Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood. Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques. It is a subject that contains some of the most beautiful and profound results in mathematics, yet many of them are accessible to higher level undergraduates.Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum. Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times. Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there.
Table of contents:
1. Curves in the Plane and in Space.- 2. How much does a Curve Curve?- 3. Global Properties of Curves.- 4. Surfaces in Three Dimensions.- 5. The First Fundamental Form.- 6. Curvature of Surfaces.- 7. Gaussian Curvature and the Gauss Map.- 8. Geodesics.- 9. Minimal Surfaces.- 10. Gauss's Theorema Egregium.- 11. The Gauss-Bonnet Theorem.- Solutions.- Index.
Table of contents:
1. Curves in the Plane and in Space.- 2. How much does a Curve Curve?- 3. Global Properties of Curves.- 4. Surfaces in Three Dimensions.- 5. The First Fundamental Form.- 6. Curvature of Surfaces.- 7. Gaussian Curvature and the Gauss Map.- 8. Geodesics.- 9. Minimal Surfaces.- 10. Gauss's Theorema Egregium.- 11. The Gauss-Bonnet Theorem.- Solutions.- Index.
From the reviews: INTERNATIONALE MATHEMATISCHE NACHRICHTEN "By the inclusion of 200 exercises with full solutions, this book has become a helpful tool for everyone teaching in its field. Summing up, it is a very good first book on the topic." Internationale Mathematische Nachrichten, Nr. 187, August 2001 MAA ONLINE "¿Pressley takes the simplest route with respect to all the technical setup: avoid it all. Instead of covariant derivatives, use derivatives with respect to local coordinates. Use moving frames without mentioning connections. Mention the Christoffel symbols very quickly, but don¿t do very much with them. For the most part, it works¿the book does include several versions of the Gauss-Bonnett theorem, allowing the professor to end the course with a bang. All in all, I was quite happy with the book."