Table Of ContentAdvanced Calculus
Textbooks in Mathematics
Series editors:
Al Boggess and Ken Rosen
Graph Theory and Its Applications, Third Edition
Jonathan L. Gross and Jay Yellen
A Concrete Introduction to Real Analysis, Second Edition
Robert Carlson
Complex Variables:
A Physical Approach with Applications and MATLAB
Steven G. Krantz
Mathematical and Experimental Modeling of Physical and Biological Processes
H.T. Banks and H.T. Tran
Introduction to Probability with Mathematica, Second Edition
Kevin J. Hastings
Linear and Nonlinear Programming with Maple:
An Interactive, Applications-Based Approach
Paul E. Fishback
Applied Functional Analysis, Third Edition
J. Tinsley Oden and Leszek Demkowicz
Elements of Advanced Mathematics, Third Edition
Steven G. Krantz
Advanced Calculus:
Theory and Practice, Second Edition
John Srdjan Petrovic
Advanced Calculus
Theory and Practice
By John Srdjan Petrovic
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2021 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-1-138-56821-1 (Hardback)
International Standard Book Number-13: 978-0-203-70514-8 (eBook)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity
of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright
holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this
form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may
rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized
in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying,
microfilming, and recording, or in any information storage or retrieval system, without written permission from the
publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,
978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface ix
1 Real Numbers 1
1.1 Axiomatic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Axioms of the Set R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Some Consequences of the Completeness Axiom . . . . . . . . . . . . . . . 8
1.4 Some Thoughts about R . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Sequences and Their Limits 15
2.1 Computing the Limits: Review . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Definition of the Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Properties of Limits . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 22
2.4 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Number e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.7 Bolzano–Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8 Limit Superior and Limit Inferior . . . . . . . .. . . . . . . . . . . . . . . 46
2.9 Some Interesting Limits . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 51
3 Continuity 57
3.1 Computing Limits of Functions: Review . . . .. . . . . . . . . . . . . . . . 57
3.2 Review of Functions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 62
3.3 Continuous Functions: A Geometric Viewpoint . . . . . . . . . . . . . . . . 65
3.4 Limits of Functions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 69
3.5 Other Limits . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 74
3.5.1 One-Sided Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.2 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5.3 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.6 Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . . . . 81
3.7 Continuity of Elementary Functions . . . . . . . . . . . . . . . . . . . . . . 86
3.8 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.9 Two Properties of Continuous Functions . . . . . . . . . . . . . . . . . . . 95
4 Derivative 101
4.1 Computing the Derivatives: A Review .. . . . . . . . . . . . . . . . . . . . 101
4.2 Derivative . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103
4.3 Rules of Differentiation . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109
4.4 Monotonicity: Local Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.5 Taylor’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.6 L’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
v
vi Contents
5 Indefinite Integral 131
5.1 Computing Indefinite Integrals: A Review . . . . . . . . . . . . . . . . . . . 131
5.2 Antiderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2.1 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.2.2 Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2.3 Binomial Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2.4 Some Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . 146
6 Definite Integral 151
6.1 Computing Definite Integrals: A Review . . . . . . . . . . . . . . . . . . . . 151
6.2 Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3 Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.4 Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.5 Properties of Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.6 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . 174
6.7 Infinite and Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.7.1 Infinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.7.2 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7 Infinite Series 189
7.1 Review of Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.2 Definition of a Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.3 Series with Positive Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.4 Root and Ratio Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.4.1 Additional Tests for Convergence . . . . . . . . . . . . . . . . . . . . 206
7.5 Series with Arbitrary Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.5.1 Additional Tests for Convergence . . . . . . . . . . . . . . . . . . . . 213
7.5.2 Rearrangement of a Series . . . . . . . . . . . . . . . . . . . . . . . . 217
8 Sequences and Series of Functions 223
8.1 Convergence of a Sequence of Functions . . . . . . . . . . . . . . . . . . . . 223
8.2 Uniformly Convergent Sequences of Functions . . . . . . . . . . . . . . . . 229
8.3 Function Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.3.1 Applications to Differential Equations . . . . . . . . . . . . . . . . . 239
8.3.2 Continuous Nowhere Differentiable Function . . . . . . . . . . . . . 242
8.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.5 Power Series Expansions of Elementary Functions . . . . . . . . . . . . . . 250
9 Fourier Series 259
9.1 Introduction . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 259
9.2 Pointwise Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . 263
9.3 Uniform Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . 270
9.4 Cesàro Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
9.5 Mean Square Convergence of Fourier Series . . . . . . . . . . . . . . . . . . 284
9.6 Influence of Fourier Series . . . . . . . . . .. . . . . . . . . . . . . . . . . . 289
10 Functions of Several Variables 293
10.1 Subsets of Rn . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 293
10.2 Functions and Their Limits . . . . . . . . .. . . . . . . . . . . . . . . . . . 299
10.3 Continuous Functions . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 305
10.4 Boundedness of Continuous Functions . . .. . . . . . . . . . . . . . . . . . 310
10.5 Open Sets in Rn . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 316
Contents vii
10.6 Intermediate Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 324
10.7 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
11 Derivatives of Functions of Several Variables 339
11.1 Computing Derivatives: Review . . . . . . . . . . . . . . . . . . . . . . . . 339
11.2 Derivatives and Differentiability . . . . . . . . . . . . . . . . . . . . . . . . 342
11.3 Properties of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
11.4 Functions from Rn to Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
11.5 Taylor’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
11.6 Extreme Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
12 Implicit Functions and Optimization 371
12.1 Implicit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
12.2 Derivative as a Linear Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
12.3 Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
12.4 Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
12.5 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
12.6 The Second Derivative Test for Constrained Optimization . . . . . . . . . . 400
12.6.1 Absolute Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
13 Integrals Depending on a Parameter 407
13.1 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
13.2 Integral as a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
13.3 Uniform Convergence of Improper Integrals . . . . . . . . . . . . . . . . . . 417
13.4 Improper Integral as a Function . . . . . . . . . . . . . . . . . . . . . . . . 423
13.5 Some Important Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
14 Integration in Rn 439
14.1 Double Integrals over Rectangles . . . . . . . . . . . . . . . . . . . . . . . . 439
14.2 Double Integrals over Jordan Sets . . . . . . . . . . . . . . . . . . . . . . . 446
14.3 Double Integrals as Iterated Integrals . . . . . . . . . . . . . . . . . . . . . 450
14.4 Transformations of Jordan Sets in R2 . . . . . . . . . . . . . . . . . . . . . 455
14.5 Change of Variables in Double Integrals . . . . . . . . . . . . . . . . . . . . 461
14.6 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
14.7 Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
15 Fundamental Theorems of Multivariable Calculus 481
15.1 Curves in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
15.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
15.3 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
15.4 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
15.5 Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
15.6 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
15.7 Differential Forms on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
15.8 Exact Differential Forms on Rn . . . . . . . . . . . . . . . . . . . . . . . . 520
16 Solutions and Answers to Selected Exercises 527
Bibliography 591
Subject Index 599
Author Index 607
Preface
This text was written for a one-semester or a two-semester course in advanced calculus. It
has several goals: to expand the material covered in elementary calculus, and to present it
in a rigorous fashion; to improve the problem-solving and proof-writing skills of students;
to make the reader aware of the historical development of calculus concepts and the reasons
behind them; and to point out the connection between various topics. In writing this book,
I was guided by my long-standing experience and interest in teaching calculus at a variety
of levels, from advanced high-school students to real and functional analysis.
For the student, my purpose was to lay out material in a precise, readable manner,
with the concepts and techniques of advanced calculus clearly presented. I wanted to give
students an understanding of higher mathematical theories together with a sense of why
these concepts matter and where they come from. My goal was to ensure that students
acquire good mathematical skills, both in proving theorems and doing careful calculations.
For the instructor, my purpose was to create a flexible, comprehensive teaching tool
using proven pedagogical techniques in mathematics. I wanted to provide instructors with
a text that they could use to teach advanced calculus effectively and efficiently in the most
appropriate manner for their particular set of students.
This book is designed to meet the needs of a one-or two-term advanced calculus course.
It is highly flexible and extremely comprehensive. It is intended not only as a successful
textbook but also as a valuable resource students can consult throughout their studies and
professional life.
Goals of an Advanced Calculus Course
An advanced calculus course has several purposes:
• Students should learn a specific set of mathematical objects, ideas and techniques, and
the role they play in calculus.
• They should become proficient in establishing further properties of these objects and
their relationships.
• They should understand the origins of particular concepts.
• They should learn to find analogies and similarities between different branches of cal
culus.
Changes in the Second Edition
I was lucky to get the input from several excellent reviewers and colleagues who have
used the first edition. I have invested a considerable amount of time and energy to follow
their suggestions and make the book more effective and student friendly. Here is the list of
changes, together with the benefits they provide:
ix
