Multivariable Analysis
This book provides a rigorous treatment of multivariable differential and integral calculus. Inverse and implicit function theorems based on total derivatives are given and the connection with solving systems of equations is included. There is an extensiv
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Satish Shirali · Harkrishan Lal Vasudeva
Multivariable Analysis
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Satish Shirali Indian Institute of Science Education and Research (Mohali) Panchkula India [email protected]
Harkrishan Lal Vasudeva Indian Institute of Science Education and Research (Mohali) Chandigarh India [email protected]
ISBN 978-0-85729-191-2 e-ISBN 978-0-85729-192-9 DOI 10.1007/978-0-85729-192-9 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Mathematics Subject Classification: 97I40, 97I50, 97I60 c Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface A thorough knowledge of multivariable analysis is an essential prerequisite for graduate studies in mathematics. The subject is presented in this book in a manner that would suit readers having a background of calculus in two and three variables, mathematical analysis in one variable, including compactness, and rudiments of matrices and determinants. The prerequisites with essential details are listed briefly in Chapter 1. In Chapter 2, after a brief discussion of the basic algebraic theory of functions defined on subsets of Rn and having values in m, the concepts of limit and continuity of these functions are defined. Also discussed is the invertibility of linear maps, which is a fundamental concern in the inverse and implicit function theorems at a subsequent stage. The chapter ends with a brief discussion of double sequences and series. Differentiation of functions from (subsets of) n into Rm, their partial derivatives and equality of ‘mixed’ partial derivatives of second order are discussed in the next chapter. The approach to the inverse and implicit function theorems in Chapter 4 is via contraction mappings in Rn. Use of compactness of a closed ball has been avoided, as it does not lend itself to the situation when Rn is replaced by an infinite-dimensional space. In the final section of t
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