Since the publication of the First Edition, many people have sent me comments, suggestions, and corrections. I have tried to take all of these into account in preparing the Second Edition, but sometimes this has proved to be impossible. One reason for this is that I want to keep the book at a level that is truly accessible to undergraduates. So, for me, some arguments simply can't be made. On the other hand, I have learned a great deal from all of the comments sent to me and, in some sense, this is the real payment for writing the book. Therefore, I want to acknowledge a few people who went beyond the call of duty to give me often extensive commentary. These folks are (in alphabetical order!): David Arnold, David Bao, Neil Bomberger, Gary Crum, Dan Drucker, Lisbeth Fajstrup, Karsten Grosse-Brauckmann, Sigmundur Gudmundsson, Greg Lupton, Takashi Kimura, Jaak Peetre, Ted Shifrin, and Peter Stiller. Thanks to all of you.
The Second Edition, of course, contains corrections to misprints and mathematical errors which found their way into the First Edition, but it also contains new material. In particular, in recent years I have become convinced of the utility of the elliptic functions in differential geometry and the calculus of variations, so I have included a simplified, straightforward introduction to these here. The main applications of elliptic functions presented here are the derivation of explicit parametrizations for unduloids and for the Mylar balloon. Such explicit parametrizations allow for the determination of differential geometric invariants such as Gauss curvature as well as an analysis of geodesics. Of course, part of this analysis involves Maple. Theseapplications of elliptic functions are distillations of joint work with Ivailo Mladenov, and I want to acknowledge that here with thanks to him for his insights and diligence concerning this work.
The Maple work found in the Second Edition once again focuses on actually doing interesting things with computers rather than simply drawing pictures. Nevertheless, in transporting the book from the AMS-TeX of the First Edition to the LaTeX2e of the Second, it has proved to be much easier to embed encapsulated Postscript files. So there are many more pictures of interesting phenomena in this edition. The pictures have all been created by me with Maple. In fact, by examining the Maple sections at the ends of chapters, it is usually pretty clear how all pictures were created. The version of Maple used for this edition is Maple 8. The Maple work in the First Edition needed extensive revision to work with Maple 8 because Maple developers changed the way certain commands work. I have been personally assured by these developers that this will not happen in the future—we will see. Should newer versions of Maple cause problems for the procedures in this book, look at my website listed for updates: www.csuohio.edu/math/oprea. One thing to pay attention to concerning this issue of Maple command changes is the following. Maple no longer supports the "linalg" package. Rather, Maple has moved to a package called "LinearAlgebra" and I have changed all Maple work in the book to reflect this. This should be stable for some time to come, no matter what new versions of Maple arise. Of course, the one thing that doesn't change is the book's focus on the solutions of differential equations as the heart of differential geometry. Because of this, Maple plays an even more important role through its "dsolve" command and its ability to solve differential equations explicitly and numerically.
Originally this book was intended for a one-quarter or one-semester course in the geometry of curves and surfaces. Now, however, it seems to have grown beyond this, so I would like to make some recommendations for instructors who do not already have their syllabi set in stone. A good one-semester course can be obtained from Chapter 1, Chapter 2, Chapter 3, and the first "half" of Chapter 5. This carries students through the basic geometry of curves and surfaces while introducing various curvatures and applying virtually all of these ideas to study geodesics. My personal predilections would lead me to use Maple extensively to foster a certain geometric intuition. I also might use material such as the industrial application of Section 5.7 as a student group project for the semester. A second semester course could focus on the remainder of Chapter 5, Chapter 6, and Chapter 7 while saving Chapter 4 on minimal surfaces or Chapter 8 on higher dimensional geometry for projects. Students then will have seen Gauss-Bonnet, holonomy, and a kind of recapitulation of geometry (together with a touch of mechanics) in terms of the Calculus of Variations. There are, of course, many alternative courses hidden within the book and I can only wish "good hunting" to all who search for them.
