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undergraduate texts in mathematics editorial board s axler k.a ribet for other titles in this series go to http www.springer.com/series/666

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sudhir r ghorpade balmohan v limaye a course in multivariable calculus and analysis with 79 figures

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sudhir r ghorpade department of mathematics indian institute of technology bombay powai mumbai 400076 india srg@math.iitb.ac.in balmohan v limaye department of mathematics indian institute of technology bombay powai mumbai 400076 india bvl@math.iitb.ac.in editorial board s axler mathematics department san francisco state university san francisco ca 94132 usa axler@sfsu.edu k a ribet mathematics department university of california at berkeley berkeley ca 947203840 usa ribet@math.berkeley.edu issn 0172-6056 isbn 978-1-4419-1620-4 e-isbn 978-1-4419-1621-1 doi 10.1007/978-1-4419-1621-1 springer new york dordrecht heidelberg london library of congress control number:2009940318 mathematics subject classification 2000 26-01 26bxx 40-01 40bxx © 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 printed on acid-free paper springer is part of springer science+business media www.springer.com

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preface calculus of real-valued functions of several real variables also known as multivariable calculus is a rich and fascinating subject on the one hand it seeks to extend eminently useful and immensely successful notions in one-variable calculus such as limit continuity derivative and integral to higher dimensions on the other hand the fact that there is much more room to move about in the n-space rn than on the real line r brings to the fore deeper geometric and topological notions that play a significant role in the study of functions of two or more variables courses in multivariable calculus at an undergraduate level and even at an advanced level are often faced with the unenviable task of conveying the multifarious and multifaceted aspects of multivariable calculus to a student in the span of just about a semester or two ambitious courses and teachers would try to give some idea of the general stokes s theorem for differential forms on manifolds as a grand generalization of the fundamental theorem of calculus and prove the change of variables formula in all its glory they would also try to do justice to important results such as the implicit function theorem which really have no counterpart in one-variable calculus most courses would require the student to develop a passing acquaintance with the theorems of green gauss and stokes never mind the tricky questions about orientability simple connectedness etc forgotten somewhere is the initial promise that we shall do unto functions of several variables whatever we did in the previous course to functions of one variable also forgotten is a reasonable expectation that new and general concepts introduced in multivariable calculus should be neatly tied up with their relics in one-variable calculus for example the area of a bounded region in the plane defined via double integrals should be related to formulas for the areas of planar regions between two curves given by equations in rectangular coordinates or in polar coordinates likewise the volume of a solid in 3-space defined via triple integrals should be related to methods for computing volumes of solids of revolution thereby resolving the mystery that the washer method and the shell method always give the same answer indeed a conscientious student is likely to face a myriad of questions

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vi preface if the promise of extending one-variable calculus to higher dimensions is taken seriously for instance why aren t we talking of monotonicity which was such a big deal in one-variable calculus do rolle s theorem and the mean value theorem which were considered very important have genuine analogues why is there no l h^pital s rule now can t we talk of convexity o and concavity of functions of several variables and in that case shouldn t it have something to do with derivatives is it still true that the processes of differentiation and integration are inverses of each other and if so then how aren t there any numerical methods for approximating double integrals and triple integrals whatever happened to infinite series and improper integrals we thought and believed that questions and concerns such as those above are perfectly legitimate and should be addressed in a book on multivariable calculus thus about a decade ago when we taught together a course at iit bombay that combined one-variable calculus and multivariable calculus we looked for books that addressed these questions and could be easily read by undergraduate students there were a number of excellent books available most notably the two volumes of apostol s calculus and the two-volume introduction to calculus and analysis by courant and john besides a wealth of material was available in classics of older genre such as the books of bromwich and hobson however we were mildly dissatisfied with some aspect or the other of the various books we consulted as a first attempt to help our students we prepared a set of notes written in a telegraphic style with detailed explanations given during the lectures subsequently these notes and problem sets were put together into a booklet that has been in private circulation at iit bombay since march 1998 goaded by the positive feedback received from colleagues and students we decided to convert this booklet into a book to begin with we were no less ambitious we wanted a self-contained and rigorous book of a reasonable size that covered one-variable as well as multivariable calculus and adequately answered all the concerns expressed above as years went by and the size of our manuscript grew we developed a better appreciation for the fraternity of authors of books especially of serious books on calculus and real analysis it was clear that choices had to be made along the way we decided to separate out one-variable calculus and multivariable calculus our treatment of the former is contained in a course in calculus and real analysis hereinafter referred to as acicara published by springer new york in its undergraduate texts in mathematics series in 2006 the present book may be viewed as a sequel to acicara and it caters to theoretical as well as practical aspects of multivariable calculus the table of contents should give a general idea of the topics covered in this book it will be seen that we have made certain choices some quite standard and some rather unusual as is common with introductory books on multivariable calculus we have mainly restricted ourselves to functions of two variables we have also briefly indicated how the theory extends to functions of more than two variables wherever it seemed appropriate we have worked out the generalizations to functions of three variables indeed as explained in the first

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preface vii chapter there is a striking change as we pass from the one-dimensional world of r and functions on r to the two-dimensional space r2 and functions on r2 on the other hand the work needed to extend calculus on r2 to calculus on the n-dimensional space rn for n 2 is often relatively routine among the unusual choices that we have made is the noninclusion of line integrals surface integrals and the related theorems of green gauss and stokes of course we do realize that these topics are very important however a thorough treatment of them would have substantially increased the size of the book or diverted us from doing justice to the promise of developing wherever possible notions and results analogous to those in one-variable calculus for readers interested in these important theorems we have suggested a number of books in the notes and comments on chapter 5 the subject matter of this book is quite classical and therefore the novelty if any lies mainly in the selection of topics and in the overall treatment with this in view we list here some of the topics discussed in this book that are normally not covered in texts at this level on multivariable calculus monotonicity and bimonotonicity of functions of two variables and their relationship with partial differentiation functions of bounded variation and bounded bivariation rectangular rolle s and mean value theorems higher-order directional derivatives and their use in taylor s theorem convexity and its relation with the monotonicity of the gradient and the nonnegative definiteness of the hessian an exact analogue of the fundamental theorem of calculus for real-valued functions defined on a rectangle cubature rules based on products and on triangulation for approximate evaluations of double integrals conditional and unconditional convergence of double series and of improper double integrals basic guiding principles and the organizational aspects of this book are similar to those in acicara we have always striven for clarity and precision we continue to distinguish between the intrinsic definition of a geometric notion and its analytic counterpart a case in point is the notion of a saddle point of a surface where we adopt a nonstandard definition that seems more geometric and intuitive complete proofs of all the results stated in the text except the change of variables formula are included and as a rule these do not depend on any of the exercises each chapter is divided into several sections that are numbered serially in that chapter a section is often divided into several subsections which are not numbered but appear in the table of contents when a new term is defined it appears in boldface definitions are not numbered but can be located using the index lemmas propositions examples and remarks are numbered serially in each chapter moreover for the convenience of readers we have often included the statements of certain basic results in one-variable calculus each of these appears as a fact and is also serially numbered in each chapter each such fact is accompanied by a reference usually to acicara where a proof can be found the end of a proof of a lemma or a proposition is marked by the symbol while the symbol 3 marks the end of an example or a remark bibliographic details about the books and articles mentioned in the text and in this preface can be found in

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viii preface the list of references citations appear in square brackets each chapter concludes with notes and comments where distinctive features of exposition are highlighted and pointers to relevant literature are provided these notes and comments may be collectively viewed as an extended version of the preface and a reader wishing to get a quick idea of what is new and different in this book might find it useful to browse through them the exercises are divided into two parts part a consisting of relatively routine problems and part b containing those that are of a theoretical nature or are particularly challenging except for the first section of the first chapter we have avoided using the more abstract vector notation and opted for classical notation involving explicit coordinates we hope that this will seem more friendly to undergraduate students while relatively advanced readers will have no difficulty in passing to vector notation and working out analogues of the notions and results in this book in the general setting of rn although we view this book as a sequel to acicara it should be emphasized that this is an independent book and can be read without having studied acicara the formal prerequisite for reading this book is familiarity with one-variable calculus and occasionally a nodding acquaintance with 2×2 and 3 × 3 matrices and their determinants it would be useful if the reader has some mathematical maturity and an aptitude for mathematical proofs this book can be used as a textbook for an undergraduate course in multivariable calculus parts of the book could be useful for advanced undergraduate and graduate courses in real analysis or for self-study by students interested in the subject for teachers and researchers this may be a useful reference for topics that are skipped or cursorily treated in standard texts we thank our parent institution iit bombay and in particular its department of mathematics for providing excellent infrastructure financial assistance received from the curriculum development cell at iit bombay is gratefully acknowledged we are indebted to rafikul alam aldric brown dinesh karia swanand khare rekha kulkarni shobhan mandal thamban nair s h patil p shunmugaraj and especially r r simha for a critical reading of parts of this book and many useful suggestions we thank maria zeltser for reviewing the entire book we are especially thankful to arunkumar patil who is mainly responsible for drawing the figures in this book thanks are also due to ann kostant and her colleagues at springer new york for excellent cooperation to c l anthony for typing a substantial part of the manuscript and to david kramer for his thorough copyediting last but not least we express our gratitude toward sharmila ghorpade and nirmala limaye for their continued support we would appreciate receiving comments suggestions and corrections a dynamic errata together with relevant information about this book will be posted at http www.math.iitb.ac.in/srg/acimc and we encourage the reader to visit this website for updates concerning this book mumbai india august 2009 sudhir ghorpade balmohan limaye

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contents 1 vectors and functions 1.1 preliminaries algebraic operations order properties intervals disks and bounded sets line segments and paths 1.2 functions and their geometric properties basic notions bounded functions monotonicity and bimonotonicity functions of bounded variation functions of bounded bivariation convexity and concavity local extrema and saddle points intermediate value property 1.3 cylindrical and spherical coordinates cylindrical coordinates spherical coordinates notes and comments exercises sequences continuity and limits 2.1 sequences in r2 subsequences and cauchy sequences closure boundary and interior 2.2 continuity composition of continuous functions piecing continuous functions on overlapping subsets characterizations of continuity continuity and boundedness continuity and monotonicity 1 2 2 4 6 8 10 10 13 14 17 20 25 26 29 30 31 32 33 34 43 43 45 46 48 51 53 55 56 57 2

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x contents continuity bounded variation and bounded bivariation continuity and convexity continuity and intermediate value property uniform continuity implicit function theorem 2.3 limits limits and continuity limit from a quadrant approaching infinity notes and comments exercises 3 57 58 60 61 63 67 68 71 72 76 77 partial and total differentiation 83 3.1 partial and directional derivatives 84 partial derivatives 84 directional derivatives 88 higher-order partial derivatives 91 higher-order directional derivatives 99 3.2 differentiability 101 differentiability and directional derivatives 109 implicit differentiation 112 3.3 taylor s theorem and chain rule 116 bivariate taylor theorem 116 chain rule 120 3.4 monotonicity and convexity 125 monotonicity and first partials 125 bimonotonicity and mixed partials 126 bounded variation and boundedness of first partials 127 bounded bivariation and boundedness of mixed partials 128 convexity and monotonicity of gradient 129 convexity and nonnegativity of hessian 133 3.5 functions of three variables 138 extensions and analogues 138 tangent planes and normal lines to surfaces 143 convexity and ternary quadratic forms 147 notes and comments 149 exercises 151 applications of partial differentiation 157 4.1 absolute extrema 157 boundary points and critical points 158 4.2 constrained extrema 161 lagrange multiplier method 162 case of three variables 164 4.3 local extrema and saddle points 167 4

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contents xi discriminant test 170 4.4 linear and quadratic approximations 175 linear approximation 175 quadratic approximation 178 notes and comments 180 exercises 181 5 multiple integration 185 5.1 double integrals on rectangles 185 basic inequality and criterion for integrability 193 domain additivity on rectangles 197 integrability of monotonic and continuous functions 200 algebraic and order properties 202 a version of the fundamental theorem of calculus 208 fubini s theorem on rectangles 216 riemann double sums 222 5.2 double integrals over bounded sets 226 fubini s theorem over elementary regions 230 sets of content zero 232 concept of area of a bounded subset of r2 240 domain additivity over bounded sets 244 5.3 change of variables 247 translation invariance and area of a parallelogram 247 case of affine transformations 251 general case 258 5.4 triple integrals 267 triple integrals over bounded sets 269 sets of three-dimensional content zero 273 concept of volume of a bounded subset of r3 273 change of variables in triple integrals 274 notes and comments 280 exercises 282 applications and approximations of multiple integrals 291 6.1 area and volume 291 area of a bounded subset of r2 291 regions between polar curves 293 volume of a bounded subset of r3 297 solids between cylindrical or spherical surfaces 298 slicing by planes and the washer method 302 slivering by cylinders and the shell method 303 6.2 surface area 309 parallelograms in r2 and in r3 311 area of a smooth surface 313 surfaces of revolution 319 6

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xii contents 6.3 centroids of surfaces and solids 322 averages and weighted averages 323 centroids of planar regions 324 centroids of surfaces 326 centroids of solids 329 centroids of solids of revolution 335 6.4 cubature rules 338 product rules on rectangles 339 product rules over elementary regions 344 triangular prism rules 346 notes and comments 360 exercises 361 7 double series and improper double integrals 369 7.1 double sequences 369 monotonicity and bimonotonicity 373 7.2 convergence of double series 376 telescoping double series 382 double series with nonnegative terms 383 absolute convergence and conditional convergence 387 unconditional convergence 390 7.3 convergence tests for double series 392 tests for absolute convergence 392 tests for conditional convergence 399 7.4 double power series 403 taylor double series and taylor series 411 7.5 convergence of improper double integrals 416 improper double integrals of mixed partials 420 improper double integrals of nonnegative functions 421 absolute convergence and conditional convergence 425 7.6 convergence tests for improper double integrals 428 tests for absolute convergence 430 tests for conditional convergence 431 7.7 unconditional convergence of improper double integrals 435 functions on unbounded subsets 436 concept of area of an unbounded subset of r2 441 unbounded functions on bounded subsets 443 notes and comments 447 exercises 449 references 463 list of symbols and abbreviations 467 index 471

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1 vectors and functions typically a first course in calculus comprises of the study of real-valued functions of one real variable that is functions f d r where d is a subset of the set r of all real numbers we shall assume that the reader has had a first course in calculus and is familiar with basic properties of real numbers and functions of one real variable for a ready reference one may refer to [22 which is abbreviated throughout the text as acicara however for the convenience of the reader relevant facts from one-variable calculus will be recalled whenever needed the basic object of our study will be the n-dimensional euclidean space rn consisting of n-tuples of real numbers namely rn x1 xn x1 xn r and real-valued functions on subsets of rn whenever we write rn it will be tacitly assumed that n n that is n is a positive integer elements of rn are sometimes referred to as vectors in n-space when n 1 in contrast the elements of r are referred to as scalars given a vector x x1 xn in rn and 1 i n the scalar xi is called the ith coordinate of x the algebraic operations on r can be easily extended to rn in a componentwise manner thus we define the sum of x x1 xn and y y1 yn to be x y x1 y1 xn yn it is easily seen that addition defined in this way satisfies properties analogous to those in r in particular the zero vector 0 0 0 plays a role similar to the number 0 in r we might wish to define the product of x1 xn and y1 yn to be x1 y1 xn yn however this kind of componentwise multiplication is not well behaved for example the componentwise product of the nonzero vectors 1 0 and 0 1 in r2 is the zero vector 0 0 and consequently the reciprocals of these nonzero vectors cannot be defined as a matter of fact there is no reasonable notion whatsoever of division in rn in general see the notes and comments at the end of this chapter moreover as explained later the order relation on r extends only partially to rn when n 1 s.r ghorpade and b.v limaye a course in multivariable calculus and analysis undergraduate texts in mathematics doi 10.1007/978-1-4419-1621-1_1 © springer science business media llc 2010 1

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2 1 vectors and functions for these reasons the theory of functions of several variables differs significantly from that of functions of one-variable however once n 1 there is not a great deal of difference between the smaller values of n and the larger values of n this is particularly true with the basic aspects of the theory of functions of several variables that are developed here with this in view and for the sake of simplicity we shall almost exclusively restrict ourselves to the case n 2 in this case the space rn can be effectively visualized as the plane also graphs of real-valued functions of two variables may be viewed as surfaces in 3-space more generally a surface in 3-space can be given by the zeros of a function of three variables with this in mind we shall also occasionally allude to r3 and to real-valued functions of three variables in the first section below we discuss a number of preliminary notions concerning vectors in rn and some important types of subsets of rn next in section 1.2 we develop some basic aspects of real-valued functions of two variables finally in section 1.3 we discuss some useful transformations or coordinate changes of the 3-space r3 1.1 preliminaries we begin with a discussion of basic facts concerning algebraic operations order properties elementary inequalities important types of subsets etc in these matters there is hardly any simplification possible by restricting to r2 and thus we will work here with rn for arbitrary n n algebraic operations we have already discussed the notion of addition of points in rn and the fact that the corresponding analogue of algebraic properties in r holds in rn more precisely this means that the following properties hold note that each of these is an immediate consequence of the corresponding properties of real numbers see for example section 1.1 of acicara a1 a2 a3 a4 x y z x y z for all x y z rn x y y x for all x y rn x 0 x for all x rn given any x rn there is x rn such that x x 0 these properties may be used tacitly in the sequel as indicated earlier we do not have a good notion of multiplication of points in rn when n 2 but we have useful notions of scalar multiplication and dot product that are defined as follows given any c r and x x1 xn rn we define cx cx1 cxn

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1.1 preliminaries 3 this is referred to as the scalar multiplication of the vector x by the scalar c geometrically speaking the scalar multiple cx corresponds to stretching or contracting the vector x according as c 1 or 0 c 1 whereas if c 0 then cx corresponds to the reflection of x about the origin followed by stretching or contracting given any x x1 xn and y y1 yn in rn the dot product also known as the inner product or the scalar product of x and y is the real number denoted by x · y and defined by x · y x1 y1 · · · xn yn the dot product permits us to talk about the angle between two vectors we shall explain this in greater detail a little later we also have an analogue of the notion of the absolute value of a real number which is defined as follows given any x x1 xn rn the norm also known as the magnitude or the length of x is the nonnegative real number denoted by |x and defined by |x x·x x2 · · · x2 n 1 geometrically speaking the norm |x represents the distance between x and the origin 0 0 0 more generally for any x y rn the norm of their difference that is |x y represents the distance between x and y a vector u in rn for which |u 1 is called a unit vector in rn for example in r2 the vectors i 1 0 and j 0 1 are unit vectors elementary properties of scalar multiplication dot product and the norm are given in the following proposition it may be remarked that the inequality in iv below is a restatement of the cauchy­schwarz inequality as given in proposition 1.12 of acicara but the proof given here is somewhat different the inequality in v is referred to as the triangle inequality proposition 1.1 given any r s r and x y z rn we have i rsx rsx rx y rx ry and r sx rx sx ii x · y y · x x y · z x · z y · z and rx · y rx · y x · ry iii |x 0 moreover |x 0 x 0 iv |x · y |x y v |x y |x |y vi |rx |r x proof properties listed in i ii and iii are obvious the inequality in iv is obvious if x 0 assume that x 0 let a x·x b x·y and c y ·y then by iii a 0 given any t r consider qt at2 2bt c in view of i and ii we have qt tx y · tx y and hence by iii qt 0 for all t r in particular upon putting t -b/a and multiplying throughout by a we obtain ac b2 0 that is b2 ac hence |b a c which proves iv the inequality in v follows from ii and iv since

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