Exploring Topics in Higher Mathematics

with Maple

This course is not a content driven course where you ingest a large amount of material because someone thinks it is good for you, although you are not always sure what its application is. This course is an attempt to give you a peek into the real world of math where one can pursue some interesting mathematical question and have fun with it, that is, if you think it is possible to have fun with math, which might have some likelihood of being true since you were somehow attracted to math in the first place. We will use a book to explore 5 areas of mathematics, seeing how we can get interesting results that are only possible with a computer algebra system. No prior expertise in Maple is required.

Did you really learn something useful in Linear Algebra? Advanced Calculus? Did you use Maple officially in either of those courses? This course will show you how essential a computer algebra system is to interesting problems in both of these areas and more.

This course was given as a
writing enriched
course MAT5930 in Spring 2000 and as MAT5920 (an applied math elective rather
than just an elective) in Spring 2004.

[The students really enjoyed this 00S experience: check out the CATS numbers
on NOVASIS under Robert Jantzen for
this course and semester. Same for 04S.]

- Spring 2000 Syllabus

[follow up clarification (read "motivational push from bob")]

- Spring 2004 Preregistration Hype [writing
intensive]

- Fall 2008 Hype, to run as an elective,

not officially writing enriched (yet, do we need to worry about this?)

No previous knowledge of Maple is necessary although it would not hurt to have a little familiarity with the software. This is not a course in Maple, but in using Maple to explore some higher mathematics, showing the strength of a computer algebra system in extending one's mathematical reach. Mathematics is primary, the tool is secondary. The necessary software code is supplied via template worksheets and instructor consulting.

This course is not a content driven course where you ingest a large amount of material because someone thinks it is good for you, although you are not always sure what its application is. This course is an attempt to give you a peek into the real world of math where one can pursue some interesting mathematical question and have fun with it, that is, if you think it is possible to have fun with math, which might have some likelihood of being true since you were somehow attracted to math in the first place.

The intent of the authors of our text (see below) is to give some structure to some areas of mathematics where technology can help you explore questions that could be interesting, that is, interesting depending on your particular mathematical prejudices of course.

This course will be run through our official Villanova home page sites. We will set up a class folder on each of our web sites linked to our home pages and use Microsoft Front Page to manage and publish our worksheets on that space. [We will learn how to do this in the class.] We will work through each chapter, one at a time, absorbing the topic setup materials, doing some routine problems associated with that set up, and then branching out to some explorative type problems. The writing enriched (whether official or not) part of the course (if you need this, prompt bob to apply for this status) will mean that we want to have nicely documented worksheets with mathematical exposition explaining what we are doing, with feedback from the instructor to improve the presentation. If this course acquires writing intensive status (if you need this, prompt bob to apply for this status), technically we need the equivalent of 20 pages of prose, which the student has had the opportunity to revise with instructor feedback. Whatever the status, one of the goals of this course is to improve the student's ability to do technical writing, taking advantage of Maple's typesetting features.

Discovering Mathematics with Maple, by R.J. Stroeker and J.F. Kaashoek (Birkhauser, Boston, 1999)

- From the publisher:
- This book introduces Maple in a rather special way; namely, it shows how a

computer algebra system most naturally fits into a mathematically oriented

work environment. The intended audience is the mixed group of

mathematicians, econometricians, engineers and other scientists, principally

those who in their chosen discipline need to work with mathematical ideas and

techniques.

Unlike other introduction to computer algebra system, it instructs the reader

about the use of Maple by posing problems of a strictly mathematical nature.

The book is intended not as a mere introduction to Maple, but also to show

how Maple — and other computer algebra systems — may enrich our

knowledge and deepen our understanding of mathematics. In that respect it

focuses on mathematics rather than on Maple, and in doing this it tries to

convince scientists who are not naturally inclined to incorporate the computer

into their daily work routine that a computer algebra system may very well

enhance their theoretical work.

**Table of Contents**

- A Tour of Maple V

[**tour**] - Functions and Sequences

[**func**] - Matrices and Vectors

[**matr**] - Counting and Summation

[**count**] - Derivative and Integral

[**deriv**] - Vector Spaces and Linear Mappings

[**vect**]

**Appendices**

A. Exercises in Experimentation

B. Hints and Answers

C. Quick Guide to Selected Maple Commands

The book explores 5 areas of mathematics, first introducing some Maple
commands

that help mechanize some operations, and then takes off with it to do something

interesting in that area of mathematics where a computer algebra system can

help do things we could not do without it at our level of expertise. The
physical

book contains a bit more exposition than the 5 worksheets for each topic.

Most of the book is on-line here, updated to
Maple 12 Standard worksheets from the original Maple 5 on the book CD, with the
**linalg** package updated to the **LinearAlgebra** package.

Each chapter X (with an abbreviation string given in brackets above) has an
introductory file

**<string>sX.mw**

and a pair of worksheet files **
<string>wXa.mw**,

that explore the topic

correlated with a pair of assignment files

giving problems to be solved related to the guided worksheet files.

Some introductory text is only to be found in the paper textbook, as is the collection

of unguided problems at the end of each chapter.

The first chapter takes a tour of Maple but then immediately does some
interesting

things students in the calculus sequence who used Maple would never have done,

including a bit of Maple programming.

The second chapter investigates some strange functions that only a computer
algebra system

allows us to study, also involving infinite sequences.

The third chapter talks about generalized inverses of matrices to step beyond
elementary

linear algebra and examines a linear filter on data sets together with least
squares fitting.

The fourth chapter uses the binomial series to do some interesting things
with

symbolic summation and testing for randomness.

The fifth chapter takes on analysis (advanced calculus!) and the crazy
Riemann

nondifferentiable function.

The last chapter examines diagonalization of matrices and the singular value
decomposition,

which it uses to do an elementary image compression.

The first appendix has some selected answers and hints.

The second appendix is the target: once guided through these 5 topics and a list of problems for each topic, some problems are presented for the students to do on their own.

We will work on these delivering self-contained reports which document the problem statement and solution.

Each student who wishes a grade higher than A- based on successful completion of the textbook workload must develop some topic of interest to his/herself of sufficient sophistication which requires a computer algebra system to investigate. For more details see:

*25-aug**-2008*:
bob jantzen