# The Mathematics of Soap Films: Explorations with Maple®

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*John Oprea*

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.

#### Readership

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

#### Reviews & Endorsements

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

# Table of Contents

## The Mathematics of Soap Films: Explorations with Maple

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface xi12 free
- Chapter 1. Surface Tension 116 free
- Chapter 2. A Quick Trip through Differential Geometry and Complex Variables 3146
- Chapter 3. The Mathematics of Soap Films 5974
- §3.1. The Connection 5974
- §3.2. The Basics of Minimal Surfaces 6075
- §3.3. Area Minimization and Soap Films 6782
- §3.4. Isothermal Parameters 7287
- §3.5. Harmonic Functions and Minimal Surfaces 7590
- §3.6. The Weierstrass- Enneper Representations 7792
- §3.7. The Gauss Map 86101
- §3.8. Stereographic Projection and the Gauss Map 91106
- §3.9. Creating Minimal Surfaces from Curves 95110
- §3.10. To Be or Not To Be Area Minimizing 105120
- §3.11. Constant Mean Curvature 114129

- Chapter 4. The Calculus of Variations and Shape 121136
- Chapter 5. Maple, Soap Films, and Minimal Surfaces 159174
- § 5.1. Introduction 159174
- §5.2. Fused Bubbles 159174
- §5.3. Capillarity: Inclined Planes 166181
- §5.4. Capillarity: Thin Tubes 172187
- §5.5. Minimal Surfaces of Revolution 175190
- §5.6. The Catenoid versus Two Disks 181196
- §5.7. Some Minimal Surfaces 192207
- §5.8. Enneper's Surface 199214
- §5.9. The Weierstrass–Enneper Representation 207222
- §5.10. Bjorling's Problem 221236
- §5.11. The Euler–Lagrange Equations 225240
- §5.12. The Brachistochrone 236251
- §5.13. Surfaces of Delaunay 243258
- §5.14. The Mylar Balloon 258273

- Bibliography 261276
- Index 265280
- Back Cover Back Cover1282