A First Course in Differential Geometry : Surfaces in Euclidean Space

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.