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 these 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.

The book will provide an invaluable resource to all those taking a first course in differential geometry, for their lecturers, and for all others interested in the subject.

Andrew Pressley is Professor of Mathematics at Kinga (TM)s College London, UK.

The Springer Undergraduate Mathematics Series (SUMS) is a series designed for undergraduates in mathematics and the sciences worldwide. From core foundational material to final year topics, SUMS books take a fresh and modern approach and are ideal for self-study or for a one- or two-semester course. Each book includes numerous examples, problems and fully worked solutions.

This volume presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum.

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.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.

In discussing the differential geometry of curves and surfaces, Elementary Differential Geometry presents the reader with both profound and insightful results. Prerequisites are kept to an absolute minimum (nothing beyond first courses in linear algebra and multivariable calculus) and the most direct and straightforward approach is used throughout.

This new edition, like the first edition, aims to present the main results in the differential geometry of curves and surfaces while keeping the prerequisites to a minimum. New sections have been added on parallel transport and its applications on map coloring (as an application of the Gauss-Bonnet theorem), on surfaces of constant negative curvature, and a number of other topics. Around 200 new exercises have been added to enhance the usefulness of the book to both teachers and students.

This revised and expanded second edition also contains: a new chapter on non-euclidean geometry, a large collection of additional exercises which can be found online, exercises with solutions available to lecturers online, additional material on Map Colouring, Holonomy and geodesic curvature and, various additions to existing sections.

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.

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.