Calculus For Scientists And Engineers

655 Pages·2019·12.579 MB·English

by  Martin Brokate, Pammy Manchanda, Abul Hasan Siddiqi

Industrial and Applied Mathematics
Martin Brokate
Pammy Manchanda
Abul Hasan Siddiqi
Calculus for
Scientists
and
Engineers
Industrial and Applied Mathematics
Editor-in-Chief
Abul Hasan Siddiqi, Sharda University, Greater Noida, India
Editorial Board
Zafer Aslan, Istanbul Aydin University, Istanbul, Turkey
Martin Brokate, Technical University, Munich, Germany
N.K. Gupta, Indian Institute of Technology Delhi, New Delhi, India
Akhtar A. Khan, Rochester Institute of Technology, Rochester, USA
René Pierre Lozi, University of Nice Sophia-Antipolis, Nice, France
Pammy Manchanda, Guru Nanak Dev University, Amritsar, India
Zuhair Nashed, University of Central Florida, Orlando, USA
Govindan Rangarajan, Indian Institute of Science, Bengaluru, India
Katepalli R. Sreenivasan, NYU Tandon School of Engineering, Brooklyn, USA
TheIndustrialandAppliedMathematicsseriespublisheshigh-qualityresearch-level
monographs, lecture notes and contributed volumes focusing on areas where
mathematics is used in a fundamental way, such as industrial mathematics,
bio-mathematics, financial mathematics, applied statistics, operations research and
computer science.
More information about this series at http://www.springer.com/series/13577
Martin Brokate Pammy Manchanda
(cid:129) (cid:129)
Abul Hasan Siddiqi
Calculus for Scientists
and Engineers
123
Martin Brokate Pammy Manchanda
Department ofMathematics Department ofMathematics
Technical University of Munich Guru NanakDev University
Munich,Bayern, Germany Amritsar, Punjab,India
AbulHasan Siddiqi
Department ofMathematics
Sharda University
Greater Noida, Uttar Pradesh, India
ISSN 2364-6837 ISSN 2364-6845 (electronic)
Industrial andAppliedMathematics
ISBN978-981-13-8463-9 ISBN978-981-13-8464-6 (eBook)
https://doi.org/10.1007/978-981-13-8464-6
©SpringerNatureSingaporePteLtd.2019
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Foreword
In my past position as Director of the Abdus Salam International Centre of
Theoretical Physics in Trieste, Italy, I was deeply involved in strengthening
advancedresearchindevelopingcountries.EventhoughtheCentre’smandatewas
postdoctoralresearch,itwasclearthatonlybyenhancingthequalityofeducationat
alllevelscanoneenableadvancedresearchatasustainablelevelofexcellence.The
Centrehadthusventured,oftenanddeliberately,intoundergraduateeducation.One
lacunawehadobservedwasthelackofgood,affordable,andmotivatingtextbooks
in science and mathematics.
Iwasthuspleasedwhentheauthorsofthisbook—allofwhomarewell-known
researchersandteachersIknowpersonally—approachedmeinthesummerof2006
with a proposal for writing a quality book on calculus for use in undergraduate
education.Theirgoalsweretomake thebookusefulforinstructioninanycountry
but priced such that students in developing countries could afford it. With this
understanding,IfacilitatedseveralvisitsoftheauthorstotheCentre,duringwhich
they actively collaborated on the book. I am pleased that the book is now being
publishedinitsneweditionbySpringerNature,andthanktheauthorsforpersisting
with the collaboration despite geographical separation.
I am satisfied that the coverage and presentation in the book are at a high level
andmeetoneofourprincipalrequirements.Theauthorshaveendeavoredtopresent
the basic concepts clearly and point to their applications in diverse fields. I hope
thatmanybrilliantyoungstudentswillbenefitfromthisbook,whichistheresultof
a continuing collaboration among the authors, on which I congratulate them
warmly.
New York, USA K. R. Sreenivasan
Courant Institute of Mathematical Sciences
New York University
v
Preface
This book is meant to be used as a first course in calculus for students of science
and engineering. It will also be useful for students of other disciplines who are
interested in learning calculus.
Weendeavoredtoexplainthebasicconceptsofcalculushandinhandwiththeir
relevancetoreal-worldproblems. We havegiven specialemphasis onapplications
without compromising rigorous analysis. Plenty of solved examples have been
given to clarify techniques related to a particular theme. In appendices, we have
discussedconceptsandthemesweregardasprerequisites,likethenumbersystem,
trigonometric functions, and analytic geometry. Moreover, proofs of some of the
theoremshavebeenincludedthereinordertonotinterrupttheflowoftheargument
inthemainbodyofthetext.Some references tootherbooksoncalculusthathave
motivated our presentation have been given in the bibliography.
The text is application oriented. Many interesting, relevant, and up-to-date
applications have been drawn from the fields of business, economics, social and
behavioral sciences, life sciences, physical sciences, and other fields of general
interest. Applications are found in the main body of the text as well as in the
exercise sets. In fact, one goal of the text is to include at least one real-life appli-
cation in each section wherever possible.
The book comprises 12 chapters. Chapter 1 is devoted to an introduction of
functionsofoneindependentvariable.Chapter2providestheconceptsoflimitand
continuityalongwiththeirphysicalandgeometricalinterpretations.Chapter3deals
with derivatives and the techniques of differentiation. Chapter 4 discusses the
optimizationofafunction,thatis,findingminimaandmaximaofafunctionoveran
interval. Moreover, applications of optimization to various real-world problems,
including a fairly large number of solved examples from business and finance, are
also studied inthischapter.Chapter 5considers sequencesandseries, inparticular
Maclaurin and Taylor series. Chapters 6 and 7 are devoted to the process of inte-
gration and its applications in business and industry, engineering problems, and
probability theory. We show with a lot of examples that the utility of integrals has
expandedfarfromtheiroriginalpurpose,thecomputationoftheareabelowacurve.
vii
viii Preface
Chapter 8 introduces functions of several variables. Concepts of level curves
(levelsets)orcontours,graphsoffunctionsoftwovariables,andequipotentialand
isothermal surfaces are developed. Physical situations represented by functions of
more than one variable are discussed. This chapter also deals with the extension
oftheconceptsoflimit,continuity,differentiability,optimization,andintegrationto
functions of several variables. Physical situations, where such extensions are
required, are discussed in detail. Often we have restricted our presentation to
functionsoftwovariablesonly,forthesakeofclarityandeasierunderstandingand
because most results which hold true for two variables can be readily extended to
functions of more than two variables.
Chapter 9 is devoted to the calculus of vector-valued functions (vector fields),
that is, functions that are defined on a domain of dimension 1, 2, or 3 and take
values in the plane or the space. Continuity, differentiability, and integration for
vector-valued functions are introduced, and the theorems of Green, Gauss, and
Stokes are discussed. Applications of vector calculus and of these theorems to
problems of science and engineering are presented.
Chapter 10 deals with Fourier methods and their applications to real-world
problems for readers who want to pursue this topic.
Chapter 11 is devoted to the introduction of ordinary and partial differential
equations. Modeling of real-world problems with these equations is explained.
Chapter 12 shows how MATLAB can be used as an aid for teaching and
learning concepts of calculus, in particular those we have discussed in this book.
Teachers may use MATLAB as a tool for vivid and precise demonstrations, while
studentsmayuseMATLABasatoolforexploringbythemselvesvariousconcepts
ofcalculus.Indeed, MATLAB isbeing usednowadayspractically in every branch
of science and engineering.
*Chapter12ismainlywrittenbyDr.A.K.Verma,AssistantProfessor,Sharda
GroupofInstitutions,Agra,andDr.Jean-MarcGinouxfromFrance.Dr.Vermahas
also drawn all figures of this book using MATLAB.
The International Centre for Theoretical Physics (ICTP), Trieste, Italy (a joint
venturebetweenUNESCOandtheItaliangovernment),hasplayedapivotalrolein
the creation of this book. Established in 1964 and renamed as the Abdus
Salam ICTP in 1997, ICTP possesses a worldwide rather unique combination of
features.Itisameetingpointofscientistsfromdevelopedanddevelopingcountries
and,inparticular,continuestoprovideyeoman’sserviceintrainingbrightscientists
from developingcountries. The authors ofthis book have been frequentvisitorsto
this Centre since 1986 and have availed the Centre’s hospitality to enhance their
academic capabilities and cooperation. During one of their visits in 2006, while
walking along the Adriatic Sea, they began to discuss the utility of writing a book
on calculus for undergraduates and agreed to write such a book on calculus with
special emphasis on the clarity of concepts and their applications in diverse fields.
Our objective is to show that, throughout all of its contents, the mathematics of
calculusisnotjustanabstractsubject,buthasrelevancetomanydifferentfieldsof
human knowledge. The next morning, Director of ICTP, Prof. K. R. Sreenivasan
wasapproachedwiththerequesttosupportthewritingofsuchabook.Hewasvery
Preface ix
prompt in approving the idea and assured to provide all kinds of facilities in
ICTP. We take this opportunity to thank him for the financial and infrastructural
support without which this book could not have been completed. In particular, we
have highly benefitted from the excellent library in ICTP.
We take this opportunity to thank Dr. Meenakshi, UGC Research Fellow, gold
medalist,andnowLectureratDevSamajCollegeforWomen,Ferozepur,whohas
gonethroughthisbookcarefullyandhasgivenseveralvaluablesuggestions.Also,
ShardaUniversitydeservesaspecialmention,asamajorpartofthetechnicalwork
was carried out at this place.
Munich, Germany Martin Brokate
Amritsar, India Pammy Manchanda
Greater Noida, India Abul Hasan Siddiqi
Contents
1 Functions and Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Function, Domain, and Range . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Various Types of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Important Examples of Functions. . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Functions as Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Algebra of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6 Proofs, Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7 Geometric Transformation of Functions . . . . . . . . . . . . . . . . . . 32
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Limit and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1 Idea and Definition of the Limit . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Evaluating Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Improper Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Definition of the Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Derivative of Elementary Functions . . . . . . . . . . . . . . . . . . . . . 59
3.3 Some Differentiation Formulas. . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Derivatives of Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5 A Basic Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.6 Differentials, Newton–Raphson Approximation. . . . . . . . . . . . . 83
3.7 Indeterminate Forms and l’Hôpital’s Rule . . . . . . . . . . . . . . . . 90
3.8 Sensitivity Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1 Extremum Values of Functions . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
xi