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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.

John Oprea: Cleveland State University, Cleveland, OH

Cover 1
Title 4
Copyright 5
Contents 8
Preface 12
Chapter 1. Surface Tension 16
§1.1. Introduction 16
§1.2. The Basics of Surface Tension 17
§1.3. Experiments with Soap Films 21
§1.4. The Laplace–Young Equation 28
§1.5. Plateau's Rules for Soap Films and Consequences 30
§1.6. A Sampling of Capillary Action 38
§1.7. Final Remarks 44
Chapter 2. A Quick Trip through Differential Geometry and Complex Variables 46
§2.1. Parametrized Surfaces 46
§2.2. Normal Curvature 52
§2.3. Mean Curvature 55
§2.4. Complex Variables 58
§2.5. Gauss Curvature 65
Chapter 3. The Mathematics of Soap Films 74
§3.1. The Connection 74
§3.2. The Basics of Minimal Surfaces 75
§3.3. Area Minimization and Soap Films 82
§3.4. Isothermal Parameters 87
§3.5. Harmonic Functions and Minimal Surfaces 90
§3.6. The Weierstrass- Enneper Representations 92
§3.7. The Gauss Map 101
§3.8. Stereographic Projection and the Gauss Map 106
§3.9. Creating Minimal Surfaces from Curves 110
§3.10. To Be or Not To Be Area Minimizing 120
§3.11. Constant Mean Curvature 129
Chapter 4. The Calculus of Variations and Shape 136
§4.1. Introduction 136
§4.2. Minimizing Integrals 139
§4.3. Necessary Conditions: Euler–Lagrange Equations 142
§4.4. Solving the Fundamental Examples 151
§4.5. Problems with Extra Constraints 161
Chapter 5. Maple, Soap Films, and Minimal Surfaces 174
§ 5.1. Introduction 174
§5.2. Fused Bubbles 174
§5.3. Capillarity: Inclined Planes 181
§5.4. Capillarity: Thin Tubes 187
§5.5. Minimal Surfaces of Revolution 190
§5.6. The Catenoid versus Two Disks 196
§5.7. Some Minimal Surfaces 207
§5.8. Enneper's Surface 214
§5.9. The Weierstrass–Enneper Representation 222
§5.10. Bjorling's Problem 236
§5.11. The Euler–Lagrange Equations 240
§5.12. The Brachistochrone 251
§5.13. Surfaces of Delaunay 258
§5.14. The Mylar Balloon 273
Bibliography 276
Index 280
A 280
B 280
C 280
D 280
E 280
F 280
G 280
H 281
I 281
L 281
M 281
N 281
O 281
P 281
R 281
S 281
T 281
W 281
Back Cover 282
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