Theoretical Methods in the Physical Sciences: An introduction to problem solving using Maple V

296 Pages·1994·7.534 MB·English

by  William E. Baylis (auth.)

Theoretical Methods
in the Physical Sciences
to Bobbye
and to Evy and Kathy
W i l l i am E. B a y l is
Theoretical Methods
in the Physical Sciences
an introduction to problem solving
using Maple V
Springer Science+Business Media, LLC
William E. Baylis
Department of Physics
University of Windsor
Windsor, Ontario
Canada N9B 3P4
Library of Congress Cataloging-in-Publication Data
Baylis, William E. (William Eric), 1939-
Theoretical methods in the physical sciences : an introduction to
problem solving using Maple V / William E. Baylis
p. cm.
Includes index.
ISBN 978-1-4684-7138-0 ISBN 978-1-4612-0275-2 (eBook)
DOI 10.1007/978-1-4612-0275-2
1. Physical sciences-Data processing. 2. Problem solving.
3. Maple (Computer file) I. Title.
Q183.9.B39 1994 94-3473
500.2,078--dc20 CIP
Printed on acid-free paper
© Springer Science+Business Media New York 1994
Originally published by Birkhäuser Boston in 1994
Copyright is not claimed for works of U.S. Government employees.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording,
or otherwise, without prior permission of the copyright owner.
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addressed directly to Springer Science+Business Media, LLC.
ISBN 978-1-4684-7138-0
Typeset by the author.
9 8 7 6 5 4 3 2 1
PREFACE
The advent of relatively inexpensive but powerful computers is af
fecting practically all aspects of our lives, but some of the greatest
influence is being felt in the physical sciences. However, university
curricula and teaching methods have responded somewhat cautiously,
having only recently come to terms with the now omnipresent calcula
tor. While many instructors at first feared that the widespread use of
pocket calculators would lead to generations of students who could not
multiply or perhaps even add, few now seriously lament the disappear
ance of slide rules, logarithm tables, and the often error-bound tedium
that such tools of the trade demand. Time that used to be spent on the
use of logarithm tables and manual square-root extraction can be prof
itably turned to earlier studies of calculus or computer programming.
Now that the calculator has been accepted into the classroom, we face a
computer-software revolution which promises to be considerably more
profound.
Modern textbooks in the physical sciences routinely assume their
readers have access not only to calculators, but often to home or even
mainframe computers as well, and the problems teachers discuss and
assign students can be more complex and often more realistic than in
the days of only pad and pencil computations. As less effort is spent
on numerical computation, more can be devoted to conceptual under
standing and to applications of the increasingly sophisticated mathe
matical methods needed for a real appreciation of recent advances in
the discipline.
While the calculator revolution was spreading through the schools,
researchers in some branches of theoretical physics were developing pro
grams to help them with the manipulation of algebraic and differential
equations. Instead of computing numbers, such programs rearrange
symbols and derive analytical solutions. The programs were initially
bulky behemoths running only on large mainframes or supercomput
ers; they were also balky and awkward to use, and their capabilities
were quite limited. Only dedicated scientists in specialized theoreti
cal fields such as general relativity made much use of them. However,
as personal computers have advanced to become more powerful than
the supercomputers of two decades ago, the software has become faster
vi PREFACE
and more encompassing, and an ever larger community of scientists has
embraced the use of symbolic manipulation and computer algebra.
During the past couple of years, mathematics packages such as
Maple, Mathematica, MathCad, Macsyma and Derive have become so
common for PC-compatible, Macintosh, Atari, and other computers as
well as for most Unix workstations that popular computer magazines,
like PC Magazine, have featured and reviewed them. These programs
generally combine abilities in symbolic manipulation with routines for
numerical approximation and plotting. It seems likely that most stu
dents now entering university programs in the physical sciences will
make some use of such mathematics packages before they receive post
graduate degrees.
The impact of these mathematics packages on university teaching
may well prove far greater than that of pocket calculators. The pack
ages will of course allow a wider variety of more complex, more realistic
problems to be solved, but they will also inevitably change the way
students approach problems. Instead of laboriously plotting data for a
single experiment and fitting it to simple curves, much more data can
be quickly visualized, fit and analyzed. Instead of grinding through
the algebra to determine the coefficients of a power-series solution to
a linear differential equation, the student can let the computer whip
out the results and spend his or her time visualizing the results and
relating them to physical phenomena. Instead of worrying about what
substitution will simplify an integral to standard form so that its so
lution can be recognized, the student can concentrate on learning and
applying new mathematical abstractions.
The introduction of powerful mathematics packages in the under
graduate curriculum will undoubtedly cause disruption and pedagog
ical debate. Just as when pocket calculators became common, there
will again be fears about the loss of facility in mathematical manipu
lation. However, as the new tools become more popular, the need for
such facility will decline, and the talents required of students making
a career in mathematics may also change. The potential gains, includ
ing a greater appreciation of mathematical abstraction and hands-on
experience with a broader selection of mathematical tools, will, once
realized, far outweigh the losses.
The term computer math conjures up visions of number crunching
PREFACE vii
on super-computers and mind-boggling numerical detail. Such aspects
play an important role, but they are not the essence of physical science.
Physical scientists make extensive use of simple models and pictures to
convey concepts that are often more firmly based in geometry than
in analysis. A course introducing theoretical methods in the physical
sciences will naturally emphasize numerical and analytical techniques,
but should not exclude the development of pictorial and geometrical
concepts. Any math package chosen for the course should have superior
graphical capabilities, and these should be used fully to present data
in visual form.
This text grew out of a one-semester course which introduced a
symbolic-mathematics package into the curriculum of the physical sci
ences at the University of Windsor. The objective of the course is not
restricted to training students to use a symbolic math program; while
that should be a practical by-product, the ability to tap artificial intel
ligence cannot replace the development of human wisdom. The most
important skill science students can acquire is to learn science on their
own through reading, deep thought and analysis, and discovery. No one
would claim that this skill is easy to acquire: it requires concentrated
effort on the part of the student, who can use all the help available.
The math package is just one useful tool in guiding students to the new
approaches and new thought processes needed by the physical scientist
to increase her or his understanding of nature.
The text is designed for a one-term course, to be taken in the first
or second year of university study. Since some familiarity with ba
sic kinematics and elementary calculus is assumed, the course may be
most effective after the student has successfully completed at least one
semester of introductory courses in university-level physics and calcu
lus. The reason for the early introduction of the material is to provide
tools that the student can use throughout his or her career. The timing
is critical since it is during the first year that many students decide (or
their professors decide for them) whether or not they were cut out to
be physical scientists. Their previous practice of memorizing equations
and plugging in appropriate numbers, which served many quite well in
high school, is no longer sufficient for academic success at the univer
sity. Now it is more important to relate equations to phenomena and
mathematical abstractions to physical experience; still, this presents
viii PREFACE
a major barrier to large numbers of students. Some of the tools and
practice in this text, from dimensional analysis and order-of-magnitude
estimates to Maple algorithms and geometric algebra, should help.
The course for which this text is designed is not meant to replace,
and in fact is quite distinct from, the traditional third- or fourth-year
course in mathematical physics. For example, although the symbolic
math package learned here can be quite useful in solving differential
equations and in studying special functions, such topics receive only
a bare introduction on these pages. Instead, the text concentrates on
areas where first- and second-year students in the physical sciences are
known to have difficulty, either because of inadequate preparation or
because of an underdeveloped problem-solving capacity.
The text may supplement-but should certainly not replace-the
text in a general physics, astronomy, chemistry, or geology course; it
makes no pretense of encompassing a systematic treatment of the first
year curriculum, and it lacks a laboratory of real physical experimen
tation which is so important for developing an "intuitive feeling" for
the subject. Virtual experiments on a computer can supplement and
extend physical experience, but they can never replace it.
This text is also not designed for a programming course. As an inte
gral part, it uses the Maple package (currently Maple V, Release 3), an
acclaimed leader in symbolic capabilities, developed since 1980 by sci
entists and mathematicians at the University of Waterloo, Ontario, the
ETH in Zurich, and elsewhere. Extensive use is made of the symbolic
manipulation, numerical-approximation, and graphical capabilities of
the package, but only minimal attention is paid to writing new proce
dures. Most of the examples are taken from elementary physics with
some applications to astronomy, chemistry, and geology. The emphasis
on physics reflects its role as the most fundamental and probably the
most difficult of the natural sciences.
Geometrical concepts are emphasized in chapters 4 and 9 on vectors.
Chapter 4 presents traditional vector manipulations of addition, scalar
multiplication, and dot and cross products. Although vector compo
nents, basis vectors and reference frames are discussed, the stress is
more on the concept of vectors as single entities. The relative position
and motion of an object with respect to an observer, or of two objects
with respect to one another, can be treated directly with vectors with-
PREFACE ix
out the explicit introduction of components. Chapter 9 extends the
vector concept to what is known as the geometric algebra of vectors
and their products. It thus opens the rich potential of Clifford-algebra
techniques to the reader.
It is assumed that students have access to the full Maple V package,
Release 2 or higher, either on an independent PC or through a network
connection, but most of the work discussed in these notes can be ade
quately carried out with the student edition of Maple V (Release 2 or
higher), as well as on Unix or Apple Macintosh platforms. No prior
knowledge of Maple or symbolic processing is needed, and indeed, this
text can serve as an introduction to many features of Maple needed by
physical scientists.
However, the present text makes no pretext of being a complete
guide to Maple, and most serious students and researchers will want
further information. A recommended source of additional information
to Maple V, Release 2, is A. Heck, Introduction to Maple (Springer
Verlag 1993). The Maple V Language Reference Manual by B. W.
Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monogan,
and S. M. Watt (Springer-Verlag 1991) is also a useful reference. The
Maple V Library Reference Manual by Char et al. (Springer-Verlag
1991) is an optional reference of Maple procedures, although most of
the information it contains is also available on-line through the Maple
Help facility.
A 3~-inch diskette has been packaged with the text for use with
computers running Windows 3.1 or later. In addition to the Maple
worksheets for each chapter, discussed below, it holds files for units,
conversion factors and physical constants (Chapter 1), for properties
of the chemical isotopes (Chapter 1), and for making calculations with
the Pauli algebra (Chapter 9). Since the files are in seven-bit ASCII, it
should be possible to transfer them and use them with Maple on other
computer platforms. It is strongly recommended that the instructor
supplement Chapter 8 on complex numbers with a copy of the freeware
fractal program FRACTINT (for DOS or MS-Windows) or XFRACT
(for Unix/X-windows). The program rapidly calculates and displays
stunningly beautiful fractal drawings, explains the history and under
lying mathematical algorithms, and allows users wide flexibility in the
design and display of the images. Condensed files containing versions
x PREFACE
of the program may be downloaded from the Graphics Developers+
Forum (GO GRAPHDEV) of CompuServe,l and recent versions are
also available on many bulletin boards and a number of FTP sites on
the Internet, for example on SIMTEL20 and its mirror sites. Diskettes
can also be ordered at nominal cost from shareware/freeware library
services. 2
About half of the sections of the current text contain examples and
exercises to be tried on a computer. The headings of these sections
begin with an asterisk (*). Much of this material, together with further
exercises, is also available in the Maple worksheets which are stored as
files on the diskette packaged with this text. There is one worksheet
for each chapter, and although the text can be read without them,
the worksheets do ease the students' introduction to Maple. Readers
whose principal motive is to gain expertise on Maple may want to spend
most of their time with the worksheets and use the text primarily as a
reference. Most students, however, will probably want to progress one
chapter at a time, first reading the text and then working through the
worksheet on a computer.
Although many students will have some familiarity with computers,
none is assumed, and a brief overview of the operation of stand-alone
or networked PC's and the use of floppy disks is given in chapter 1 of
these notes. Readers who are new to Windows may find it beneficial to
go through the Windows tutorial so that they know how to manipulate
windows, move through menus and select options, and cut or copy text
with some combination of the mouse (or trackball) and the keyboard.
No programming knowledge is needed.
It should be understood that reading in the physical sciences is very
different from reading a novel. To be effective, it must not be a passive
lCompuServe is a registered trademark of CompuServe Incorporated, Columbus,
Ohio. The file FRAINT.EXE is a self-expanding file which contains FRACTINT
for MS-DOS systems, whereas the files WINFRA.ZIP and XFRACT.ZIP contain
zipped versions of FRACTINT for MS-Windows and X-windows, respectively. All
three files are located in the GRAPHDEV Library named "Fractal Sources." The
current version number 18.2.
2For example, they are available from the Public (Software) Library, P.O. Box
35705, Houston, TX 77235-5705, USA. Phone [phone] or (in the USA) 800-
242-4775. Ask for item #9113.