Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

*(c) Copyright 1994,1995,1996, 2000 by Evans M. Harrell II and James V. Herod. All rights reserved.

version of 9 January 2000

This is a WWW textbook written by Evans M. Harrell II and James V. Herod, both of Georgia Tech. In case it should move from this location someday, a link will be maintained at

http://www.math.gatech.edu/~harrell/
The book is targeted at engineering students who have had two years of calculus, introductory linear algebra, and introductory ordinary differential equations. It has been used at Georgia Tech for a basic introduction to partial differential equations, Fourier series and other orthogonal series, integral equations, and linear operator theory, and our students have given the Web format high marks.

In the past Georgia Tech was on the quarter system, and the text was used for two distinct 10-week courses. The "red syllabus" has been used for Math 4582, which emphasizes Fourier series, other orthogonal series, and the method of separation of variables. It is essentially Chapters I-XII, which were entirely written by Evans Harrell. (More about the red syllabus.)

The "green syllabus" has been used for Math 4348, which emphasizes integral operators and the method of Green functions; it follows Chapters I,II, and XII-XIX. These chapters originated with extensive classnotes by Jim Herod, with additions and Web translation provided by Evans Harrell. Note: If the emphasis in a course is to be integral equations and Green functions, the instructor may wish to begin with a lecture on the Model Problems of Chapter XIII to provide motivation, before Chapter I.

Two recommended 15-week course syllabuses are:

• An extended red syllabus by developing the notion of a Green function, with Chapters XIII and XVI-XVIII.
• The "yellow syllabus," which covers Fourier and orthogonal series and then continues as with the green syllabus, i.e., Chapters I-V (or IV), followed by XIII-XX. Many teachers will prefer to move Chapter XVII, on classification theory, closer to the beginning of this course.
• A variant on the yellow syllabus would be to continue from the red syllabus with Chapter XVII and then the lecture notes on the method of characteristics by Jim Herod. While those notes have not been incorporated into the Web text, a minimally Web formatted version is available as a link from the Table of Contents.

These syllabuses assume little review. If students are less well prepared than we expect ours to be, a semester-long course could be made by reviewing linear algebra or ordinary differential equations before the red or green syllabus. For the green syllabus one of the appendices could be used to review linear algebra. In our experience, this is not feasible in a 10-week course.

We regard the text as suitable for students who are quite comfortable with calculus and are mainly interested in problem solving. For that reason, we do not stress proofs, although we try to give careful statements of theorems and to discuss the technical assumptions. Also, we do not spend much time with material like methods to calculate the integrals arising in Fourier analysis, choosing instead to appeal to software to do some calculations. Currently, this is usually done with Mathematica, although some parts of the text are also available using Maple. It is not at all necessary to have previous experience with mathematical software, since the calculations in the text are self-explanatory. On the other hand, there are links to many more detailed calculations done in Mathematica notebooks, which will be useful for those who are familiar with Mathematica or who are learning it. Many Georgia Tech students have also found Matlab to be useful for calculations like those in this text.

Both we and our students have enjoyed this experiment in Web publishing. While it is still not clear what effect the World Wide Web will have on publishing, we believe that the effect will be very great. Experiments like this are valuable and necessary to learn the potential of the new forms of communication, and to help us reach this potential we welcome your commentary. In particular, if you would like to use the book at another university, we would be quite interested, and are ready to discuss the mechanics of doing this.

By the way, the Web implementation of most of this text, including all of the red syllabus, should work with any browser, including text-only browsers like lynx. We have made an effort to keep the text as widely accessible as possible, although access controls will probably be implemented in the future.

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Why use a Web textbook?

Some may hesitate to use a Web text because they like the convenience of a printed text which can be read without being at a computer or terminal, and for that reason it may be reassuring a nicely formatted hard copy is available. This can be obtained by purchasing disks from Evans Harrell or by downloading files from the Web (at this time files can be downloaded only for the red syllabus). The format of these files is Microsoft Word. (Mac 5.0 or Windows 6.0 on the disk, or rtf files on the Web. Microsoft's rtf format is a text-only format which should be properly interpreted and opened by Word.) Locally at Georgia Tech printed copies are available for photocopying.

This takes care of the only obvious disadvantage to using a Web text, but there are many distinct advantages:

• The Web text is many times less expensive than a printed text, because distribution and marketing are nearly free. This in itself is reason enough to prefer a well-written Web text.
• The Web text is frequently updated and allows the instructor and the student rapid access to the authors in case of serious difficulties. There is no reason to wait for new additions, with their delays and high costs.
• The index and internal links allow rapid cross-referencing of definitions, equations, illustrative exercises, and further material, which allow the text to be read at different levels.
• There are also many external links to useful sites, such as the home pages for Mathematica and Maple. (For entertainment value, some links surreptitiously lead the reader to external sites which are not so directly useful.)
• Finally, links in the text allow a student to perform live calculations with Mathematica while reading the text. If the student has mathematical software on his or her own machine, it is possible to configure set up the latest version of Netscape to launch Mathematica or Maple automatically when appropriate
For an author Web publishing offers several advantages, especially in connection with copyright.

Finally, a word about reading the book on the Web. Web browsers as of this date are not terribly uniform. The book has been prepared with a word processor and has been viewed with Netscape 3.0 and later. We went to some effort to make the interface somewhat universal, especially by inserting text alternatives to the graphics, for those with non-graphic Web browsers. Much of the book has been viewed with lynx, but not all. Please inform the authors of any problems you have viewing the book.

Good luck and happy browsing!

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