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.Additional ISBNs1108424937, 1108600913, 1108441025, 9781108424936, 9781108600910, 9781108441025A First Course in Differential Geometry: Surfaces in Euclidean Space is written by Lyndon Woodward; John Bolton and published by Cambridge University Press. ISBNs for A First Course in Differential Geometry are 9781108582049, 1108582044 and the print ISBNs are 9781108424936, 1108424937. Additional ISBNs include 1108424937, 1108600913, 1108441025, 9781108424936, 9781108600910, 9781108441025.