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Solving Problems in Mathematical Analysis, Part II Problem Books in Mathematics Tomasz Radożycki Solving Problems in Mathematical Analysis, Part II Definite, Improper and Multidimensional Integrals, Functions of Several Variables and Differential Equations Problem Books in Mathematics Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA More information about this series at http://www.springer.com/series/714 Tomasz Radozycki˙ Solving Problems in Mathematical Analysis, Part II Definite, Improper and Multidimensional Integrals, Functions of Several Variables and Differential Equations Tomasz Radozycki˙ Faculty of Mathematics and Natural Sciences, College of Sciences Cardinal Stefan Wyszynski´ University Warsaw, Poland Scientific review for the Polish edition: Jerzy Jacek Wojtkiewicz Based on a translation from the Polish language edition: “Rozwiazujemy ˛ zadania z analizy matematycznej” cz˛es´c´ 2 by Tomasz Radozycki˙ Copyright ©WYDAWNICTWO OSWIA-´ TOWE “FOSZE” 2013 All Rights Reserved. ISSN 0941-3502 ISSN 2197-8506 (electronic) Problem Books in Mathematics ISBN 978-3-030-36847-0 ISBN 978-3-030-36848-7 (eBook) https://doi.org/10.1007/978-3-030-36848-7 Mathematics Subject Classification: 00-01, 00A07, 34-XX, 34A25, 34K28, 53A04, 26Bxx © Springer Nature Switzerland AG 2020 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. 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. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface The second part of this book series covers, roughly speaking, the material of the second semester course of mathematical analysis held by university departments of sciences. I tried to preserve the character of the first part, presenting very detailed solutions of all problems, without abbreviations or understatements. In contrast to authors of typical problem sets, my goal was not to provide the reader many examples for his own work in which only final results would be given (although such problems are also provided at the ends of chapters), but to clarify all steps of the solution to a given problem: from choosing the method, through its technical implementation to the final result. A priority for me was the simplicity of argumentation although I am aware that the price for this is sometimes the loss of full precision. Concerned about the volume of the book and knowing that it is intended for students who have already passed the first semester of mathematical analysis, I let myself sometimes—in contrast to the first part—skip the details of very elementary transformations. The inspiration that guided me to prepare this particular set of problems is given in the preface of the first part. The theoretical summaries placed at the beginning of each chapter only serve to collect the theorems and formulas that will be used. Theory of the mathematical analysis should be learned from other textbooks or lectures. The problems contained in this book can also be studied without reading these theoretical summaries since all necessary notions are repeated informally in the solution where applicable. Warsaw, Poland Tomasz Radozycki˙ v Contents 1 Exploring the Riemann and Definite Integral ........................... 1 1.1 Examining the Integrability of Functions ........................... 2 1.2 Finding Riemann Integrals by Definition ........................... 12 1.3 Finding Limits of Certain Sequences................................ 20 1.4 Calculating Various Interesting Definite Integrals .................. 25 1.5 Explaining Several Apparent Paradoxes ............................ 37 1.6 Using the (Second) Mean Value Theorem .......................... 45 1.7 Exercises for Independent Work..................................... 49 2 Examining Improper Integrals............................................ 53 2.1 Investigating the Convergence of Integrals by Definition .......... 54 2.2 Using Different Criteria .............................................. 60 2.3 Using Integral Test for Convergence of Series ...................... 70 2.4 Exercises for Independent Work..................................... 74 3 Applying One-Dimensional Integrals to Geometry and Physics ...... 75 3.1 Finding Lengths of Curves........................................... 76 3.2 Calculating Areas of Surfaces ....................................... 83 3.3 Finding Volumes and Surface Areas of Solids of Revolution ...... 91 3.4 Finding Various Physical Quantities ................................ 98 3.5 Exercises for Independent Work..................................... 108 4 Dealing with Functions of Several Variables ............................ 111 4.1 Finding Images and Preimages of Sets .............................. 112 4.2 Examining Limits and Continuity of Functions .................... 115 4.3 Exercises for Independent Work..................................... 123 5 Investigating Derivatives of Multivariable Functions .................. 125 5.1 Calculating Partial and Directional Derivatives..................... 126 5.2 Examining Differentiability of Functions ........................... 130 5.3 Exercises for Independent Work..................................... 140 vii viii Contents 6 Examining Higher Derivatives, Differential Expressions, and Taylor’s Formula ...................................................... 141 6.1 Verifying the Existence of the Second Derivatives ................. 142 6.2 Transforming Differential Expressions and Operators ............. 147 6.3 Expanding Functions ................................................. 158 6.4 Exercises for Independent Work..................................... 165 7 Examining Extremes and Other Important Points ..................... 167 7.1 Looking for Global Maxima and Minima of Functions on Compact Sets...................................................... 168 7.2 Examining Local Extremes and Saddle Points of Functions ....... 175 7.3 Exercises for Independent Work..................................... 186 8 Examining Implicit and Inverse Functions .............................. 187 8.1 Investigating the Existence and Extremes of Implicit Functions... 188 8.2 Finding Derivatives of Inverse Functions ........................... 201 8.3 Exercises for Independent Work..................................... 205 9 Solving Differential Equations of the First Order ...................... 207 9.1 Finding Solutions of Separable Equations .......................... 208 9.2 Solving Homogeneous Equations ................................... 216 9.3 Solving Several Specific Equations ................................. 221 9.4 Solving Exact Equations ............................................. 234 9.5 Exercises for Independent Work..................................... 249 10 Solving Differential Equations of Higher Orders ....................... 251 10.1 Solving Linear Equations with Constant Coefficients .............. 252 10.2 Using Various Specific Methods..................................... 265 10.3 Exercises for Independent Work..................................... 277 11 Solving Systems of First-Order Differential Equations ................ 279 11.1 Using the Method of Elimination of Variables...................... 280 11.2 Solving Systems of Linear Equations with Constant Coefficients ........................................................... 286 11.3 Exercises for Independent Work..................................... 310 12 Integrating in Many Dimensions.......................................... 313 12.1 Examining the Integrability of Functions ........................... 315 12.2 Calculating Integrals on Given Domains ............................ 322 12.3 Changing the Order of Integration................................... 328 12.4 Exercises for Independent Work..................................... 334 13 Applying Multidimensional Integrals to Geometry and Physics ...... 337 13.1 Finding Areas of Flat Surfaces ...................................... 338 13.2 Calculating Volumes of Solids....................................... 345 13.3 Finding Center-of-Mass Locations .................................. 354 Contents ix 13.4 Calculating Moments of Inertia...................................... 361 13.5 Finding Various Physical Quantities ................................ 368 13.6 Exercises for Independent Work..................................... 378 Index ..............................................................................
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