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The Mathematics of Soap Films: Explorations with Maple®
John Oprea, Cleveland State University, OH

Student Mathematical Library
2000; 266 pp; softcover
Volume: 10
ISBN-10: 0-8218-2118-0
ISBN-13: 978-0-8218-2118-3
List Price: US$36
Institutional Members: US$28.80
All Individuals: US$28.80
Order Code: STML/10
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Nature tries to minimize the surface area of a soap film through the action of surface tension. The process can be understood mathematically by using differential geometry, complex analysis, and the calculus of variations. This book employs ingredients from each of these subjects to tell the mathematical story of soap films.

The text is fully self-contained, bringing together a mixture of types of mathematics along with a bit of the physics that underlies the subject. The development is primarily from first principles, requiring no advanced background material from either mathematics or physics.

Through the Maple® applications, the reader is given tools for creating the shapes that are being studied. Thus, you can "see" a fluid rising up an inclined plane, create minimal surfaces from complex variables data, and investigate the "true" shape of a balloon. Oprea also includes descriptions of experiments and photographs that let you see real soap films on wire frames.

The theory of minimal surfaces is a beautiful subject, which naturally introduces the reader to fascinating, yet accessible, topics in mathematics. Oprea's presentation is rich with examples, explanations, and applications. It would make an excellent text for a senior seminar or for independent study by upper-division mathematics or science majors.

® Waterloo Maple, Inc., Ontario, Canada.


Advanced undergraduates, graduate students, and mathematicians interested in the mathematics of soap films.


"Easy reading and it is a pleasure to follow the mathematics while looking at the corresponding pictures obtained by the use of [the software]."

-- Mathematical Reviews

"This book attempts to fill a long-time gap in the literature, and in important ways, achieves a great success ... a book like Oprea's has been sorely needed ... includes physical and experimental motivation, together with accessible undergraduate mathematics, it could also be called soap bubble mathematics for the masses ...

"[The author] provides no more and no less than is necessary to completely derive the mathematical theory of minimal surfaces. Other strengths of the book include the breadth of topics ... the amount of detail included in worked examples and the general readability. Finally the computer component is an added advantage ... some nicely-developed explorations ...

"I am very enthusiastic about this book! It would make an excellent text for an undergraduate course in minimal surface theory. ... Enough detail is included so that this book would also be suitable for an independent study. The next time I teach undergraduate differential geometry, my plan is to first teach a lead-in course using Oprea's book. This provides students with easy access to soap film mathematics ..."

-- MAA Online

Table of Contents

  • Surface tension
  • A quick trip through differential geometry and complex variables
  • The mathematics of soap films
  • The calculus of variations and shape
  • Maple, soap films and minimal surfaces
  • Bibliography
  • Index
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