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Gavriil Paltineanu Ileana Bucur Mariana Zamfir Differential Calculus for Engineers Differential Calculus for Engineers · · Gavriil Paltineanu Ileana Bucur Mariana Zamfir Differential Calculus for Engineers GavriilPaltineanu IleanaBucur DepartmentofMathematicsandComputer DepartmentofMathematicsandComputer Science Science TechnicalUniversityofCivilEngineering TechnicalUniversityofCivilEngineering Bucharest Bucharest Bucharest,Romania Bucharest,Romania MarianaZamfir DepartmentofMathematicsandComputer Science TechnicalUniversityofCivilEngineering Bucharest Bucharest,Romania ISBN 978-981-19-2552-8 ISBN 978-981-19-2553-5 (eBook) https://doi.org/10.1007/978-981-19-2553-5 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Preface ThisbookisbasedonlecturesgivenbytheauthorsattheFacultyofGeodesyand theFacultyofCivilEngineeringfromTechnicalUniversityofCivilEngineeringof Bucharest. The material presented in this book exceeds the content of the spoken lessons and so it is also useful for other engineering specialties and even for students in mathematics. The book provides the engineering disciplines with necessary information of differentialcalculusoffunctionswithoneandseveralvariables. The paper includes all the basic notions and results of differential calculus that are taught today in higher technical education, namely: sequences and series of numbers,sequences,andseriesoffunctions,powerseries,elementsoftopologyon n-dimensional space, limits of functions, continuous functions, partial derivatives offunctionsofseveralvariables,Taylor’sformula,extremaofafunctionofseveral variables(freeorwithconstrains),changeofvariables,dependentfunctions. Thestyleofthepaperisdirectandfewreferencesaremadetopreviousknowl- edge.Theselectionofthematerialmadebytheauthors,thesimplest,oftenoriginal, versionschosenofthedemonstrationspresentedaswellasthenumerousexamples andexercisescompletelysolvedconstitutethesecretofthesuccessofthiswork. We tried to offer the fundamental material concisely and without distracting details.Wefocusedonthepresentationofbasicideasinordertomakeindetailed andacomprehensibleaspossible.Besidesstudentsintechnicalfacultiesandthose startingamathematicalcourse,thebookmaybeusefultoengineersandscientists whowishtorefreshtheirknowledgeaboutsomeaspectsofdifferentialcalculus. Bucharest,Romania GavriilPaltineanu IleanaBucur MarianaZamfir v Contents 1 SequencesofRealNumbers ..................................... 1 1.1 RealNumbers ............................................ 1 1.2 RealNumberSequences .................................... 4 1.3 ExtendedRealNumberLine ................................ 10 2 RealNumberSeries ............................................ 13 2.1 ConvergentandDivergentSeries ............................ 13 2.2 SerieswithPositiveTerms .................................. 17 2.3 SerieswithArbitraryTerms ................................. 24 2.4 ApproximatingtheSumofaLeibniz’sSeries .................. 27 2.5 AbsolutelyandConditionallyConvergentSeries ............... 28 2.6 OperationsonConvergentSeries ............................ 31 2.7 SequencesandSeriesofComplexNumbers ................... 32 3 SequencesofFunctions(FunctionalSequences) ................... 37 3.1 SimpleandUniformlyConvergence .......................... 37 3.2 ThePropertiesoftheUniformlyConvergentFunctional Sequences ................................................ 42 4 SeriesofFunctions(FunctionalSeries) ........................... 47 4.1 SimpleandUniformConvergence ........................... 47 4.2 PropertiesoftheUniformlyConvergentSeriesofFunctions ..... 49 4.3 PowerSeries .............................................. 51 4.4 Taylor’sFormula .......................................... 59 4.5 Taylor’sandMaclaurin’sSeries ............................. 65 4.6 Elementary Functions. Euler’s Formulas. Hyperbolic TrigonometricFunctions ................................... 70 5 FunctionsofSeveralVariables ................................... 73 5.1 VectorSpaceRn.BasicNotionsandNotations ................. 73 5.2 ConvergentSequencesofVectorsinRn ....................... 75 5.3 TopologyElementsonRn .................................. 76 5.4 LimitsofFunctionsofSeveralVariables ...................... 87 vii viii Contents 5.5 ContinuousFunctionsofSeveralVariables .................... 93 5.6 PropertiesofContinuousFunctionsDefinedonCompact orConnectedSets ......................................... 97 5.7 LinearContinuousMapsfromRn toRm ...................... 103 6 DifferentialCalculusofFunctionsofSeveralVariables ............. 107 6.1 PartialDerivatives.DifferentiabilityofaFunctionofSeveral Variables ................................................. 107 6.2 DifferentiabilityofVectorFunctions.JacobianMatrix .......... 117 6.3 DifferentiabilityofCompositeFunctions ...................... 119 6.4 TheFirstOrderDifferenentialandItsInvarianceForm .......... 125 6.5 TheDirectionalDerivative.TheDifferentialOperators: Gradient,Divergence,CurlandLaplacian ..................... 128 6.6 PartialDerivativesandDifferentialsofHigherOrders ........... 134 6.7 Second-OrderPartialDerivativesofFunctionsComposed ofTwoVariables .......................................... 142 6.8 ChangeofVariables ....................................... 145 6.9 Taylor’sFormulaforFunctionsofSeveralVariables ............ 150 6.10 LocalExtremaofaFunctionofSeveralVariables .............. 154 6.11 LocalInversionTheorem ................................... 159 6.12 RegularTransformations ................................... 161 6.13 ImplicitFunctions ......................................... 162 6.14 LocalConditionalExtremum ................................ 171 6.15 DependentandIndependentFunctions ........................ 180 References ........................................................ 183 Index ............................................................. 185 Chapter 1 Sequences of Real Numbers 1.1 RealNumbers FurtherweshalldenotebyNthesetof naturalnumbersi.e.: N={0,1,2,...,n,...}andbyN∗ =N\{0}. OnthesetofnaturalnumbersNaredefinedtwoarithmeticoperations:theaddition noted by “+” and the multiplication noted by “·”. The other two basic arithmetic operationsi.e.subtractionanddivisionarenotpossibleinN. TomakepossiblethesubtractionoperationthenegativenumbersareaddedtoN, thusobtainingthesetof integers: Z={...,−n,...,−2,−1,0,1,2,...,n,...}. Thenextextensionofnumbers,whichmakespossiblealsothedivisionoperation, is the set of rational numbers denoted by Q, namely the numbers of the form p, q where p,q ∈Z,q (cid:4)=0,pandqrelativelyprime(i.e.onlyintegerfactordividesboth ofthemis1).Allrationalnumberscanbeexpressedasrepeatingdecimalfractions. Alsoweremarkthat(Q,+,·)isanAbelian(commutative)field. Fromancienttimesithasobservedthatthesetofrationalnumbersisnotsufficient rich to allow the express of the m√easure of any size in na√ture. For example, the diagonal of a square of side 1 is 2, and it is known that 2 ∈/ Q. Thus, it was necessary to extend the set of rational numbers and it was created the set of real numbersdenotedbyR.Realnumbersotherthanrationalarecalledirrational.Unlike rational numbers, irrational numbers can be represented by infinite non-repeating decimalfractions. Wedonotintendtopresentheretheconstructionofrealnumbers.Wewilljust saythatwecanconstructasetRwhichcontainstherationalnumbers,onwhichtwo operationsaredefinited:theadditionnotedby“+”andthemultiplicationnotedby“·”, andanorderrelation,denotedby“≤”,sothattherealnumbersystem(R,+,·,≤) ©TheAuthor(s),underexclusivelicensetoSpringerNatureSingaporePteLtd.2022 1 G.Paltineanuetal.,DifferentialCalculusforEngineers, https://doi.org/10.1007/978-981-19-2553-5_1 2 1 SequencesofRealNumbers is a total ordered commutative field, which additionally satisfies the following properties: (A) (Archimedes’axiom) Foranyx ∈Randanyy ∈R,y >0,thereexistsn ∈Nsuchthatny > x. (B) (Cantor’saxiom) Everynon-emptysubsetof RwithanupperboundinR,hasaleastupper boundinR(seeDefinition1.1.1below). Thelastpropertyiswhatdifferentiatestherealnumbersfromtherationalnumbers. Forexample,thesetofrationalnumberswithsquarelessthan2hasaration√alupper bound in Q, namely 3, but no rational least upper bound, because this is 2, and √ 2 2∈/ Q. Therefore from algebraic point of view, (R,+) is a commutative group with identityelement0,and(R\{0},·)isacommutativegroupwithidentityelement1.In additionwehavethedistributivelaw: x ·(y+z)= x ·y+x ·z, ∀x,y,z ∈R. Theorderrelation“≤”istotal,namelyforanyx,y ∈Rwehaveeitherx ≤ yor y ≤ x.Alsotheorderrelationiscompatiblewithalgebraicsrtucturei.e.: 1. ifx(cid:8) ≤ y(cid:8) andx(cid:8)(cid:8) ≤ y(cid:8)(cid:8) thenx(cid:8)+x(cid:8)(cid:8) ≤ y(cid:8)+y(cid:8)(cid:8), 2. ifx ≤ y andα >0thenα·x ≤α·y. Fromthefactthat(R,+,·,≤)isatotalorderedcommutativefieldresultsallthe knownrulesofcalculationwithrealnumbers. Remark 1.1.1 It can show that the Archimedes’ axiom is equivalent with the followingproperty: ∀x ∈R, thereis[x]∈Zsuchthat[x]≤ x <[x]+1. ([x]iscalledtheintegerpartofx). Proposition1.1.1 Foranyx,y ∈R,x < y,thereexistsr ∈Qsuchthatx <r < y. TheproofresultsfromArchimedes’axiom. From Proposition 1.1.1 we deduce that between two real numbers there are an infinitudeofrationnumbers. ThefollwingpropositionfollowingfromCantor’saxiom. Proposition1.1.2 Foranyx,y ∈R,x < y,thereexistsatlastirrationalnumberz suchthat x <z < y. From Proposition 1.1.2 it results that between two real numbers there are an infinitudeofirrationalnumbers. 1.1 RealNumbers 3 Definition 1.1.1 A set X of real numbers is said to be bounded above (below) if thereexistsarealnumberb(respectivelya)suchthateachelementx ∈ X satisfies theinequality: x ≤b(respectivelya ≤ x) In this case, the number b (respectively a) is called the upper bound (lower bound). Obviously,ifbisanupperboundofX,thenanynumberb(cid:8) ≥bisalsoanupper bound, therefore every bounded above set has an infinitude of upper bounds. The argumentconcerningthelowerboundsofasetboundedbelowissimilar. Definition1.1.2 AsetX ofrealnumbersissaidtobeboundedifhasbothanupper boundandalowerbound,thatis,therearetworealnumbersaandbsuchthat: a ≤ x ≤b, ∀x ∈ X The least of all the upper bounds of a the set X bounded above is called the supremumofthatsetandisdenotedby M =supX. ThegreatestofallthelowerboundsofasetXboundedbelowiscalledtheinfimum ofthatsetandisdenotedbym =inf X. Remark1.1.2 ArealnumberM isthesupremumofthesubsetX ofRifandonlyif thefollowingtworequirementsaresatisfied: 1. x ≤ M,∀x ∈ X; 2. ∀ε >0,∃xε ∈ X suchthat M −ε < xε. Proof “⇒”Indeed,ifM =supX,thenMisanupperboundofX,whenceitresults (1).Since M istheleastupperboundof X,itfollowsthatforanyε >0, M − εis notanupperboundof X,hencethereexistsxε ∈ X suchthat M −ε < xε. “⇐”Let M ∈ Rbewiththeproperties(1)and(2).From(1)itresultsthatM is anupperboundofX.Let M(cid:8) < M andε = M −M(cid:8) >0.From(2)wededucethat thereexistsxε ∈ X suchthatxε > M−ε = M(cid:8).ThereforeM(cid:8)isnotanupperbound ofX,so M =supX. Remark1.1.3 ArealnumbermistheinfimumofthesubsetX ofR,ifandonlyif thefollowingtworequirementsaresatisfied: 1. m ≤ x,∀x ∈ X; 2. ∀ε >0,∃xε ∈ X suchthatxε <m+ε. Remark1.1.4 ItcanshowthattheCantor’saxiomisequivalentwiththefollowing property: If{a }and{b }aretworationalnumbersequenceswiththeproperties: n n 1. a ≤a ≤...≤a ≤...≤b ≤...≤b ≤b , 1 2 n n 2 1