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. ® Waterloo Maple, Inc., Ontario, Canada.
Author(s): John Oprea
Publisher: Amer Mathematical Society
Year: 2000
Language: English
Pages: 284
Cover page......Page 1
Title page......Page 5
Contents......Page 9
Preface......Page 13
1.1. Introduction......Page 17
1.2. The Basics of Surface Tension......Page 18
1.3. Experiments with Soap Films......Page 22
1.4. The Laplace-Young Equation......Page 29
1.5. Plateau's Rules for Soap Films and Consequences......Page 31
1.6. A Sampling of Capillary Action......Page 39
1.7. Final Remarks......Page 45
2.1. Parametrized Surfaces......Page 47
2.2. Normal Curvature......Page 53
2.3. Mean Curvature......Page 56
2.4. Complex Variables......Page 59
2.5. Gauss Curvature......Page 66
3.1. The Connection......Page 75
3.2. The Basics of Minimal Surfaces......Page 76
3.3. Area Minimization and Soap Films......Page 83
3.4. Isothermal Parameters......Page 88
3.5. Harmonic Functions and Minimal Surfaces......Page 91
3.6. The Weierstrass-Enneper Representations......Page 93
3.7. The Gauss Map......Page 102
3.8. Stereographic Projection and the Gauss Map......Page 107
3.9. Creating Minimal Surfaces from Curves......Page 111
3.10. To Be or Not To Be Area Minimizing......Page 121
3.11. Constant Mean Curvature......Page 130
4.1. Introduction......Page 137
4.2. Minimizing Integrals......Page 140
4.3. Necessary Conditions: Euler-Lagrange Equations......Page 143
4.4. Solving the Fundamental Examples......Page 152
4.5. Problems with Extra Constraints......Page 162
5.2. Fused Bubbles......Page 175
5.3. Capillarity: Inclined Planes......Page 182
5.4. Capillarity: Thin Tubes......Page 188
5.5. Minimal Surfaces of Revolution......Page 191
5.6. The Catenoid versus Two Disks......Page 197
5.7. Some Minimal Surfaces......Page 208
5.8. Enneper's Surface......Page 215
5.9. The Weierstrass-Enneper Representation......Page 223
5.10. Björling's Problem......Page 237
5.11. The Euler-Lagrange Equations......Page 241
5.12. The Brachistochrone......Page 252
5.13. Surfaces of Delaunay......Page 259
5.14. The Mylar Balloon......Page 274
Bibliography......Page 277
Index......Page 281
