This volume has consistently been a leading text in its field since its initial release. It has remained in continuous print, reflecting its enduring relevance and significance in scholarly and professional discourse.   One of the most widely used texts in its field, this volume has been continuously in print since its initial 1976 publication. Many examples and exercises, some with hints and answers, enhance the clear, well-written exposition. Prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables.   Comprehensive Coverage : The volume introduces a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. It provides a thorough understanding of key concepts and techniques. - Clear Exposition : The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. - Examples and Exercises : The book includes numerous examples and exercises that enhance the learning experience. These practical applications solidify understanding and provide opportunities for practice and self-assessment. - Hints and Answers : Hints and answers to selected problems are provided, allowing readers to check their work and gain further insights into the subject matter. - Updated Edition : This second edition has been corrected, revised, and updated by the author, ensuring that the content remains current and relevant. - Suitable for Advanced Study : While suitable for advanced undergraduates, this text is also appropriate for graduate students of mathematics. Prerequisites include an undergraduate course in linear algebra and some familiarity with calculus of several variables. About the author: Manfredo P. do Carmo was a Brazilian mathematician and authority in the very active field of differential geometry. He was an emeritus researcher at Rio's National Institute for Pure and Applied Mathematics and the author of Differential Forms and Applications. One of the most widely used texts in its field, this volume introduces the differential geometry of curves and surfaces in both local and global aspects. The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Many examples and exercises enhance the clear, well-written exposition, along with hints and answers to some of the problems. The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text's prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume. Dover revised and updated republication of the edition originally published by Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976. www.doverpublications.com Manfredo P. do Carmo was a Brazilian mathematician and authority in the very active field of differential geometry. He was an emeritus researcher at Rio's National Institute for Pure and Applied Mathematics and the author of Differential Forms and Applications. Differential Geometry of Curves & Surfaces Revised & Updated By Manfredo P. Do Carmo Dover Publications, Inc. Copyright © 2016 Manfredo P. do Carmo All rights reserved. ISBN: 978-0-486-80699-0 CHAPTER 1 Curves 1-1. Introduction The differential geometry of curves and surfaces has two aspects. One, which may be called classical differential geometry, started with the beginnings of calculus. Roughly speaking, classical differential geometry is the study of local properties of curves and surfaces. By local properties we mean those properties which depend only on the behavior of the curve or surface in the neighborhood of a point. The methods which have shown themselves to be adequate in the study of such properties are the methods of differential calculus. Because of this, the curves and surfaces considered in differential geometry will be defined by functions which can be differentiated a certain number of times. The other aspect is the so-called global differential geometry. Here one studies the influence of the local properties on the behavior of the entire curve or surface. We shall come back to this aspect of differential geometry later in the book. Perhaps the most interesting and representative part of classical differential geometry is the study of surfaces. However, some local properties of curves appear naturally while studying surfaces. We shall therefore use this first c