Table Of ContentThe Maple Handbook
Maple V Release 3
Darren Redfern
The
e
a----
Handbook
Maple V Release 3
With 16 illustrations
Springer-Verlag
New York Berlin Heidelberg London
Paris Tokyo HongKong Barcelona Budapest
Darren Redfern
Practical Approach
P.O. Box 1007
Stratford, ON N5A 6W4
Canada
Library of Congress Cataloging-in-Publication Data
Redfern, Darren.
The Maple handbook / Darren Redfern. --2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN-I3: 978-0-387-9433 I -2 e-ISBN-I3: 978-1 -4684-0229-2
DOl: 10. I 007/978-I -4684-0229-2
1. Maple (Computer file) 2. Mathematics --Data processing.
1. Title.
QA76.95.R43 1994
510'.285'53 --dc20 93-25796
Maple is a registered trademark of Waterloo Maple Software.
Printed on acid-free paper.
@1993, 1994 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part
without the written permission of the publisher (Springer-Verlag New York Inc., 175
Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection
with reviews or scholarly analysis. Use in connection with any form of information
storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed is forbidden.
The use of descriptive names, trade names, trademarks, etc., in this publication,
even if the former are not especially identified, is not to be taken as a sign that
such names, as understood by the Trade Marks and Merchandise Marks Act, may
accordingly be used freely by anyone.
Production managed by Karen Phillips; manufacturing supervised by Vincent Scelta.
Photocomposed copy prepared from the author's u\TI¥' file.
987654321
Contents
Introduction ........................................... 1
Getting Started with Maple ........................... 9
Calculus ............................................. 35
Linear Algebra ...................................... 69
Solving Equations .................................. 111
Polynomials and Common Transformations ...... 145
Geometry ........................................... 189
Combinatorics and Graph Theory ................. 231
Number Theory .................................... 265
Statistics ............................................ 299
Standard Functions and Constants ................. 331
Expression Manipulation .......................... 363
Plotting ............................................. 383
Programming and System Commands ............ 415
Miscellaneous ...................................... 461
Index ............................................... 515
Introduction
How to Use This Handbook
The Maple Handbook is a complete reference tool for the Maple
language, and is written for all Maple users, regardless of their dis
cipline or field(s) of interest. All the built-in mathematical, graphic,
and system-based commands available in Maple V Release 3 are
detailed herein.
Please note that The Maple Handbook does not teach about the
mathematics behind Maple commands. If you do not know the
meaning of such concepts as definite integral, identity matrix, or
prime integer, do not expect to learn them here. As well, while
the introductory sections to each chapter taken together do provide
a basic overview of the capabilities of Maple, it is highly recom
mended that you also read a more thorough tutorial such as In
troduction to Maple by Andre Heck or First Leaves: A Tutorial
Introduction to Maple V.
Overall Organization
One of the main premises of The Maple Handbook is that most
Maple users approach the system to solve a particular problem (or
set of problems) in a specific subject area. Therefore, all commands
are organized in logical subsets that reflect these different cate
gories (e.g., calculus, algebra, data manipulation, etc.) and the com
mands within a subset are explained in a similar language, creating
a tool that allows you quick and confident access to the information
necessary to complete the problem you have brought to the system.
This design goes hand in hand with the fact that there are many
Maple commands that behave differently when faced with the vary
ing data types and constructs inherent in different subject areas.
When this is the case, the commands have separate entries in each
relevant subset. A good example of this is the op command (which
2 Introduction
deals with the internal structure of data objects). The results of the
op command are sufficiently different when dealing with expres
sions, matrices, plots, and procedures to warrant an entry in each
corresponding section, as well as one in the section on manipulat
ing data structures. The overall philosophy is to make you travel as
little as possible to access information necessary to your work.
In addition, because there is much information about Maple that is
very difficult to express on a purely command-by-command basis,
each subject is prefaced with a short introductory section, written
to represent an active Maple session where input, output, text, and
graphics are combined. There is also an introductory session, titled
Getting Started with Maple, which is intended to get those readers
not already familiar with Maple off to a quick start.
Cross Referencing
One of the most important goals of The Maple Handbook is to
provide pointers to appropriate information so that you are able to
solve your problems quickly and efficiently. The Maple Handbook
is rife with valuable references, presented in such a way to be ac
cessible yet not clutter the infoITIlation to which they are attached.
There are three types of references present within The Maple Hand
book. Firstly, each command listing has a See also section that
points you to commands within that section, or in other sections,
that contain related information. Secondly, in case you know the
name of a command but are unsure to which subject area it belongs,
there is a complete alphabetical index of all Maple commands at
the end of the handbook. Finally, most command listings contain
essential page references to supplementary information or illumi
nating examples contained in the official Maple manuals (First
Leaves: A Tutorial Introduction to Maple V, Maple V Language
Reference Manual, Maple V Library Reference Manual).
Individual Command Entries
While the information contained in the entries for each individ
ual command is unique, the format in which the information is
presented is identical across all entries. The following fictitious
example illustrates the various elements that are used throughout
this book.
Introduction 3
acommand(expr, var)
Performs a command on expr with respect to var.
Output: If expr contains only numeric values, an expression se
quence of integers is returned. Otherwise, a list of symbolic values
is returned.
Argument options: (expr. var=a. . b) to limit the computation to
var ranging from a to b. a must be greater than b. • ([exprl •
...• exprn]. [varl •...• yarn]) to perform the calculation on exprl
through exprn with respect to varl through varn , respectively. An
expression sequence of n lists is returned.
Additional information: If var is previously assigned to anything
other than its own name, an error message is returned.
See also: bcommand. cpackage[dcommand].
anothercommand
= = =
FL 12-14 LA 43 LI 386
The command call, acommand(expr. var), gives the command
name as well as its most common type of parameter sequence. The
command name can be of several different forms but always repre
sents how the command can be typed into Maple to be immediately
executed:
acommand-a command that is in the standard library
apackage[acommand]-a command that is in a specialty pack
age. To access these commands they must either be entered in their
long form (as given) or they must follow an appropriate with com
mand. Unfortunately, in Maple V Release 3, there are still a few
packages where the long form (i.e., apackage[acommandJ) does
not provide access to the command. In these cases, the acommand
call must be preceded by with(apackage) or with(apackage.
acommand). The individual command entries always list the long
form of the command name. If that doesn't appear to work, con
sult the command entry for the entire package that precedes the
individual entries.
acommand(aparameter)-a command that has a unique function
when combined with a particular type of parameter (including spe
cial optional arguments).
'acommand/bcommand'-used for specialized second-level
commands that combine the functionality of acommand and bcom
mand. Backquotes ensure that special character I is not translated
as a division sign. While there are many, many such second-level
commands in Maple, very few of them should be used directly by
you and are therefore not detailed in this book.
4 Introduction
The command call's parameter sequence (as well as the parameter
sequences found in othet elements of a command entry) normally
represents placeholders for the actual input you would use when
calling the command. For example, a placeholder of int could be
replaced with the integer 2, 754, or -11. When a parameter se
quence contains an element that appears in italics, for example nu
meric, it means that the word numeric is to be entered as is in the
command, not replaced with some other value. Such elements most
frequently occur with prede:fmed options and input values.
Whenever possible, the expected data type of a parameter is spec
ified with one of the following abbreviations.
A an array data type
boolean true or false
complex a complex value or expression
eqn an equation
expr an expression
exprseq an expression sequence
float, # a floating-point value
filename a file name
fnc a function
fnc(var) a function in variable var
ineq an inequation
int an integer
list, [] a list data type
list[typed a list with elements of type typel
n, m, i, j, posint a positive integer
name a name to be assigned a value
num, a, b a numeric value
M a two-dimensional array, matrix
option one of a set of predefined options
poly a polynomial expression
proc a procedure
pt a point (e.g., [a,b])
rat a rational expression or value
ratpoly a rational polynomial
RootOf a root notation
s, series a series expression
set, {} a set data type
set{typel} a set with elements of type typel
subexpr a subexpression
T a table data structure
V a one-dimensional array, vector
var an unassigned variable
Introduction 5
The above abbreviations deal with parameter types that are encoun
tered across all disciplines covered by The Maple Handbook. There
are also dozens of other data types that are specific to individual ar
eas; these are detailed in the introductions to each chapter. [Note:
when a sequence of parameters is represented with ... , for exam
ple, exprl, ... , exprn, do not confuse this with Maple's ellipsis
operator .. -One is a short-hand representation, while the other is
a language structure.]
Following the command call is a short description of how the com
mand works on the given parameter sequence. This is meant to give
you enough information to use the command in most instances. If
more information is needed, the Argument options and Additional
information sections should be read.
The Output listing gives some idea of what type of output (i.e., what
data types) to expect from the most common calling structure. This
is extremely helpful when you either want to dissect the answer
for further use or correctly include the command within another
command or a procedure.
The Argument options listing provides valid variations to the pa
rameter sequence and brief explanations of their functioning. If any
alternate parameter sequence is of paramount importance, there is
an individual command listing to discuss it. • characters appear
in this section to separate multiple parameter sequences.
The Additional information section lists just that-additional in
formation about the command. This could include, among other
things, special pointers to other command entries, brief descrip
tions of algorithms used to compute the command, or warnings
about dangerous combinations of parameters. • characters ap
pear in this section to distinguish separate items.
The See also section gives pointers to other command listings within
that section (in normal typeface) or in other sections (in italic type
face) which contain related information or work in conjunction
with the initial command. When searching for a command in an
other section, it is best to consult the index at the back of this book
for an exact page location.
The Manual Cross References at the end of each command listing
provide page numbers from the Maple manuals where more infor
mation related to the command is available. FL = First Leaves, LA
= Language Reference Manual, LI = Library Reference Manual.
6 Introduction
Maple's On-line Help System
All versions of Maple come with an exhaustive on-line help facility
containing hundreds of pages of detailed descriptions. These help
pages have been updated during the many years of Maple devel
opment and contain much information that is valuable and much
that is esoteric. On many platforms, there are on-line help topic
browsers that allow you to navigate easily between various help
pages.
The Maple Handbook adds to this facility. By trimming down the
information presented to you, it allows you to save many hours of
needless reading. The on-line help pages for Maple were written by
many different researchers/developers and lack a common voice
and a completely unifying style. By putting all the information in
one voice, with consistent handling of similar situations through
out, The Maple Handbook furthers comprehension.
Please keep in mind that on-line help is always available to you
when you are running Maple. The Maple Handbook is meant to
offer you fast and efficient access to information about Maple
the on-line help pages can be used to broaden that knowledge at
your leisure. To view the on-line help for any Maple command,
simply enter the command name prepended with a question mark
character. For example, the command ?factor displays the on-line
help page for the factor command.
Where to Go for More Information
The range of Maple books, courseware, and third-party applica
tions is constantly growing. Apart from the standard Maple man
uals (First Leaves, etc.) there are books on using Maple in subject
areas from linear algebra to calculus to differential equations; and
there are several more volumes currently being written by authors
in the academic and commercial fields. The next few years will see
an explosion in the number and variety of Maple materials.
In addition, there have been many scholarly papers, reports, and
theses written around the Maple system. For more information on
these or other Maple materials, contact the vendor who sold you
Maple. They have access to much information that will be of inter
est to you.
