使用Mathematica解决科学计算问题

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"Computing with Mathematica - Second Edition" 《计算与Mathematica》是一本专为想要学习如何利用Mathematica解决数学、计算机科学、工程学以及其它科学领域问题的读者设计的书籍。这本书配有一张CD-ROM，旨在教授读者在这些对数学和计算技能至关重要的领域中有效运用Mathematica。 Mathematica是一种强大的符号计算和数值计算软件，由Wolfram Research开发。它广泛用于教育、研究和工业应用中，提供了丰富的数学函数、图形绘制能力、数据分析工具和编程语言功能。通过使用Mathematica，用户可以进行复杂的数学运算，创建交互式应用程序，以及构建和可视化各种科学模型。书中可能涵盖了以下知识点： 1. **Mathematica基础**：包括Mathematica的界面介绍，如何启动和运行程序，以及基本的输入语法和命令。 2. **数学运算**：如代数、微积分、线性代数、概率统计等领域的高级运算，展示了Mathematica在解决这些数学问题时的强大功能。 3. **符号计算**：Mathematica能够处理符号表达式，执行符号积分、微分、求解方程和系统等。 4. **数值计算**：包括数值积分、数值解方程、矩阵运算以及优化算法等内容。 5. **可视化**：Mathematica提供了丰富的图形绘制工具，用户可以创建二维和三维图形，用于数据可视化和理论模型的展示。 6. **编程与脚本**：Mathematica支持编程，可以编写自定义函数，实现复杂逻辑和算法，这在解决实际问题时非常有用。 7. **应用案例**：书中可能会包含多个来自不同学科领域的实例，演示如何使用Mathematica解决具体问题。 8. **数据处理与分析**：Mathematica能处理大量数据，进行统计分析，提供数据挖掘和预测功能。 9. **交互式应用**：利用Mathematica的Notebook环境，用户可以创建交互式的教学材料和应用程序。然而，值得注意的是，该书的有限保修和免责声明指出，出版商及产品相关人员不对产品的性能或使用结果提供任何保证，产品仅按“原样”销售。这意味着购买者需理解软件可能存在缺陷，并且在90天内发现磁盘有缺陷时，可以选择更换磁盘或退款。《计算与Mathematica》是帮助读者掌握这款强大计算工具的宝贵资源，无论是在学术研究还是专业工作中，都能提升在数学和科学计算方面的效率和能力。

Preface xv

In Part V we provide suggestions for the student projects. We include the time table, the

expectations for the written and oral presentation of a project, and the evaluation criteria. There

are 25 student projects.

New to this Edition

At the suggestion of the reviewers of the ﬁrst edition, we changed the order of some topics,

added a few new topics, and adapted others. The most signiﬁcant change from the ﬁrst edition

is that this new edition is based on Mathematica 4.1, which led to the inclusion of material

not available in earlier versions. Speciﬁcally, in Part I we discuss the basic features of the Help

Browser, added a notebook about the features of notebooks, preferences, and hyperlinks, and

added a notebook on basic typesetting and the usage of the built-in palettes. At the end of

Part II we added a notebook that discusses substitution rules, options in built-in functions,

and the deﬁnition of optional arguments. These topics were moved from Part IV in the ﬁrst

edition. We also added an applications notebook that uses substitutions interactively to create

the basic graphics object for the cover of the printed text. In Part III we added more exercises.

Part IV starts with a notebook where we discuss the deﬁnition of options in functions in more

detail than in Part II. The second notebook in Part IV discusses how hyperlinks, buttons, and

palettes are constructed. The packages notebook discusses ﬁles, directories, package ﬁles, and

the initialization mechanism. The last notebook introduces the import and export mechanisms

as well as data and notebook conversions.

Mathematica 4.1 allows for two-dimensional typing and editing with the use of palettes. We

decided to use this feature in text cells for easier readability but not in input cells. A student

using the notebooks is not expected to edit the text in the text cells but is expected to work with

the input cells and make changes as needed. Linear editing in input cells seems less cumbersome

than two-dimensional editing from the palettes.

Course Materials

All materials were developed in Mathematica and exist in electronic form and with identical

content for the Macintosh and Windows platforms. A copy of Mathematica 4.1 is desirable for

work with the notebooks. If version 4.1 is not available, it is possible to use Mathematica 3.1, but

in this case the features that are new in version 4.1 will not be accessible and some of the code

and formatting will not work. The materials make no use of special fonts or special functions

and will run with the version of Mathematica 4.1 as it is shipped from Wolfram Research. A copy

of Mathematica is not included.

CD-ROM Seventeen electronic Mathematica notebooks are the core of the course materials,

the interactive text for the course. The CD-ROM contains these 17 notebooks for version 4.1 of

Mathematica in four folders, Part I–IV. A few of the notebooks require associated notebooks

or folders, which can always be found in the same folder as the notebook itself. All notebooks

can be used with Windows as well as the Macintosh platforms as long as a copy of either

Mathematica 4.1 or higher is running on the computer. The CD-ROM also contains, in the folder

xvi Preface

for Part V, the Project Support Folder. It includes notebooks for some of the suggested student

projects. A copy of Mathematica is not included on the CD-ROM.

Printed Materials The text, Computing with Mathematica, is a printed version of the 17 notebooks

on the CD-ROM, which are the core of the course. It is intended as easily accessible reference

material for the student as well as the instructor. The notebooks are printed in unevaluated form.

The text also contains, in Part V, a printed version of the student projects.

An Instructor’s Guide for Computing with Mathematica contains sample solutions to all the

exercises in the notebooks. Because of the nature of the materials, solutions are not unique and

other (and better) solutions are always possible. The Instructor’s Guide is available in electronic

form but not in printed form. It is not included on the CD-ROM but can be obtained from the

publisher by qualiﬁed instructors.

The Student Project

When we teach this course, the student project is the most important component of the course.

All students are expected to complete a project that they may choose from their major ﬁeld of

study. Students can work individually or in teams of two. The course structure allows about four

to ﬁve weeks for the completion of the project. Students write up their project as a Mathematica

notebook. They give a 20-minute oral presentation explaining their project and demonstrating

their notebook to the entire class. A list of suggestions for student projects is available in Part V.

Each suggested project is introduced with a goal statement for the project, a list of resources

for the project, and a list of prerequisites. Students can choose a project from this list, modify a

suggested project, or they can design their own. We have found that the most successful projects

were those designed by students themselves, because they were able to choose a topic that they

really wanted to work on.

A Note to the Instructor

The exercises in each notebook introduce a variety of mathematical topics that should be familiar

to students with the stated prerequisites for the course. This variety may create a need for

some instructor guidance if the students don’t quite have the mathematical background that is

expected. The instructor may have to provide some of the necessary background information

on topics that come up in the exercises. The text makes no attempt to teach the mathematics that

may be needed in the exercises. With few exceptions, a standard text on Calculus and Analytic

Geometry, a standard text on Linear Algebra, or a standard text on Discrete Mathematical Structures

should be all that is needed. The Instructor’s Guide for Computing with Mathematica contains

suggestions for solutions to all the exercises. Exercises are placed at the end of each notebook

rather than interspersed in the text as one might expect in an interactive medium. This choice

was made to allow for a collection of more challenging exercises based on the material in a

whole notebook instead of having only minor interactive exercises based on small parts of a

notebook.

Preface xvii

A Note to Students

This course assumes that you remember what you learned in calculus, in linear algebra, and in

high school geometry. When you work on the exercises, occasionally you may have to get out a

calculus or linear algebra textbook and revisit some topics that you studied in earlier courses.

A Sample Course

At the University of Michigan-Dearborn a semester allows for approximately 14 weeks of classes

and one week for ﬁnal examinations. The following course outline is based on the syllabus that

has been used for several semesters at the University of Michigan-Dearborn and, with minor

variations, at Eastern Michigan University.

The enrollment in the class is limited to 20 students. Students’ performance was assessed

through homework, a Mathematica proﬁciency examination (midterm exam), and the ﬁnal

project.

Weeks 1 and 2: Part I; do all exercises in the notebooks of Part I. Get proﬁcient with the

use of the campus network, the course management system, electronic communication, and ﬁle

transfer mechanisms for homework assignments.

Weeks 3, 4, and 5: Part II; do all exercises in the notebook Values, Variables, and Assignments.

In the notebook Functions do exercises 1–7 and either 8 or 9. From the notebook Recursive

Deﬁnitions choose a selection of exercises, for example, numbers 1, 2, 3, 5, 8, 9, 12, and 13,

or, alternatively, numbers 1, 2, 3, 4, 6, 7, 10, and 11. From Substitution Rules and Optional

Arguments do exercises 1 and 2.

Weeks 6 and 7: Part III; do at least exercises 1, 2, 3, either 4 or 5, and 6 in List Processing

Functions.

Week 8: Part III; do a selection of exercises in Iteration with Loops, for example, numbers 1,

2, 3, 5, and one from 6, 7, or 8. Students choose a topic for the ﬁnal project during this week.

Week 9: Proﬁciency examination. Students also collect materials for the ﬁnal project and study

the necessary mathematics for their ﬁnal project. Students submit a two-page proposal for the

ﬁnal project with an outline of what they want to accomplish. The proposal also has to contain

a detailed time table for completing the project.

Week 10: Part III, section 3, Modules and Local Variables; do four of the exercises. Students

begin work on the ﬁnal project.

Weeks 11 and 12: Students work on their ﬁnal projects and study sections from Part IV

relevant to their projects.

Week 13: Students continue work on the projects, they also start writing their ﬁnal project

report and prepare for the oral presentation of the ﬁnal project.

Week 14: Project presentations. Students continue writing their ﬁnal project report and

prepare for the oral presentation.

xviii Preface

The instructor should be prepared to give a lot of individual help and guidance during weeks

9, 11, 12, and 13.

The class is scheduled in a departmental teaching computer lab for three hours per week.

During the ﬁrst week students have to attend class to get organized, to understand the course

management system, the campus networks, electronic communication, and ﬁle transfer mech-

anisms for homework assignments. During this time the instructor also gives several short

lectures about the mathematics that is needed to do the homework assignments. Later in the

semester students do not need to come to the departmental teaching computer lab to work on

their assignments. They will typically work on their home computers or in the campus computer

labs and come to the departmental teaching computer lab during the scheduled time slots only

when they have difﬁculty with an assignment. Some students choose always to work in the

departmental teaching computer lab during the scheduled time. The instructor will schedule

several mandatory class days during the semester as needed. The proﬁciency (midterm) exam is

taken in the departmental teaching computer lab, and all students are required to attend when

oral presentations of ﬁnal projects are scheduled.

Homework is submitted electronically, and corrections and homework scores are sent out

electronically as well. At the beginning of each week the instructor sends out a Class Update out-

lining the goals for the week and announcing deadlines. Deadlines for homework are enforced

to keep students on schedule.

Acknowledgments

The ﬁrst edition of this book and CD-ROM, published in 1998, was supported by the National

Science Foundation with Grant No. DUE 9451360. Kevin Burke, then a faculty member at Siena

Heights College, participated in the design of the project and the National Science Founda-

tion grant. Students and faculty at the University of Michigan-Dearborn, Eastern Michigan

University, and Siena Heights College who used preliminary versions and the ﬁrst edition of

this material made many excellent suggestions that led to improvements in the text and the

CD-ROM. The detailed and thorough reviews that were received from Thomas Cusick, SUNY

at Buffalo; Stephen Brick, University of South Alabama; Don Hazlewood, Southwest Texas State;

Anthony Peressini, University of Illinois at Urbana; and Bengt Fornberg, University of Colorado

led to the rearrangement of some topics and the inclusion of new material. It has been a pleasure

to work with the editorial staff at Academic Press. Thanks to all!

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