The Mathematics of Soap Films: Explorations with Maple (Student Mathematical Library, Vol. 10) (Student Mathematical Library, V. 10)

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. 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 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.--jacket. Ch. 1. Surface Tension -- The Basics Of Surface Tension -- Experiments With Soap Films -- The Laplace-young Equation -- Plateau's Rules For Soap Films And Consequences -- A Sampling Of Capillary Action -- Ch. 2. A Quick Trip Through Differential Geometry And Complex Variables -- Parametrized Surfaces -- Normal Curvature -- Mean Curvature -- Complex Variables -- Gauss Curvature -- Ch. 3. The Mathematics Of Soap Films -- The Connection -- The Basics Of Minimal Surfaces -- Area Minimization And Soap Films -- Isothermal Parameters -- Harmonic Functions And Minimal Surfaces -- The Weierstrass-enneper Representations -- The Gauss Map -- Stereographic Projection And The Gauss Map -- Creating Minimal Surfaces From Curves -- To Be Or Not To Be Area Minimizing -- Constant Mean Curvature -- Ch. 4. The Calculus Of Variations And Shape -- Minimizing Integrals -- Necessary Conditions: Euler-lagrange Equations -- Solving The Fundamental Examples -- Problems With Extra Constraints -- Ch. 5. Maple, Soap Films, And Minimal Surfaces -- Fused Bubbles -- Capillarity: Inclined Planes -- Capillarity: Thin Tubes -- Minimal Surfaces Of Revolution -- The Catenoid Versus Two Disks -- Some Minimal Surfaces -- Enneper's Surface -- The Weierstrass-enneper Representation -- Bjorling's Problem -- The Euler-lagrange Equations -- The Brachistochrone -- Surfaces Of Delaunay -- The Mylar Balloon. John Oprea. Includes Bibliographical References And Index. 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 MapleR 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 Maple in this case is not a brand of detergent from Vermont perhaps but a software product that allows users to create shapes. Oprea (mathematics, Cleveland State U.) offers a textbook for a college junior or senior seminar or an independent study course. He expects readers to know calculus and be familiar with differential equations, but provides the necessary notions from differential geometry and complex variables. Annotation c. Book News, Inc., Portland, OR (booknews.com)