Geometry of Surfaces in \({\mathbb {R}}^3\)
Using normal sections we define normal curvature, principal curvatures and Gaussian curvature. Geometric interpretations and a method of calculating the Gaussian curvature using parametrization are given.
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Seán Dineen
Multivariate Calculus and Geometry Third Edition
Springer Undergraduate Mathematics Series
Advisory Board M. A. J. Chaplain University of Dundee, Dundee, Scotland, UK K. Erdmann University of Oxford, Oxford, England, UK A. MacIntyre Queen Mary University of London, London, England, UK E. Süli University of Oxford, Oxford, England, UK M. R. Tehranchi University of Cambridge, Cambridge, England, UK J. F. Toland University of Cambridge, Cambridge, England, UK
For further volumes: http://www.springer.com/series/3423
Seán Dineen
Multivariate Calculus and Geometry Third Edition
123
Seán Dineen School of Mathematical Sciences University College Dublin Dublin Ireland
ISSN 1615-2085 ISSN 2197-4144 (electronic) ISBN 978-1-4471-6418-0 ISBN 978-1-4471-6419-7 (eBook) DOI 10.1007/978-1-4471-6419-7 Springer London Heidelberg New York Dordrecht Library of Congress Control Number: 2014936739 Ó Springer-Verlag London 1998, 2001, 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my four godchildren Anne-Marie Dineen, Donal Coffey, Kevin Timoney, Eoghan Wallace
Preface to First Edition
The importance assigned to accuracy in basic mathematics courses has, initially, a useful disciplinary purpose but can, unintentionally, hinder progress if it fosters t
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