...How can we ameliorate the quantum leap from introductory calculus and linear algebra to more abstract methods in both pure and applied mathematics? There is a subject which can take students of mathematics to the next level of development and this subject is, at once, intuitive, calculable, useful, interdisciplinary, and, most importantly, interesting. Of course, I'm talking here about Differential Geometry....
- -from the introduction of Differential Geometry and its Applications, by John Oprea.
The above quote expresses a philosophy that applies to John Oprea's latest book, The Mathematics of Soap Films: Explorations with Maple®. This book attempts to fill a long-time gap in the literature, and, in important ways, achieves great success.
I have often given talks entitled "Fun with Bubbles," to try to expose undergraduates to some beautiful mathematics at an elementary level. I have found it hard to subsequently provide eager students with accessible references. Osserman's A Survey of Minimal Surfaces came closest. While it is a graduate text, it is well-written and organized so as to be accessible to undergraduates in the first chapters, with more background assumed as the reader progresses. Although there are numerous other good books devoted to minimal surfaces, they all assume the background of a graduate student from page one. (I mention here my favorite of these, Minimal Surfaces I and II, by Dierkes, Hildebrandt, Küster, and Wohlrab, for the interested reader.)
Prior to Oprea's new book, the available undergraduate texts included only a few lessons on the mathematics of soap films (e.g., in Do Carmo's, Gray's, and Oprea's books on differential geometry), or focused solely on physical aspects of soap films (e.g., Boys' and Isenberg's books). Because of the beauty of soap films and their mathematics, a book like Oprea's has been sorely needed. Because it includes physical and experimental motivation, together with accessible undergraduate mathematics, it could well be called soap bubble mathematics for the masses.
Students who have learned multivariable calculus, separable differential equations, vectors and cross-products, and Taylor series expansions will be able to understand everything in Oprea's book. All of the other necessary mathematical theory (some complex variables, some differential geometry of curves and surfaces, and some topics from the calculus of variations) is developed from scratch. This is what I view as one of the main contributions of Oprea's book: he 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 (including Bjorling's problem, and variational problems with constraints, such as the hanging chain and the mylar balloon), the amount of detail included in worked examples and the general readability. Finally, the computer component is an added advantage, and does for minimal surfaces what Gray's computer-based book did for undergraduate differential geometry.
Chapter 1 gives a general, motivational introduction to the behavior of soap films. Many beautiful physical problems and properties are described here, and there are some nice pictures. I found the many suggested experiments to be intriguing. This chapter is fun to read, which is important. However, there are a number of ways in which I think the chapter could be improved. For example, the statement on page one, that "molecules near the surface are drawn into the liquid and the surface of the liquid displays a 'curvature'," is somewhat mysterious. I would like to see more explanation and/or better pictures for this and other vague statements in the first chapter. Additionally, our best students may be bothered by inaccurate statements such as the one on page five, that area minimization "is, in fact, equivalent to energy minimization here." I suggest a statement such as the following: "Because the soap film is so thin, surface tension is the dominant force, and thus energy minimization is essentially the same as area minimization." However, such problems are restricted to the first chapter. Because a knowledgeable professor will be able to clear up any ambiguities for her students, I maintain that the book is still an excellent text.
Chapter 2 contains a bare bones introduction to parametrized surfaces, complex variables, and various surface curvatures (normal, mean, and Gauss) in local coordinates. This chapter is particularly well-written, with great care taken not to lose students merely through terminology. A simple example of this is the first sentence in the proof of Theorem 2.5.3, Gauss' famous "Theorema Egregium". Oprea states "Because mixed partial derivatives are equal no matter the order of differentiation, ...." instead of just saying that mixed partials commute.
The presentation in this entire chapter is very appealing, but the development of the geometry of surfaces is especially elegant, from the barest minimum of results on curves. The third chapter continues in a similar style, with a generous amount of detail, and develops the mathematics of minimal surfaces. Most of the exercises in Chapters 2 and 3 are computational. I would like to see more thoughtful problems here, although I admit this is a tall order, and only Do Carmo has really attempted it. (Gray's book has a good number of problems about minimal surfaces, but they are almost all solvable by calculation.)
Chapter 4 is my favorite chapter in Oprea's book. The calculus of variations is developed by considering several problems. The brachistochrone, a hanging chain, and the isoperimetric problem are explored at length. In this chapter, there are some excellent and challenging problems (beyond calculation) that good students will enjoy.
Chapter 5 contains computer code and exercises to accompany and expand upon the first four chapters. In contrast to Gray's book, Oprea's book could be used (and could have been written) without using computational mathematics. But including the computer applications provides a nice complement to the analytical proofs. All necessary code is provided to allow the student to create color computer-generated pictures of the surfaces examined in earlier chapters. Black and white pictures of the expected computer results are provided, and are adequate for those lacking computer access. It is also not necessary to have any prior experience with Maple. It would have been helpful, however, if a short glossary of useful commands had been provided. For example, I had to dig deep in the chapter to find the command to change the mesh/grid size (grid=[10,10] ), which allowed me to obtain better (more refined) pictures. Oprea handles this well in his differential geometry book, and I hope he will eventually incorporate that approach in this book. Also, a minor change — indenting continued lines in the provided code — would improve its readability.
Since Oprea provides all of the code and the output, it is not clear to me what benefit there is in having students tediously re-create this from scratch. It would be nice if a disk or access to a web site with the basic code, could be provided. (Gray provided code on a web site to accompany his book.) This is a golden opportunity for fun exploration and conjecture-building. By making changes and elaborating from a basic set of code already available, students could focus on understanding the behavior of minimal surfaces, and frustration from typing could be minimized. There are some nicely-developed explorations in this chapter, including those of the "fused" double bubble, catenoids that are not minimizing, specific Euler-Lagrange equations, and Delaunay surfaces.
I am very enthusiastic about this book! It would make an excellent text for an undergraduate course in minimal surface theory. Good, standard courses in single- and multi-variable calculus are the only necessary prerequisites. 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 soap film book. This provides students with easy access to soap film mathematics, and should provide ample motivation to continue with the standard differential geometry.
Boys, C. V., Soap Bubbles: Their Colours and Forces Which Mold Them, Dover Publications, Inc., 1959.
Dierkes, U., S. Hildebrandt, A. Küster, O. Wohlrab, Minimal Surfaces I & II, Springer-Verlag, 1992.
Do Carmo, Manfredo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., 1976.
Gray, Alfred, Modern Differential Geometry of Curves and Surfaces with Mathematica®, Second edition, CRC Press, 1998. Web site with computer code for the book is at http://math.cl.uh.edu/~gray/ .
Isenberg, Cyril, The Science of Soap Films and Soap Bubbles, Dover Publications, Inc. 1992.
Oprea, John, Differential Geometry and its Applications, Prentice-Hall, Inc., 1997.
Osserman, Robert, A Survey of Minimal Surfaces, Dover Publications, Inc., 1986.
Helen Moore is a lecturer in the Mathematics Department at Stanford University. Her research interests include differential geometry, geometric analysis, minimal surfaces, and modeling with differential equations. Outside of research, her interests include Ultimate Frisbee, acoustic guitar, and science.