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Cambridge University Press 978-0-521-72149-3 — Elementary Differential Geometry Christian Bär Frontmatter More Information ElementaryDifferentialGeometry Thelinkbetweenthephysicalworldanditsvisualisationisgeometry.Thiseasy-to-read, generously illustrated textbook presents an elementary introduction to differential geometry with emphasis on geometric results. Avoiding formalism as much as possi- ble, the author harnesses basic mathematical skills in analysis and linear algebra to solve interesting geometric problems, which prepare students for more advanced study in mathematics and other scientific fields such as physics and computer science. The wide range of topics includes curve theory, a detailed study of surfaces, cur- vature, variation of area and minimal surfaces, geodesics, spherical and hyperbolic geometry, the divergence theorem, triangulations, and the Gauss–Bonnet theorem. The section on cartography demonstrates the concrete importance of elementary dif- ferential geometry in applications. Clearly developed arguments and proofs, colour illustrations, and over 100 exercises and solutions make this book ideal for courses and self-study. The only prerequisites are one year of undergraduate calculus and linear algebra. Christian Bär is Professor of Geometry in the Institute for Mathematics at the University of Potsdam, Germany. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-72149-3 — Elementary Differential Geometry Christian Bär Frontmatter More Information Elementary Differential Geometry ¨ ChristianBar Universität Potsdam, Germany © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-72149-3 — Elementary Differential Geometry Christian Bär Frontmatter More Information University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314-321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi - 110025, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521721493 Originally published in German as Elementare Differentialgeometrie by Walter de Gruyter 2001 © Walter de Gruyter GmbH & Co. KG 2000 First published in English by Cambridge University Press 2010 English translation © C. Bär 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Reprinted with corrections 2011 A catalogue record for this publication is available from the British Library Library of Congress Cataloging in Publication data Bär, Christian. Elementary differential geometry / Christian Bär. p. cm. ISBN 978-0-521-89671-9 (Hardback) – ISBN 978-0-521-72149-3 (Pbk.) 1. Geometry, Differential–Textbooks. I. Title. QA641.B325 2010 516.3´6–dc22 2010001343 ISBN 978-0-521-89671-9 Hardback ISBN 978-0-521-72149-3 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-72149-3 — Elementary Differential Geometry Christian Bär Frontmatter More Information Contents Preface pagevii Notation xi 1 Euclideangeometry 1 1.1 Theaxiomaticapproach 1 1.2 TheCartesianmodel 13 2 Curvetheory 22 n 2.1 CurvesinR 22 2.2 Plane curves 34 2.3 Spacecurves 57 3 Classicalsurfacetheory 81 3.1 Regular surfaces 81 3.2 Thetangentplane 93 3.3 Thefirstfundamentalform 98 3.4 Normalfieldsandorientability 103 3.5 Thesecondfundamentalform 106 3.6 Curvature 110 3.7 Surface area and integration on surfaces 126 3.8 Someclassesofsurfaces 132 4 Theinnergeometryofsurfaces 149 4.1 Isometries 149 4.2 Vector fields and the covariant derivative 152 4.3 RiemanncurvaturetensorandTheoremaEgregium 160 4.4 Riemannianmetrics 168 4.5 Geodesics 171 4.6 Theexponentialmap 183 4.7 Parallel transport 192 4.8 Jacobi fields 196 v © in this web service Cambridge University Press www.cambridge.org
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