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Differential geometry is the study of curved spaces using the techniques of calculus. It is a mainstay of undergraduate mathematics education and a cornerstone of modern geometry. It is also the language used by Einstein to express general relativity, and so is an essential tool for astronomers and theoretical physicists. This introductory textbook originates from a popular course given to third year students at Durham University for over twenty years, first by the late L. M. Woodward and later by John Bolton (and others). It provides a thorough introduction by focusing on the beginnings of the subject as studied by Gauss: curves and surfaces in Euclidean space. While the main topics are the classics of differential geometry - the definition and geometric meaning of Gaussian curvature, the Theorema Egregium, geodesics, and the Gauss-Bonnet Theorem - the treatment is modern and student-friendly, taking direct routes to explain, prove and apply the main results. It includes many exercises to test students' understanding of the material, and ends with a supplementary chapter on minimal surfaces that could be used as an extension towards advanced courses or as a source of student projects. 'An excellent introduction to the subject, suitable for learners and lecturers alike. The authors strike a perfect balance between clear prose and clean mathematical style and provide plenty of examples, exercises and intuitive diagrams. The choice of material stands out as well: covering the essentials and including interesting further topics without cluttering. This wonderful book again reminded me of the beauty of this topic!' Karsten Fritzsch, Gottfried Wilhelm Leibniz Universitat Hannover, Germany 'How to present a coherent and stimulating introduction to a mathematical subject without getting carried away into bloating it by our love for the subject? This book not only expresses the authors' enthusiasm for differential geometry but also condenses decades of teaching experience: it focuses on few milestones, covering the required theory in an efficient and stimulating way. It will be a pleasure to teach/learn alongside this text.' Udo Hertrich-Jeromin, Technische Universitat Wien, Austria 'This is an attractive candidate as a text for an undergraduate course in classical differential geometry and should certainly be given serious consideration by any instructor teaching such a course.' Mark Hunacek, Department of Mathematics, Iowa State University

A First Course in Differential Geometry: Surfaces in Euclidean Space

Preface; 1. Curves in Rn; 2. Surfaces in Rn; 3. Smooth maps; 4. Measuring how surfaces curve; 5. The Theorema Egregium; 6. Geodesic curvature and geodesics; 7. The Gauss-Bonnet theorem; 8. Minimal and CMC surfaces; 9. Hints or answers to some exercises; Index.