Advanced Calculus A Geometric View
With a fresh geometric approach that incorporates more than 250 illustrations, this textbook sets itself apart from all others in advanced calculus. Besides the classical capstones--the change of variables formula, implicit and inverse function theorems,
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James J. Callahan
Advanced Calculus A Geometric View
James J. Callahan Department of Mathematics and Statistics Smith College Northampton, MA 01063 USA [email protected]
Editorial Board S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA [email protected]
ISSN 0172-6056 ISBN 978-1-4419-7331-3 e-ISBN 978-1-4419-7332-0 DOI 10.1007/978-1-4419-7332-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010935598 Mathematics Subject Classification (2010): 26-01, 26B12, 26B15, 26B10, 26B20, 26A12 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
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To my teacher, Linus Richard Foy
Preface
A half-century ago, advanced calculus was a well-defined subject at the core of the undergraduate mathematics curriulum. The classic texts of Taylor [19], Buck [1], Widder [21], and Kaplan [9], for example, show some of the ways it was approached. Over time, certain aspects of the course came to be seen as more significant—those seen as giving a rigorous foundation to calculus—and they became the basis for a new course, an introduction to real analysis, that eventually supplanted advanced calculus in the core. Advanced calculus did not, in the process, become less important, but its role in the curriculum changed. In fact, a bifurcation occurred. In one direction we got calculus on n-manifolds, a course beyond the practical reach of many undergraduates; in the other, we got calculus in two and three dimensions but still with the theorems of Stokes and Gauss as the goal. The latter course is intended for everyone who has had a year-long introduction to calculus; it often has a name like Calculus III. In my experience, though, it does not manage to accomplish what the old advanced calculus course did. Multivariable calculus naturally splits into three parts: (1) several functions of one variable, (2) one function of several variables, and (3) several functions of several variables. The first two are well-developed in Calculus III, but the third is really t
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