of differential or integral equations,
• Evaluation and manipulation of special functions,
• Plotting and visualization of functions of interest,
• Reduction of expressions to numerical form at user-chosen levels of numerical
precision.
These capabilities are of great importance for both the developers and the users of
mathematical methods in science and engineering. Symbolic methods are inherently
slower computationally than the purely numerical methods that they supplement,
but they are greatly superior for obtaining qualitative understanding of functions
and mathematical processes. They also enable the processing of algebraic operations
that are too cumbersome or error-prone to be carried out by hand.
It is important to realize that the availability of symbolic computing cannot be
a substitute for a knowledge of basic mathematical analysis. A computer algebra
program only provides answers to the specific mathematical questions asked by the
user. At least for the foreseeable future, the choice of mathematical methods for a
scienceorengineeringproblemremainsthejobofthescientistorengineer.Itturnsout
that this task usually cannot even be turned over to a mathematician. A knowledge
of the essence of the scientific problem is ordinarily a prerequisite to its successful
solution.
This book is not primarily about numerical methods for solving problems in
applied mathematics, as that area has reached great sophistication and deserves
study as a topic of its own. Rather, our objective here is to introduce the reader
tosymboliccomputationtoolsthatwillbeextremelyusefulingainingunderstanding
of the mathematical processes and functions that we introduce, while at the same
time providing access to standard methods for carrying out numerical calculations.
The application of advanced mathematical methods to science and engineering is
greatly facilitated by this multifaceted approach, and such an approach is becoming
anessentialpartoftheeducationofscientistsandengineers.Accordingly,themission
of this introductory chapter is to introduce and develop familiarity with some ideas
relevant to symbolic computation, thereby preparing the reader for the use of such
systems in the remainder of the book and beyond.
1.1 COMPUTING: HISTORICAL NOTE
AlthoughmechanicalmachinestododigitalcomputationdatebackasfarasBabbage
(1791–1871) and continued to develop until the mid-20th century (with prominent
manufacturers then being Friden, Monroe, and others), one of the key advances that