The Mathematics of Soap Films: Explorations With Maple (Student Mathematical Library, Vol. 10) (Student Mathematical Library, V. 10) by John Oprea (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐John Oprea’s “The Mathematics of Soap Films: Explorations with Maple” has five chapters and about 250 pages. The book “is about the mathematics which describes the geometric properties of soap films. Using…plane geometry, differential geometry, complex analysis and the calculus of variations, we…understand why soap films take the shapes they do…” Chapter one, surface tension, has 30 pages, which gives an introduction to surface tension. The surface tension “leads a soap film to minimizes its surface area…[and] allows us to study soap film shapes from a purely mathematical viewpoint.” Chapter two, A Quick Trip through Differential Geometry and Complex Variables, has 20 pages, which gives a quick trip on parameterized surfaces, curvature, and complex variables. Chapter three, The Mathematics of Soap Films, has 55 pages, which makes the connection between surface tension, pressure, curvature, and establishes the theorem “Every soap film is a physical model of a minimal surface.” The Weierstrass-Enneper representation is the heart-and-soul of minimal surface theory which allows us to create minimal surfaces by just choosing holomorphic functions. Since then, the soap films can be studied not only from the theory of minimal surface but also on computers. Chapter five, Maple, Soap Films, and Minimal Surface, has 100 pages, more than one-third of the pages of the book are utilized to demonstrate the use of the computer software Maple on studying soap films and minimal surface. Chapter four, The Calculus of Variations and Shape, has 25 pages, which presents the spirit of the calculus of variations–minimizing integrals and the necessary conditions: Euler-Lagrange Equation. A section of the calculus of variations with extra constraints is also provided for further studies. “There is much to be learned in making computers actually do interesting things in mathematics.” After spending a week reading the book, I am able to write a Maple program solving a challenged mathematical problem (number theory related) posed on a university’s web site (I never write a Maple program prior reading this book.), even though the book is calculus related.

⭐Great Value tutorial for studying minimal surfaces in a pragmatic way creating opportunities to experiment with soap films in abundance.

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